Semilocal Categories and Modules with Semilocal Endomorphism Rings 978-3-030-23283-2

Table of contents :
Contents......Page 7
Preface......Page 11
List of Symbols......Page 15
1.1 Commutative Monoids......Page 17
1.2 Preordered Groups, Positive Cones......Page 32
1.3 The Monoid V(C),Discrete Valuations, Krull Monoids......Page 34
1.4 Essential Morphisms......Page 43
1.5 Further Results on Krull Monoids*......Page 44
1.6 Some Further Notions About Commutative Monoids*......Page 47
1.7 Appendix to Chapter 1: Sets and Classes*......Page 55
1.8 Notes on Chapter 1......Page 62
2.1 Semisimple Rings and Modules......Page 65
2.2 Free Rings and Free Algebras......Page 66
2.3 Ranks of Free Modules......Page 68
2.4 Projective Modules and Radicals......Page 71
2.5 Projective Covers, Injective Envelopes......Page 79
2.6 The Monoid V(R)......Page 81
2.7 Some Universal Constructions in Ring Theory*......Page 85
2.8 The Grothendieck Group......Page 95
2.9 Direct Limits of Projectives, Inverse Limits of Injectives*......Page 99
2.10 Notes on Chapter 2......Page 104
3.1 Goldie Dimension of Lattices and Modules......Page 106
3.2 Dual Goldie Dimension of a Module......Page 108
3.3 Semilocal Rings......Page 113
3.4 Local Morphisms......Page 117
3.5 Modules with Semilocal Endomorphism Rings......Page 122
3.6 Notes on Chapter 3......Page 124
4.1 Preadditive Categories......Page 126
4.2 Products, Coproducts, Biproducts of Objects......Page 129
4.3 Additive Categories......Page 133
4.4 Splitting Idempotents......Page 134
4.5 Product and Coproduct of Categories......Page 139
4.6 Embedding into an Additive Category......Page 141
4.7 Weak Coproduct of Additive Categories......Page 144
4.8 Constructions Equivalent to the Construction of Weak Coproduct......Page 148
4.9 Ideals in a Preadditive Category......Page 149
4.10 The Krull–Schmidt Property......Page 151
4.11 Annihilating a Class of Objects......Page 156
4.12 Local Functors, Almost Local Functors and Amenable Semisimple Categories......Page 159
4.13 An Application: Endomorphism Rings of Finitely Generated Modules over Commutative Semilocal Rings......Page 164
4.14 Semilocal Categories......Page 166
4.15 Some Realization Theorems......Page 168
4.16 Notes on Chapter 4......Page 175
5.1 The Spectral Category of a Grothendieck Category......Page 179
5.2 Nonsingular Modules......Page 181
5.3 The Functor P and Its Right Derived Functors......Page 184
5.4 First Applications of Spectral Categories......Page 187
5.5 Finitely Copresented Objects......Page 191
5.6 The Dual Construction to the Construction of the Spectral Category......Page 196
5.7 Applications of the Category (Mod-R)......Page 201
5.8 Finitely Presented Modules over a Semilocal Ring......Page 204
5.9 Notes on Chapter 5......Page 207
6.1 The Functor Auslander–Bridger Transpose......Page 209
6.2 Projective Modules with a Semiperfect Endomorphism Ring......Page 214
6.3 Auslander–Bridger Modules......Page 217
6.4 The Auslander–Bridger Transpose......Page 221
6.5 The Functors−⊗R R/J(R) and TorR
1 (−,R/J(R))......Page 223
6.6 Notes on Chapter 6......Page 227
7.1 Maximal Ideals......Page 228
7.2 Simple Additive Categories......Page 232
7.3 Maximal Ideals Exist in Semilocal Categories......Page 234
7.4 The Monoid V(C) for a Semilocal Category C......Page 241
7.5 Comparing Ideals of Endomorphism Rings of Distinct Objects......Page 244
7.6 Notes on Chapter 7......Page 246
8.1 Rings and Modules of Type n......Page 247
8.2 The Canonical Functor A→A/I1×· · ·×A/In......Page 251
8.3 Semilocal Categories and Local Functors......Page 254
8.4 Weak Krull–Schmidt Theorem for Additive Categories......Page 257
8.5 Bipartite Digraphs and Hall’s Theorem......Page 260
8.6 Functors into Amenable Semisimple Categories......Page 262
8.7 Biuniform Modules......Page 263
8.8 Further Results on Uniform, Couniform, and Biuniform Modules......Page 269
8.9 Infinite Direct Sums of Biuniform Modules......Page 273
8.10 Uniqueness of the Monogeny Classes......Page 275
8.11 Completely Prime Ideals in a Category......Page 276
8.12 Failure of the Krull–Schmidt uniqueness for finitely presented modules over serial rings......Page 281
8.13 Cyclically Presented Modules over Local Rings......Page 284
8.14 Kernels of Morphisms Between Indecomposable Injective Modules......Page 287
8.15 Couniformly Presented Modules......Page 290
8.16 Epi-isomorphism and Lower Isomorphism of Auslander–Bridger Modules......Page 292
8.17 Dual Auslander–Bridger Modules......Page 294
8.18 Notes on Chapter 8......Page 298
9.1 Associated Ideals and Finite Type......Page 302
9.2 A Family of Dimensions......Page 306
9.3 Examples of Modules of Finite Type......Page 313
9.4 Direct Summands of Cyclically Presented Modules over Rings of Finite Type......Page 317
9.5 Maximal Ideals in the Endomorphism Ring......Page 321
9.6 Weak Krull–Schmidt Theorem for DCP Modules......Page 324
9.7 Kernels of Morphisms Between Injective Modules of Finite Type......Page 331
9.8 Notes on Chapter 9......Page 336
Chapter 10 The Krull–Schmidt Theorem in the Case Two......Page 338
10.1 The Weak Krull–Schmidt Theorem and the Decomposition V (C)
∼=
V (C1) ⊕ V (C2)......Page 339
10.2 The Graph G(C)......Page 341
10.3 Relations Between the Generators of the Monoid V(G)......Page 343
10.4 Two Examples......Page 348
10.5 Another Realization Theorem......Page 352
10.6 Complete Graphs......Page 354
10.7 Condition (DSP)......Page 355
10.8 Completely Prime Ideals: The Functorial Point of View......Page 365
10.9 Direct Summands of Serial Modules of Finite Goldie Dimension......Page 369
10.10 Direct Summands of Direct Sums of Modules Whose Endomorphism Rings Have at Most Two Maximal Right Ideals......Page 374
10.11 Under Further Hypotheses, All Direct Summands of a Finite Direct Sum of Modules of Type ≤ 2 Are Direct Sums of Modules of Type ≤ 2......Page 378
10.12 Examples......Page 385
10.13 Notes on Chapter 10......Page 389
11.1 Small,ℵ0-Small, Quasismall Modules......Page 390
11.2 A Further Necessary Condition for Isomorphism of Direct Sums of Families of Uniserial Modules......Page 395
11.3 The Submodules Um and Ue......Page 398
11.4 Existence of Non-quasismall Uniserial Modules......Page 401
11.5 Ordered Groups......Page 405
11.6 Skew Polynomials and the Ore Condition......Page 406
11.7 A Nearly Simple Chain Domain......Page 408
11.8 The Weak Krull–Schmidt Theorem for Serial Modules (the Infinite Case)......Page 414
11.9 A Graphic Visualization......Page 423
11.10 Quasismall Modules and Pure-Projective Modules......Page 426
11.11 Nearly Simple Chain Domains and Their Modules......Page 430
11.12 Exceptional Chain Rings and Their Modules......Page 437
11.13 Dubrovin’s Example......Page 439
11.14 Notes on Chapter 11......Page 442
Chapter 12 Some Open Problems......Page 445
Bibliography......Page 453
Index......Page 467

Citation preview

Progress in Mathematics 331

Alberto Facchini

Semilocal Categories and Modules with Semilocal Endomorphism Rings

Progress in Mathematics Volume 331

Series Editors Antoine Chambert-Loir, Université Paris-Diderot, Paris, France Jiang-Hua Lu, The University of Hong Kong, Hong Kong SAR, China Michael Ruzhansky, Imperial College, London, UK Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York, USA

More information about this series at http://www.springer.com/series/4848

Alberto Facchini

Semilocal Categories and Modules with Semilocal Endomorphism Rings

Alberto Facchini Department of Mathematics “Tullio Levi-Civita” University of Padova Padova, Italy

ISSN 2296-505X (electronic) ISSN 0743-1643 Progress in Mathematics ISBN 978-3-030-23283-2 ISBN 978-3-030-23284-9 (eBook) https://doi.org/10.1007/978-3-030-23284-9 Mathematics Subject Classification (2010): 16D70, 16L30, 16S50, 18E05, 18E15, 20M14 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

This book is dedicated to Gena Puninski Exegi monumentum aere perennius regalique situ pyramidum altius, quod non imber edax, non Aquilo inpotens possit diruere aut innumerabilis annorum series et fuga temporum. Non omnis moriar. . . (Horace, Odes, III, 30, 1–6)

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

1 Monoids, Krull Monoids, Large Monoids 1.1 Commutative Monoids . . . . . . . . . . . . . . . . . . 1.2 Preordered Groups, Positive Cones . . . . . . . . . . . 1.3 The Monoid V (C), Discrete Valuations, Krull Monoids 1.4 Essential Morphisms . . . . . . . . . . . . . . . . . . . 1.5 Further Results on Krull Monoids* . . . . . . . . . . . 1.6 Some Further Notions About Commutative Monoids* 1.7 Appendix to Chapter 1: Sets and Classes* . . . . . . . 1.8 Notes on Chapter 1 . . . . . . . . . . . . . . . . . . .

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2 Basic Concepts on Rings and Modules 2.1 Semisimple Rings and Modules . . . . . . . . . . . . . . . 2.2 Free Rings and Free Algebras . . . . . . . . . . . . . . . . 2.3 Ranks of Free Modules . . . . . . . . . . . . . . . . . . . 2.4 Projective Modules and Radicals . . . . . . . . . . . . . . 2.5 Projective Covers, Injective Envelopes . . . . . . . . . . . 2.6 The Monoid V (R) . . . . . . . . . . . . . . . . . . . . . . 2.7 Some Universal Constructions in Ring Theory* . . . . . . 2.8 The Grothendieck Group K0 (R) . . . . . . . . . . . . . . 2.9 Direct Limits of Projectives, Inverse Limits of Injectives* 2.10 Notes on Chapter 2 . . . . . . . . . . . . . . . . . . . . .

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3 Semilocal Rings 3.1 Goldie Dimension of Lattices and Modules . . 3.2 Dual Goldie Dimension of a Module . . . . . . 3.3 Semilocal Rings . . . . . . . . . . . . . . . . . 3.4 Local Morphisms . . . . . . . . . . . . . . . . . 3.5 Modules with Semilocal Endomorphism Rings 3.6 Notes on Chapter 3 . . . . . . . . . . . . . . .

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4 Additive Categories 4.1 Preadditive Categories . . . . . . . . . . . . . . . . . . . . . . 4.2 Products, Coproducts, Biproducts of Objects . . . . . . . . . 4.3 Additive Categories . . . . . . . . . . . . . . . . . . . . . . . 4.4 Splitting Idempotents . . . . . . . . . . . . . . . . . . . . . . 4.5 Product and Coproduct of Categories . . . . . . . . . . . . . 4.6 Embedding into an Additive Category . . . . . . . . . . . . . 4.7 Weak Coproduct of Additive Categories . . . . . . . . . . . . 4.8 Constructions Equivalent to the Construction of Weak Coproduct . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Ideals in a Preadditive Category . . . . . . . . . . . . . . . . 4.10 The Krull–Schmidt Property . . . . . . . . . . . . . . . . . . 4.11 Annihilating a Class of Objects . . . . . . . . . . . . . . . . . 4.12 Local Functors, Almost Local Functors and Amenable Semisimple Categories . . . . . . . . . . . . . . . . . . . . . . 4.13 An Application: Endomorphism Rings of Finitely Generated Modules over Commutative Semilocal Rings . . . . . . . . . 4.14 Semilocal Categories . . . . . . . . . . . . . . . . . . . . . . . 4.15 Some Realization Theorems . . . . . . . . . . . . . . . . . . . 4.16 Notes on Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . 5 Spectral Category and Dual Construction 5.1 The Spectral Category of a Grothendieck Category . 5.2 Nonsingular Modules . . . . . . . . . . . . . . . . . 5.3 The Functor P and Its Right Derived Functors . . . 5.4 First Applications of Spectral Categories . . . . . . 5.5 Finitely Copresented Objects . . . . . . . . . . . . . 5.6 The Dual Construction to the Construction of the Spectral Category . . . . . . . . . . . . . . . . . 5.7 Applications of the Category (Mod-R) . . . . . . . 5.8 Finitely Presented Modules over a Semilocal Ring . 5.9 Notes on Chapter 5 . . . . . . . . . . . . . . . . . .

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182 187 190 193

6 Auslander–Bridger Transpose, Auslander–Bridger Modules 6.1 The Functor Auslander–Bridger Transpose . . . . . . . . . . 6.2 Projective Modules with a Semiperfect Endomorphism Ring 6.3 Auslander–Bridger Modules . . . . . . . . . . . . . . . . . . . 6.4 The Auslander–Bridger Transpose . . . . . . . . . . . . . . . 6.5 The Functors − ⊗R R/J(R) and TorR 1 (−, R/J(R)) . . . . . . 6.6 Notes on Chapter 6 . . . . . . . . . . . . . . . . . . . . . . .

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ix

Contents

7 Semilocal Categories and Their Maximal Ideals 7.1 Maximal Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Simple Additive Categories . . . . . . . . . . . . . . . . . . . . 7.3 Maximal Ideals Exist in Semilocal Categories . . . . . . . . . . 7.4 The Monoid V (C) for a Semilocal Category C . . . . . . . . . . 7.5 Comparing Ideals of Endomorphism Rings of Distinct Objects 7.6 Notes on Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . 8 Modules of Type ≤ 2. Uniserial Modules 8.1 Rings and Modules of Type n . . . . . . . . . . . . . . 8.2 The Canonical Functor A → A/I1 × · · · × A/In . . . 8.3 Semilocal Categories and Local Functors . . . . . . . 8.4 Weak Krull–Schmidt Theorem for Additive Categories 8.5 Bipartite Digraphs and Hall’s Theorem . . . . . . . . 8.6 Functors into Amenable Semisimple Categories . . . . 8.7 Biuniform Modules . . . . . . . . . . . . . . . . . . . . 8.8 Further Results on Uniform, Couniform, and Biuniform Modules . . . . . . . . . . . . . . . . . . . . 8.9 Infinite Direct Sums of Biuniform Modules . . . . . . 8.10 Uniqueness of the Monogeny Classes . . . . . . . . . . 8.11 Completely Prime Ideals in a Category . . . . . . . . 8.12 Failure of the Krull–Schmidt Uniqueness for Finitely Presented Modules over Serial Rings . . . . . . . . . . 8.13 Cyclically Presented Modules over Local Rings . . . . 8.14 Kernels of Morphisms Between Indecomposable Injective Modules . . . . . . . . . . . . . . . . . . . . 8.15 Couniformly Presented Modules . . . . . . . . . . . . 8.16 Epi-isomorphism and Lower Isomorphism of Auslander–Bridger Modules . . . . . . . . . . . . . . . 8.17 Dual Auslander–Bridger Modules . . . . . . . . . . . . 8.18 Notes on Chapter 8 . . . . . . . . . . . . . . . . . . .

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9 Modules of Finite Type 9.1 Associated Ideals and Finite Type . . . . . . . . . . . . . . . . . 9.2 A Family of Dimensions . . . . . . . . . . . . . . . . . . . . . . . 9.3 Examples of Modules of Finite Type . . . . . . . . . . . . . . . . 9.4 Direct Summands of Cyclically Presented Modules over Rings of Finite Type . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Maximal Ideals in the Endomorphism Ring . . . . . . . . . . . . 9.6 Weak Krull–Schmidt Theorem for DCP Modules . . . . . . . . . 9.7 Kernels of Morphisms Between Injective Modules of Finite Type 9.8 Notes on Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . .

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306 310 313 320 325

x

Contents

10 The Krull–Schmidt Theorem in the Case Two 10.1 The Weak Krull–Schmidt Theorem and the Decomposition V (C) ∼ = V (C1 ) ⊕ V (C2 ) . . . . . . . . . . . . . . . . . . . . . . 10.2 The Graph G(C) . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Relations Between the Generators of the Monoid V (G) . . . 10.4 Two Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Another Realization Theorem . . . . . . . . . . . . . . . . . . 10.6 Complete Graphs . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Condition (DSP) . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Completely Prime Ideals: The Functorial Point of View . . . 10.9 Direct Summands of Serial Modules of Finite Goldie Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.10 Direct Summands of Direct Sums of Modules Whose Endomorphism Rings Have at Most Two Maximal Right Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.11 Under Further Hypotheses, All Direct Summands of a Finite Direct Sum of Modules of Type ≤ 2 Are Direct Sums of Modules of Type ≤ 2 . . . . . . . . . . . . . . . . . . . . . . 10.12 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.13 Notes on Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . 11 Serial Modules of Infinite Goldie Dimension 11.1 Small, ℵ0 -Small, Quasismall Modules . . . . . . . . . . . 11.2 A Necessary Condition for Families of Uniserial Modules 11.3 The Submodules Um and Ue . . . . . . . . . . . . . . . . 11.4 Existence of Non-quasismall Uniserial Modules . . . . . . 11.5 Ordered Groups . . . . . . . . . . . . . . . . . . . . . . . 11.6 Skew Polynomials and the Ore Condition . . . . . . . . . 11.7 A Nearly Simple Chain Domain . . . . . . . . . . . . . . 11.8 The Weak Krull–Schmidt Theorem for Serial Modules (the Infinite Case) . . . . . . . . . . . . . . . . . . . . . . 11.9 A Graphic Visualization . . . . . . . . . . . . . . . . . . . 11.10 Quasismall Modules and Pure-Projective Modules . . . . 11.11 Nearly Simple Chain Domains and Their Modules . . . . 11.12 Exceptional Chain Rings and Their Modules . . . . . . . 11.13 Dubrovin’s Example . . . . . . . . . . . . . . . . . . . . . 11.14 Notes on Chapter 11 . . . . . . . . . . . . . . . . . . . . .

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12 Some Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

Preface Let R be an associative ring. This monograph deals with the right R-modules MR whose endomorphism ring End(MR ) is semilocal. More generally, let A be any preadditive category. The volume deals with the objects A of A whose endomorphism ring EndA (A) is semilocal. In particular, we consider the behavior of these objects as far as direct-sum decompositions are concerned. In commutative algebra, a ring is semilocal if it has only finitely many maximal ideals. A not necessarily commutative ring R with Jacobson radical J(R) is semilocal if R/J(R) is a semisimple Artinian ring, that is, R/J(R) is isomorphic to a finite direct product of rings of matrices over division rings. For a commutative ring R, the factor ring R/J(R) is semisimple Artinian if and only if R has finitely many maximal ideals. We call any preadditive category with a nonzero object such that the endomorphism ring of every nonzero object is a semilocal ring a semilocal category. Let us give a brief schematic presentation of where the theory presented in this book originated from, how it developed, and where it finds its applications nowadays. (a) Commutative rings first: the commutative background, valuation rings with zero divisors, uniserial modules over commutative rings, almost maximal rings. The interest in the last twenty years in modules with semilocal endomorphism rings and their direct-sum decompositions arose with the solution of a number of problems in the last forty years. Recall that a module M is said to be uniserial if for any submodules A and B of M either A ⊆ B or B ⊆ A. A serial module is a module that is a direct sum of uniserial modules, and a ring R is serial if both the right module RR and the left module R R are serial modules. The first case that was considered was that of modules over commutative rings, in the seventies and eighties. Just to cite some important discoveries, I will mention the results due to [V´amos 1969, V´ amos 1990] and his student [Gill], for instance the relations between almost maximality and uniseriality of indecomposable injective modules. In 1973, [Albu and N˘ ast˘asescu 1973, p. 21] conjectured that the endomorphism ring of a uniserial module over a commutative local ring is a valuation ring (i.e., a commutative ring that is uniserial as a module over itself). The conjecture was disproved by [Shores and Lewis]. From the eighties on, [Fuchs] and [Fuchs and Salce] began a systematic study of modules over valuation rings. Over a valuation ring, uniserial modules play a predominant role. Endomorphism rings of uniserial xi

xii

Preface

modules over a commutative ring R are local, so that endomorphism rings of serial R-modules of finite Goldie dimension, that is, direct sums of finitely many uniserial modules, are semilocal. (b) Noncommutative serial rings. Uniserial modules and, more generally, serial modules of finite Goldie dimension over an arbitrary ring, that is, over a ring that is not necessarily commutative, also have semilocal endomorphism rings. In 1975, [Warfield 1975] and [Drozd] independently proved that every finitely presented module over a serial ring is a direct sum of uniserial modules. Warfield asked whether such a direct-sum decomposition is unique up to isomorphism. In other words, Warfield asked whether the Krull–Schmidt theorem holds for direct sums of uniserial modules. Warfield’s problem was solved completely in [Facchini 1996], proving, among other things, that the decomposition into indecomposables of a serial module of finite Goldie dimension is not necessarily unique up to isomorphism. (c) Chain rings. A deep investigation of chain rings, that is, not necessarily commutative rings for which the sets of right ideals and of left ideals are both linearly ordered under inclusion, began in the same years thanks to [Brungs and T¨ orner]. The results were gathered together in [Bessenrodt, Brungs, and T¨orner]. (d) Artinian modules. [Camps and Dicks] proved in 1993 a conjecture due to Pere Menal [Menal], showing that the endomorphism ring of any Artinian module is semilocal. Two years later, [Facchini, Herbera, Levy, and V´ amos] answered another question about Artinian modules posed by W. Krull in 1932 [Krull 1932a]. The question was the following. According to the classical Krull–Schmidt theorem, if MR is a right module of finite composition length over a ring R and MR = A1 ⊕ · · · ⊕ An = B1 ⊕ · · · ⊕ Bm are two decompositions of MR as direct sums of indecomposable submodules, then n = m and, after a suitable renumbering of the summands, Ai ∼ = Bi for every i = 1, . . . , n. Recall that a module is of finite composition length if and only if it is Noetherian and Artinian. It is easy to construct examples in which such a uniqueness up to isomorphism of direct-sum decompositions does not hold if we suppose that the module MR is not of finite composition length, but only Noetherian. So Krull [Krull 1932a, pp. 37–38] asked whether the result remains true for Artinian modules. [Facchini, Herbera, Levy, and V´amos] showed in 1995 that though every Artinian module has a decomposition into indecomposables, such a direct-sum decomposition is not necessarily unique up to isomorphism. Apart from Artinian modules, several other classes of modules with a semilocal endomorphism ring were discovered in those years; see [Herbera and Shamsuddin] and [Facchini and Herbera 2006]. For instance, finitely presented modules over a semilocal ring have a semilocal endomorphism ring. We now know that these modules have direct-sum decompositions that follow a particular kind of geometric regularity, because they are described by a suitable Krull monoid. (e) Uniserial modules over noncommutative rings. In the same paper where Warfield’s problem was solved proving a weak Krull–Schmidt theorem for ser-

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xiii

ial modules and introducing the so-called monogeny classes and epigeny classes [Facchini 1996], endomorphism rings of uniserial modules were described. Endomorphism rings of uniserial modules are semilocal rings with at most two maximal ideals. In the subsequent paper [Dung and Facchini 1997], two interesting classes of modules were discovered: quasismall modules and non-quasismall modules. (f) In the years from 2001 to 2006, Puninski and Pˇr´ıhoda solved major problems in the theory of uniserial modules, in particular about direct summands of serial modules, the existence of non-quasismall uniserial modules, and a weak Krull–Schmidt theorem for an arbitrary family of uniserial modules. The main papers are [Puninski 2001a], [Puninski 2001b], [Puninski 2004], [Pˇr´ıhoda 2006b], and [Pˇr´ıhoda 2006c]. One of the major algebraic tools employed by Puninski, which appeared for the first time in [Puninski 2001a], was the class of nearly simple chain domains. This is a class of chain domains that were considered first by Dubrovin in [Dubrovin 1980] and [Dubrovin 1983]. (g) Subsequent present applications. From that moment semilocal rings found applications in module theory, in particular in the study of modules over semilocal rings ([Dubrovin and Puninski] and [Pˇr´ıhoda and Puninski]) and group rings [Kukharev and Puninski], in model theory [Puninski, Puninskaya, and Toffalori], [Gregory], in homological algebra and in tilting theory [Bazzoni, Herzog, Pˇr´ıhoda, ˇ Saroch, and Trlifaj]. Here is a brief description of the material treated in this book. In Chapters 1 and 2, we present some elementary basic notions concerning commutative monoids and not necessarily commutative rings. In particular, we present Krull monoids. They are the analogue for commutative monoids of commutative Krull domains. A commutative integral domain R is a Krull domain if and only if the multiplicative monoid R∗ := R \ {0} is a Krull monoid [Krause]. In Chapter 3, we consider semilocal rings and modules with a semilocal endomorphism ring. If R is a semilocal ring and MR is a finitely presented module, then End(MR ) is a semilocal ring [Facchini and Herbera 2006, Proposition 3.1]. We present some necessary notions about preadditive categories (Chapter 4), spectral categories [Gabriel and Oberst], and the construction dual to that of spectral categories (Chapter 5). Then we study the Auslander–Bridger transpose of a module and Auslander–Bridger modules (Chapter 6). In Chapter 7, we introduce maximal ideals in preadditive categories, in particular maximal ideals in semilocal categories. In Chapters 8 and 9, we present the behavior of uniserial modules and, more generally, of modules of type n. A ring R is of type n if R/J(R) is the direct product of n division rings. Thus a ring has finite type if and only if it is semilocal and every idempotent in R/J(R) is central. A ring has type 1 if and only if it is a local ring. A ring has type 2 if and only if it has exactly two maximal right ideals, which are necessarily two-sided (Proposition 8.6). By definition, a module is of type n if its endomorphism ring is of type n. For instance, uniserial modules are of type ≤ 2. We study what happens if, in the Krull–Schmidt theorem, instead of considering modules whose endomorphism rings are local, i.e., modules of type 1, we

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consider modules of type 2. The behavior of direct sums of finitely many modules whose endomorphism rings have type 2 is completely described by a graph whose connected components are either complete graphs or complete bipartite graphs (Chapter 10). Finally, most of Chapter 11 is devoted to presenting some wonderful results due to Puninski and Pˇr´ıhoda about serial modules of infinite Goldie dimension, that is, arbitrary direct sums of uniserial modules. In Chapter 12, we pose some questions and present some open problems. The author wrote in 1998 a book entitled Module Theory: Endomorphism Rings and Direct-Sum Decompositions in Some Classes of Modules. In that book, published by Birkh¨auser, the author presented the mathematical literature on the topic up to 1998. Now he presents part of the research developed in the subsequent twenty years. Clearly, Module Theory covered a portion of the material presented in this new book. There is an overlap of less than 50 pages in the two books. Essentially, part of the contents of Chapters 2, 4, and 9 of the old volume also appear in Chapters 2, 3, and 8 of the new one (essentially: projective covers, Goldie dimension, and dual Goldie dimension of lattices and modules, elementary properties of semilocal rings, elementary properties of modules with a semilocal endomorphism ring, finite direct sums of biuniform modules, arbitrary direct sums of uniserial modules). This overlap has been deliberately chosen in order to have an easier to follow exposition and to have a self-contained new monograph. Also, the last chapter of the old book contained 21 problems that were open in 1998. In Chapter 12 of this new book, we present the current state of the art of those 21 problems. Several people have worked, with me or alone, on these topics. I have already mentioned above Warren Dicks, Rosa Camps, Larry Levy, Dolors Herbera, Peter V´amos, and Pavel Pˇr´ıhoda and Gennadi Puninski with their extremely brilliant contributions. In addition to these, let me recall here Afshin Amini, Babak Amini, Ahmad Shamsuddin, my PhD students Luca Diracca, Nicola Girardi, Marco Perone, and Federico Campanini, Roger Wiegand and his school in the commutative case, Adel Alahmadi, Pere Ara, Francesco Barioli, Roberta Corisello, S¸ule Eceandez Alonso, Mar´ıa Jos´e Arroyo Paniagua, vit, M. Tamer Ko¸san, Rogelio Fern´ Francisco Raggi, Jos´e R´ıos Montes, Franz Halter-Koch, Ali Moradzadeh-Dekordi, Nguyen Viet Dung, and Luigi Salce. I am extremely grateful to all of them. I have tried to give credit for all the results to their authors, either in the text, before the statement, or in the notes at the end of each chapter. The sections in the first two chapters marked with an asterisk can be omitted at first reading. Probably the reader can skip the first chapter completely at first reading and pass directly to Chapters 2 and 3. In this book, we will consider unital right modules over associative rings with identity. We refer the reader to [Anderson and Fuller] for the standard terminology of module theory that we make use of. Padova, 1 January 2019

Alberto Facchini

List of Symbols an associative ring with identity 1 = 0. a right module over the ring R. a left module over the ring R. “is defined by,” “is equal by definition to.” the Jacobson radical of the ring R. the identity mapping A → A of a set (a module) A, and the identity morphism of an object A of a category C. |A| the cardinality of the set A. the restriction of a mapping f : B → C to a subset A of B. f |A A⊆B A is a subset of B. A⊂B A is a proper subset of B. A\B set-theoretic difference. ˙ A∪B disjoint union. the set of nonnegative integers 0, 1, 2, 3, . . . N0 N the set of positive integers 1, 2, 3, 4, . . . Z the set of integers. ω the first infinite ordinal number. the ring of all n × n matrices with entries in the ring R. Mn (R) Mod-R the category of all right modules over the ring R. R -Mod the category of all left modules over the ring R. proj-R the full subcategory of Mod-R whose objects are all finitely generated projective right R-modules. R-proj the full subcategory of R -Mod whose objects are all finitely generated projective left R-modules. vect-k the category of all right vector spaces over a division ring k. dim(MR ) the Goldie dimension of an R-module MR . codim(MR ) the dual Goldie dimension of an R-module MR . U (R) the group of units (invertible elements) of the ring R. r.annR (x) the right annihilator of an element x of a ring R, that is, the set of all elements s ∈ R with xs = 0. l.annR (x) the left annihilator of an element x of a ring R, that is, the set of all elements s ∈ R with sx = 0. the direct sum ⊕i∈I Mi with Mi = M for all i ∈ I. M (I) R MR RM := J(R) 1A

xv

xvi

MI N ≤s MR Ob(C) T2 Mat(C) C add(C)

List of Symbols

 the direct product i∈I Mi with Mi = M for all i ∈ I. N is a superfluous submodule of MR . the class of objects of a category C. the class of all indecomposable right R-modules of type 2. closure of the preadditive category C under finite direct sums (see p. 128), free additive category. closure of the preadditive category C under direct summands. closure of the preadditive category C under finite direct sums and direct summands.

Chapter 1

Monoids, Krull Monoids, Large Monoids In this chapter, we review what we will need in the rest of the book as far as commutative monoids are concerned. This will show how much we assume of the reader. The contents of Sections 1.5 and 1.7 are exceptions: they are completely independent of the rest of the chapter. In Section 1.5, some in-depth examinations of the theory of Krull monoids are cited, and Section 1.7 contains some considerations of foundations of category theory. Both sections may be skipped by the uninterested reader or at a first reading.

1.1 Commutative Monoids We begin with the first elementary notions about commutative monoids. Very little will be necessary for us. Hence we give here a really elementary presentation of the concepts we need. Much deeper treatises and monographs about this topic are the books by [Clifford and Preston 1967], [Halter-Koch 1998], and [Geroldinger and Halter-Koch]. Monoids and Their Morphisms An (additive) monoid M is a set with a binary operation (addition) + : M × M → M,

(x, y) → x + y,

that is associative (that is, x+ (y + z) = (x+ y)+ z for every x, y, z ∈ M ) and has a zero element , denoted by 0, that is, an element 0 ∈ M such that x + 0 = 0 + x = x for every x ∈ M . The zero element in a monoid is necessarily unique. In this book, all the monoids we will consider are commutative, that is, x + y = y + x for every x, y ∈ M . In other words, “monoid” and “commutative monoid” will have the same meaning for us. © Springer Nature Switzerland AG 2019 A. Facchini, Semilocal Categories and Modules with Semilocal Endomorphism Rings, Progress in Mathematics 331, https://doi.org/10.1007/978-3-030-23284-9_1

1

2

Chapter 1. Monoids, Krull Monoids, Large Monoids

A monoid morphism (or homomorphism) is a mapping f of a monoid M into a monoid N such that f (0) = 0 and f (x+ y) = f (x)+ f (y) for every x, y ∈ M . Notice that we want monoid morphisms to respect the zero elements. The composite mapping of two monoid morphisms is a monoid morphism. A monoid isomorphism is a monoid morphism that is also a bijection. An endomorphism of a monoid M is a monoid morphism of M into itself. If M and N are monoids, N is a homomorphic image of M if there is a morphism M → N that is a surjective mapping. (We will show on page 10, that epimorphisms in the category of commutative monoids are not necessarily surjective mappings.) The inverse mapping of a monoid isomorphism is a monoid isomorphism. Submonoids and Algebraic Preorder A subset N of a commutative additive monoid M is a submonoid of M if it is closed under the addition of M and contains the zero element of M . For a monoid M , we will denote by U (M ) the set of all elements a ∈ M with an opposite in M , that is, U (M ) := { x ∈ M | there exists y ∈ M with x + y = 0 } (if such an element y ∈ M exists, it is unique, is denoted by −x, and is called the opposite of x). The subset U (M ) turns out to be a submonoid of M , and is an abelian group, often (improperly) called the group of units of M . We will say that M is reduced if U (M ) = {0}, that is, if x + y = 0 implies x = y = 0 for every x, y ∈ M . A preorder on a set X is a reflexive and transitive relation on X. We will denote a preordered set, that is, a set X endowed with a preorder ρ, by a pair (X, ρ). The main examples of preorders are equivalence relations (= symmetric preorders) and partial orders (= antisymmetric preorders). In the following exercise, we describe the structure of preorders on a set X, showing that conversely, every preorder can be constructed from an equivalence relation and a partial order. Exercise 1.1. Let X be a set. (a) Let ρ be a preorder on X. Define a relation ∼ρ on X by setting, for every x, y ∈ X, x ∼ρ y if xρy and yρx. Show that ∼ρ is an equivalence relation on X. (b) Define a relation ≤ρ on the quotient set X/∼ρ = { [x]∼ρ | x ∈ X } by setting, for every x, y ∈ X, [x]∼ρ ≤ρ [y]∼ρ if xρy. Show that the relation ≤ρ is well defined and is a partial order on X/∼ρ . (c) Show that there is a one-to-one correspondence between the set of all preorders ρ on X and the set of all pairs (∼, ≤), where ∼ is an equivalence relation on X and ≤ is a partial order on the quotient set X/∼. The correspondence associates to each preorder ρ on X the pair (∼ρ , ≤ρ ). Conversely, for any such pair (∼, ≤), the corresponding preorder ρ(∼,≤) on X is defined, for every x, y ∈ X, by xρ(∼,≤) y if [x]∼ ≤ [y]∼ . Thus, a preorder ρ on X can be seen essentially as a partition X/∼ρ of X such that any two elements in the same block of the partition are ρ each to the other, and the partition X/∼ρ is partially ordered by the partial order on X/∼ρ induced by the preorder ρ.

1.1. Commutative Monoids

3

Exercise 1.2. Show that the category of finite preordered sets is isomorphic to the category of finite topological spaces, the full subcategory of Top, whose objects are the topological spaces with only finitely many points. (If X is a finite topological space, the corresponding preorder ≤ on X is defined by x ≤ y if x belongs to the closure of the subset {y} of X. Conversely, every closed set in a finite topological space X is a union of closures of points.) More generally, the category of preordered sets is isomorphic to the category of Alexandrov topological spaces, the full subcategory of Top whose objects are the topological spaces whose topology is an Alexandrov topology. A topology is Alexandrov if the intersection of any family of open subsets is an open set (equivalently, if the union of any family of closed subsets is a closed subset). If M is a commutative additive monoid, a preorder ≤ on M is translationinvariant if for every x, y, z ∈ M , x ≤ y implies x + z ≤ y + z. There is a natural translation-invariant preorder on any commutative additive monoid M , called the algebraic preorder on M , defined, for all x, y ∈ M , by x ≤ y if there exists z ∈ M such that x+z = y. If x is an element of a monoid M and n ≥ 0, we can inductively define the multiple nx setting 0x := 0 and nx := (n − 1)x + x. An element u of a commutative monoid M is an order-unit if for every x ∈ M there exists an integer n ≥ 0 such that x ≤ nu. For example, let M be the monoid N0n of all n-tuples of nonnegative integers. The algebraic preorder on M is the componentwise order, that is, (x1 , . . . , xn ) ≤ (y1 , . . . , yn ) if xi ≤ yi for every i = 1, . . . , n, and an element (u1 , . . . , un ) of N0n is an order-unit if and only if ui > 0 for every i = 1, . . . , n. A submonoid N of a monoid M is said to be divisor-closed if x ∈ M , y ∈ N , and x ≤ y in the algebraic preorder ≤ of M implies x ∈ N . The term “divisorclosed” becomes clear if we move on to multiplicative notation. More precisely, if the operation in the commutative monoid M is multiplication instead of addition, then the algebraic preorder on M is the relation | (divides), and a submonoid N of M is divisor-closed if for every element y ∈ N , it contains all divisors of y in M . The group of units U (M ) of an arbitrary commutative monoid M is a divisor-closed submonoid of M contained in all divisor-closed submonoids of M . For every x ∈ M , we will denote by [[x]] the smallest divisor-closed submonoid of M containing x. The submonoid [[x]] is the set of all y ∈ M with y ≤ nx for some n ≥ 0. Let X be a subset of a monoid M . Let F be the family of all submonoids of M that contain X. The family F is always nonempty, because, for instance, it contains M itself. The intersection of all the submonoids in F is the smallest submonoid of M that contains X. It is called the submonoid of M generated by X and is denoted by [X]. It is easily seen that if X = ∅, then [X] = {0} is the zero submonoid of M . If X = ∅, then [X] = { x1 + · · · + xn | n ≥ 1 and xi ∈ X for i = 1, . . . , n } (sums of finitely many elements of X, possibly with repetitions). A subset X of a monoid M is a set of generators of M if [X] = M . A monoid M is finitely generated if it has a finite set of generators, and cyclic if it has a set of generators with one element.

Chapter 1. Monoids, Krull Monoids, Large Monoids

4

Kernels and Congruences If f : M → N is a monoid morphism, the kernel of f is the equivalence relation ∼f on the set M defined, for every x, y ∈ M , by x ∼f y if f (x) = f (y).1 A congruence on a monoid M is an equivalence relation ∼ on the set M such that x ∼ y and z ∼ w implies x + z ∼ y + w for every x, y, z, w ∈ M . Equivalently, an equivalence relation ∼ on a monoid M is a congruence if x ∼ y implies x + z ∼ y + z for every x, y, z ∈ M . It is easily verified that the kernel ∼f of any monoid morphism f : M → N is a congruence on the monoid M . If M is a monoid and ∼ is a congruence on M , the factor monoid M/∼ is the set of all congruence classes [x]∼ := { y ∈ M | y ∼ x }, where x ranges over M , with the addition inherited from that of M : [x]∼ + [y]∼ := [x + y]∼

for every x, y ∈ M.

This operation on M/ ∼ is well defined, as is easily verified. It is the unique operation on the quotient set M/∼ that makes the canonical projection π : M → M/ ∼, defined by π(x) = [x]∼ for every x ∈ M , a monoid morphism. Every congruence on a monoid is the kernel of a monoid morphism. The first two examples of congruences on a monoid M are the identity congruence = and the trivial congruence ω on M , defined by xωy for every x, y ∈ M . Any intersection of congruences on M is a congruence on M . (Here, when we say intersection we mean that a congruence on M is viewed as a subset of M × M .) Hence the set of all congruences on M , partially ordered by the relation ≤ defined by ∼ ≤ ≡ if x ∼ y implies x ≡ y for every x, y ∈ M , is a complete lattice, whose greatest element is the trivial congruence ω and whose least element is the identity congruence =. Given any subset P of the Cartesian product M × M , the intersection ∼P of all the congruences containing P is the smallest congruence on M containing P , that is, the smallest congruence ∼ on M such that x ∼ y for every (x, y) ∈ P . The congruence ∼P is called the congruence on M generated by P or the congruence on M generated by the relations x ∼P y, (x, y) ∈ P . It can be described as follows. Given a subset P of M × M , set P −1 := { (x, y) | (y, x) ∈ P }, Δ := { (x, x) | x ∈ M }, P1 := P ∪ P −1 ∪ Δ, and P2 := { (x + z, y + z) | (x, y) ∈ P1 , z ∈ M }. The congruence ∼P generated by P is the transitive closure of the relation P2 . Thus x ∼P y if and only if there exist n ≥ 1, x1 , . . . , xn , y1 , . . . , yn , z1 , . . . , zn ∈ M such that (1) x = x1 + z1 , (2) either (xi , yi ) ∈ P or (yi , xi ) ∈ P or xi = yi for every i = 1, . . . , n, (3) yi +zi = xi+1 +zi+1 for every i = 1, . . . , n − 1, (4) yn + zn = y. If ∼1 and ∼2 are two relations on M (that is, two subsets of the Cartesian product M × M ), the composition of ∼1 and ∼2 , denoted by ∼1 ◦ ∼2 , is defined, for all x, y ∈ M , by x(∼1 ◦ ∼2 )y if there exists z ∈ M such that x ∼1 z and z ∼2 y. This composition is an associative operation in the set of all relations on M . In the lattice of all congruences of M , ∼1 ∧ ∼2 is the set-theoretic intersection ∼1 ∩ ∼2 1 Here

we have adopted the terminology used in monoid theory and universal algebra. The equivalence relation ∼f is called a kernel pair in category theory.

1.1. Commutative Monoids

5

(here again, we are viewing congruences on M as suitable subsets of M × M ), and ∼1 ∨ ∼2 is the union, for k ≥ 1, of all ∼1 ◦ ∼2 ◦ ∼1 ◦ ∼2 ◦ · · · ◦ ∼1 ◦ ∼2 (2k factors alternately equal to ∼1 and ∼2 ). The congruences ∼1 and ∼2 if ∼1 ◦ ∼2 is equal to ∼1 ◦ ∼2 . In this case, ∼1 ∨ ∼2 coincides with ∼1 ◦ ∼2 . A subset P of M × M is a set of generators for a congruence ∼ of the monoid M if ∼P = ∼. A congruence ∼ on a monoid M is finitely generated if it has a finite set of generators. Before giving a third example of congruence, we introduce some further notation and terminology. An element x of a monoid M is said to be idempotent if x + x = x. A monoid M is archimedean if for every pair (x, y) of elements of M with y ≤ 0 there exists a positive integer n such that x ≤ ny. Equivalently, this means that M is either {0} or has exactly two divisor-closed submonoids. More generally, for any x, y in a commutative monoid M , define x  y if there exist positive integers n and m such that x ≤ ny and y ≤ mx. Equivalently, x  y if and only if [[x]] = [[y]]. We leave to the reader to prove, as an exercise, that  is the least congruence on M such that every element in the quotient monoid M/ is idempotent. The equivalence classes of M modulo  are additively closed subsets of M , called the archimedean components of M . As a fourth example of congruence, recall that for any monoid M , U (M ) denotes the abelian additive group of all the elements of M with an opposite in M . The relation ∼ on M , defined, for every x, y ∈ M , by x ∼ y if there exists z ∈ U (M ) with x = y+z, turns out to be a congruence on M . The congruence class [x]∼ is the coset x + U (M ) := { x + z | z ∈ U (M ) }. We will denote by Mred the factor monoid M/∼. The monoid Mred is always a reduced monoid, i.e., does not have nonzero elements with an opposite element. Thus every commutative monoid M is an extension of the reduced monoid Mred by the abelian group U (M ). As a fifth example, define an equivalence ∼ on any commutative monoid M by setting, for every x, y ∈ M , x ≡ y if there exists z ∈ M with x + z = y + z. It is easily seen that ≡ is a congruence on M , called the stable congruence, and that the factor monoid M/≡ is a cancellative monoid. Recall that a monoid N is cancellative if x + z = y + z implies x = y for every x, y, z ∈ M . More precisely, ≡ is the smallest congruence on M with M/≡ cancellative. Theorem 1.3 (Fundamental isomorphism theorem). Let f : M → N be a monoid morphism. Let ∼f be its kernel and π : M → M/ ∼f the canonical projection. Then: (a) There exists a unique mapping f: M/∼f → N making the diagram M π

/N y< y y yy yy f y y f

 M/∼f

commute, that is, such that, f ◦ π = f .

Chapter 1. Monoids, Krull Monoids, Large Monoids

6

(b) The mapping f is an injective mapping and is a monoid morphism. (c) If f is an onto mapping, then f is an isomorphism. ∼ f (M ). Here f (M ) Thus, if f : M → N is a monoid morphism, then M/∼f = is the image of f , and is a submonoid of N . More generally, the image f (M  ) of any submonoid M  of a monoid M via any monoid morphism f : M → N is a submonoid of N . We are ready for the description of cyclic monoids. This is done in the next subsection and follows from the study of the congruences of the monoid N0 . The Additive Monoid N0 of Natural Numbers Consider the additive monoid N0 whose elements are the natural numbers 0, 1, 2, 3, . . . . Fix k and n in N0 with n ≥ 1, and define the relation ∼k,n on N0 by setting, for every x, y ∈ N0 , ⎧ ⎪ ⎨x = y x ∼k,n y if or ⎪ ⎩ x ≥ k, y ≥ k, and x ≡ y (mod n). Here x ≡ y (mod n) means that x and y are integers congruent modulo n, that is, n divides x − y in Z. It is easily verified that ∼k,n is a congruence on N0 . In the factor monoid N0 /∼k,n = {[x]∼k,n | x ∈ N0 }, the elements are [0]∼k,n , [1]∼k,n , . . . , [k + n − 1]∼k,n of N0 /∼k,n . They are distinct elements, so that N0 /∼k,n is a monoid with exactly k + n elements. Notice that [0]∼k,n = {0}, [1]∼k,n = {1}, [2]∼k,n = {2}, ... [k − 2]∼k,n = {k − 2}, [k − 1]∼k,n = {k − 1}, [k]∼k,n = {k, k + n, k + 2n, k + 3n, . . . }, [k + 1]∼k,n = {k + 1, k + 1 + n, k + 1 + 2n, k + 1 + 3n, . . . }, . .. [k + n − 2]∼k,n = {k + n − 2, k + n − 2 + n, k + n − 2 + 2n, . . . }, [k + n − 1]∼k,n = {k + n − 1, k + n − 1 + n, k + n − 1 + 2n, . . . }. In the additive monoid N0 , the congruences are exactly the identity = and the congruences ∼k,n , where k ≥ 0 and n ≥ 1. The congruence ∼k,n is the principal

7

1.1. Commutative Monoids

congruence generated by the relation (k, k+n). As on page 4, we can partially order the set of all congruences Cong(N0 ) of N0 , setting ∼ ≤ ∼ if for every x, y ∈ N0 , x ∼ y implies x(∼ )y. Then Cong(N0 ) has = as its least element and the trivial congruence ∼0,1 as its greatest element. One has ∼k,n ≤ ∼k ,n if and only if k  ≤ k and n | n. Thus the partially ordered set Cong(N0 ) \ {=} is order isomorphic to the opposite of the product (N0 , ≤) × (N, |) with the componentwise order, where ≤ is the usual order on the set N0 of nonnegative integers and | is the partial order “divides” on the set N = N0 \ {0} of positive integers. The partially ordered set Cong(N0 ) is a lattice. The monoid N0 is cyclic, generated by 1. The monoids N0 /∼k,n are cyclic, generated by [1]∼k,n . Conversely, every cyclic monoid is isomorphic to either N0 or N0 /∼k,n for some k, n ∈ N0 , n ≥ 1. [2] [3]

[1]

[0]

[n−1] [n−2]

Figure 1.1: The cyclic group Z/nZ. Recall that finite cyclic groups are isomorphic to Z/nZ for some n and that the most natural representation of Z/nZ is that in Figure 1.1. Finite cyclic monoids have a slightly different behavior. The representation of N0 /∼k,n , analogous to that of Z/nZ in Figure 1.1, is that in Figure 1.2, i.e., N0 /∼k,n consists of a cycle of length n with a tail of length k that begins in [0]∼k,n . [k+3]

[k+n−2]

[k+2] [k+n−1] [k+1]

[0]

[1]

[2]

[3]

[k−2]

[k−1]

[k]

Figure 1.2: The cyclic monoid N0 /∼t, . We leave to the reader to show that a monoid M is a cyclic reduced monoid if and only if it is isomorphic to either N0 or the trivial monoid with one element

8

Chapter 1. Monoids, Krull Monoids, Large Monoids

or the monoid N0 /∼k,n for some k ≥ 1, n ≥ 1. For k = 0, the monoids N0 /∼0,n are the cyclic groups Z/nZ. Prime Ideals, Localizations and Reduced Localizations An ideal of a commutative monoid M is a subset I of M such that x ∈ I and y ∈ M imply x + y ∈ I. A prime ideal of a commutative monoid M is a subset P of M such that M \ P is a divisor-closed submonoid. That is, for any x, y ∈ M , one has x + y ∈ P if and only if either x ∈ P or y ∈ P . The union of any family of prime ideals of a commutative monoid M is a prime ideal of M , so that the set Spec(M ) of all prime ideals of M , partially ordered by set inclusion, is a complete lattice whose greatest element is the prime ideal M \ U (M ) and whose least element is the empty ideal ∅. In particular, a commutative monoid has one prime ideal if and only if M is an abelian group. It is possible to endow Spec(M ) with a spectral topology, in a similar fashion as spectra of commutative rings are endowed with the Zariski topology. See the Notes at the end of this chapter (Section 1.8). Example 1.4. It is easily checked that every cyclic reduced monoid M has only the trivial prime ideals ∅ and M \ U (M ) = M \ {0}. The prime ideals of a commutative monoid M are exactly the subsets P of M for which there exists a homomorphism ϕ of M into a reduced monoid N with P = { x ∈ M | ϕ(x) = 0 }. Equivalently, the prime ideals of a monoid M are exactly the subsets P of M for which there exists a congruence ∼ on M with M/∼ reduced and P = { x ∈ M | x ∼ 0 }. The proof of these two facts is left as an exercise to the reader. Hint: If P is a prime ideal of a monoid M , consider the congruence ∼ of M defined, for every x, y ∈ M , by x ∼ y if either both x and y belong to P or both x and y belong to M \ P . Then M/∼ is a monoid with two elements, which are M \ P , that is, the zero of M/∼, and P . This monoid M/∼ is isomorphic to the reduced monoid with two elements N0 /∼1,1 . The localization of a commutative monoid M at a prime ideal P is similar to that of commutative rings. If P is a prime ideal of M , consider the Cartesian product M × (M \ P ), that is, the set of all pairs (x, s) with x, s ∈ M and s ∈ / P. Define an equivalence relation ≡ on M × (M \ P ) by setting (x, s) ≡ (x , s ) if there exists an element t ∈ M \ P such that x + s + t = x + s + t. Let x − s denote the equivalence class of (x, s) modulo the equivalence relation ≡. The localization MP of M at P is the monoid whose elements are all x − s with x ∈ M and s ∈ M \ P , and in which the addition is defined by (x − s) + (x − s ) = (x + x ) − (s + s ). There is a canonical homomorphism f : M → MP , defined by f (x) = x − 0 for every x ∈ M . For instance, we have already seen that every monoid M has a unique least prime ideal ∅ and a unique greatest prime ideal P := M \ U (M ). The localization

1.1. Commutative Monoids

9

M∅ of M at its empty prime ideal ∅ is an abelian group, which is usually called the Grothendieck group of M or the group of differences or the enveloping group of M , and denoted by G(M ). If M is cancellative, MP ⊆ M∅ for every prime ideal P of M (more precisely, there is an embedding of monoids MP → M∅ ). The localization MP of M at P := M \ U (M ) is isomorphic to M . Proposition 1.5. Let M be a commutative monoid and P a prime ideal. For every prime ideal Q of M contained in P , set QP := { x − y ∈ MP | x ∈ Q, y ∈ M \ P }. Then the prime ideals of MP are in one-to-one correspondence (Q ↔ QP ) with the prime ideals of M contained in P . Proof. We leave to the reader the direct proofs of the following elementary facts. If A is a prime ideal of MP , then f −1 (A) is a prime ideal of M contained in P . Moreover, (f −1 (A))P = A. Conversely, if Q is a prime ideal of M contained in P , then QP is a prime ideal of MP . Furthermore, f −1 (QP ) = Q.  We conclude this subsection with an operation that does not have an analogue for commutative rings. For every prime ideal P of a commutative monoid M , the monoid (MP )red = MP /U (MP ) is called the reduced localization of M at P . If x, x ∈ M and s, s ∈ M \ P , then x − s + U (MP ) = x − s + U (MP ) in (MP )red if and only if there exist elements t, t ∈ M \ P such that x + t = x + t . For every prime ideal P, there is a canonical homomorphism ϕ : M → (MP )red , defined by ϕ(x) = x − 0 + U (MP ), which is surjective. Its kernel is the congruence ∼P on M defined, for every x, y ∈ M , by x ∼P y if there exist z, t ∈ M \ P such that x + z = y + t. Hence we could have equivalently defined the reduced localization (MP )red of a commutative monoid M at a prime ideal P as the factor monoid M/∼P . For instance, the largest prime ideal of a commutative monoid M is M \ U (M ), and the smallest one is ∅. We leave to the reader to show that the reduced localization of M at the prime ideal M \ U (M ) is Mred , and the reduced localization of M at the prime ideal ∅ is the trivial monoid with one element. Proposition 1.6. Let M be a commutative monoid and π : M → Mred = M/U (M ), π : x → x + U (M ), the canonical projection. Then the prime ideals of M are in one-to-one correspondence (Q ↔ π(Q)) with the prime ideals of Mred . Proof. We leave to the reader the direct proofs of the following facts. If Q is a prime ideal of M , then f (Q) is a prime ideal of Mred . Also, f −1 (f (Q)) = Q. Conversely, if Q is a prime ideal of Mred , then f −1 (Q ) is a prime ideal of M . Moreover, f (f −1 (Q )) = Q .  The Category of Commutative Monoids Clearly, commutative monoids form a category CMon, whose objects are all commutative monoids, whose morphisms are the monoid morphisms between commutative monoids, and whose composition is the composition of mappings.

10

Chapter 1. Monoids, Krull Monoids, Large Monoids

Notice that not all epimorphisms in CMon (that is, the morphisms f : M → N such that for every pair g, h : N → P of monoid morphisms, gf = hf implies g = h) are necessarily onto mappings. For instance, consider the embedding f : N0 → Z of the monoid N0 into the abelian additive group Z of integers. If g, h : Z → P are monoid morphisms of Z into a commutative monoid P , then the images of g and h are contained in the group U (P ). Let ε : U (P ) → P be the embedding, so that there are monoid morphisms g  , h : Z → U (P ) with g = εg  and h = εh . Then g  and h are group morphisms, and gf = hf implies g(1) = h(1), so that g  (1) = h (1). Thus the two group morphisms g  , h : Z → U (P ) coincide. It follows that g = h. So the embedding f : N0 → Z is an epimorphism in the category of commutative monoids that is not an onto mapping. Monomorphisms in the category CMon of commutative monoids (that is, the morphisms f : M → N such that for every pair g, h : P → M of monoid morphisms, f g = f h implies g = h) are exactly the monoid morphisms that are injective mappings. To see this, let f : M → N be a monoid morphism that is not injective. Let x1 = x2 be elements of M with f (x1 ) = f (x2 ). Let g, h : N0 → M be the monoid morphisms defined by g(a) = ax1 and h(a) = ax2 for every a ∈ N0 . Then g = h, because g(1) = x1 = h(1) = x2 . But f g = f h, because f g(a) = f (ax1 ) = af (x1 ) = af (x2 ) = f (ax2 ) = f h(a) for every a ∈ N0 . Thus f is not a monomorphism. There is a canonical monoid homomorphism ψM : M → G(M ). It is defined by ψM (x) = x − 0 for every x ∈ M . The kernel of ψM is the smallest congruence ∼ on the monoid M with M/∼ cancellative. So when M is cancellative, the mapping ψM turns out to be an injective mapping, that is, a monoid embedding. Every monoid morphism f : M → M  induces a group morphism G(f ) : G(M ) → G(M  ), defined by G(f )(x − y) = f (x) − f (y). Hence G is a functor of the category CMon into the category Ab of abelian groups. A category related to CMon is the category of commutative monoids with order-unit. It is defined as follows. Its objects are the pairs (M, u), where M is a commutative monoid and u ∈ M is an order-unit. The morphisms f : (M, u) → (M  , u ) are the monoid homomorphisms f : M → M  such that f (u) = u . Given a diagram (1.1) M ψ

M

ϕ

 / M 

of homomorphisms of commutative monoids (or of commutative monoids with order-unit), the pullbacks of diagram (1.1) in the category Set and in the CMon (or in the category of commutative monoids with order-unit) essentially coincide. They are in one-to-one correspondence with the subset (isomorphic to the submonoid, submonoid with order-unit) of the product M × M  whose elements are all the pairs (x, x ) with x ∈ M , x ∈ M  , and ϕ(x) = ψ(x ).

1.1. Commutative Monoids

11

Direct Products and Direct Sums of Monoids In this monograph, we will often be concerned with an indexed set (or indexed family) { Ai | i ∈ I } of objects. Here, by saying that we have an indexed set (indexed family), we mean that I is a set and the Ai ’s are objects for all i ∈ I, possibly with Ai = Aj for somei = j. If { Mi | i ∈ I } is an indexed family of monoids, its direct product i∈I Mi is the Cartesian product, whose elements are the indexed sets (xi )i∈I with xi ∈ Mi for every i ∈ I, with addition defined componentwise by  (xi )i∈I + (yi )i∈I = (xi + yi )i∈I . There is an onto morphism πj : i∈I Mi → Mj for each j ∈ I. Like any product in any category, the direct product of monoids can be characterized, that is, determined up to isomorphism, by the following Theorem 1.7 (Universal property of direct product). Let { Mi | i ∈ I } be an indexed set of monoids. For every monoid N and every indexed  set { fi : N → Mi | i ∈ I } of morphisms, there is a unique morphism f : N → i∈I Mi such that πi f = fi for every i ∈ I.  If I ={1, 2, . . . , n} is finite, we also write M1 × M2 × · · · × Mn or ni=1 Mi instead of i∈I Mi . The coproduct in the category CMon of two commutative additive monoids M and N is their external direct sum M ⊕ N , which is the Cartesian product with componentwise addition, hence coincides with their direct product M × N . If { Mi | i ∈ I } is an indexed set of commutative additive monoids, the coproduct in CMon is the external direct sum i∈I Mi , that is, the set of all indexed families (xi )i∈I with xi∈ Mi for every i ∈ I and xi = 0 for almost all i. Thus i∈I Mi is a submonoid of i∈I Mi , and they coincide if and only if the set { i ∈ I | Mi = {0} } is finite. andez-Alonso, Section 6], are The next results, taken from [Facchini and Fern´ essentially folklore, and the proofs are elementary and left to the reader. Recall that we say that two congruences ∼ and ≡ permute if ∼ ◦ ≡ is equal to ≡ ◦ ∼ (page 5). Proposition 1.8. Let M be a monoid. There are canonical one-to-one correspondences between the following sets: (a) The set of all pairs (S, T ) of submonoids of M with the property that every element of M can be written in a unique way in the form s + t for suitable s ∈ S, t ∈ T . (b) The set of all pairs (S, T ) of submonoids of M for which there exist monoids M  , M  and a monoid isomorphism ϕ : M  ⊕ M  → M such that S = ϕ(M  ) and T = ϕ(M  ). (c) The set of all pairs (S, T ) of submonoids of M with the property that the mapping S ⊕ T → M , defined by (s, t) → s + t for all (s, t) ∈ S × T , is an isomorphism. (d) The set of all pairs (∼, ∼ ) of congruences of M for which the canonical morphism M → M/∼ ⊕ M/ ∼ is a monoid isomorphism.

12

Chapter 1. Monoids, Krull Monoids, Large Monoids

(e) The set of all pairs (∼, ∼ ) of congruences of M that are complements in the lattice of all congruences of M (that is, such that ∼ ∧ ∼ is the equality = M and ∼ ∨ ∼ is the trivial congruence ω), and such that ∼ and ∼ permute. If S and T are submonoids of M satisfying the first three equivalent conditions of Proposition 1.8, we say that the monoid M is the internal direct sum of its submonoids S and T , and write M = S ⊕ T . If a monoid M is the internal direct sum of two submonoids S and T , there is a canonical isomorphism between M and the external direct sum of S and T (this is condition (c) in Proposition 1.8). If a reduced monoid M is the internal direct sum of its submonoids S and S  , then S and S  are necessarily divisor-closed submonoids of M . From now on, it will no longer be necessary to distinguish between internal direct sum and external direct sum. It will be clear from the context. If I is a set, {Mi | i ∈ I} is an indexed set of additive monoids, and i∈I Mi is its direct sum, then there is an injective morphism εj : Mj → ⊕i∈I Mi for each j ∈ I. Direct sum is coproduct in the category CMon, in the sense that it satisfies the following universal property, and is characterized by it; that is, it is determined up to isomorphism by the following universal property. Theorem 1.9 (Universal property of direct sum). Let { Mi | i ∈ I } be an indexed set of monoids. For every monoid N and every indexed set { fi : Mi → N | i ∈ I } of morphisms, there is a unique morphism f : ⊕i∈I Mi → N such that f εi = fi for every i ∈ I. Free Commutative Monoids We have already said on page 3 what a set of generators of a monoid is. Let us see what a free set of generators is. Definition 1.10. Let X be a set of generators of a monoid M . The set X is called a free set of generators if for every n ≥ 1, m ≥ 1, and x1 , . . . , xn , y1 , . . . , ym ∈ X with x1 + · · · + xn = y1 + · · · + ym , one has that n = m and there exists a permutation σ of {1, 2, . . . , n} with xi = yσ(i) for every i = 1, 2, . . . , n. A commutative monoid M is said to be free if it has a free set of generators. If X = ∅ is a subset of a monoid M , then X is a set of generators of M if and only if every element of M can be written as a sum of elements of X, equivalently as a sum of multiples of elements of X, that is, as a linear combination of elements of X with coefficients in N0 . Moreover, X is a free set of generators of M if and only if every element of M can be written as a linear combination of distinct elements of X in a unique way up to a permutation of the elements. For instance, N0 is a free commutative monoid with free set of generators {1}. (X) Let X be an arbitrary set. Let N0 be the set of all mappings f : X → N0 (X) such that f (x) = 0 for almost all x ∈ X. Then N0 is a commutative monoid

1.1. Commutative Monoids

13

with respect to the operation + defined by (f + g) (x) = f (x) + g (x) (X)

for every f, g ∈ N0 and every x ∈ X. (It is the direct sum of the indexed set { Mx | x ∈ X }, where Mx = N0 for every x ∈ X.) For every fixed x0 ∈ X, let δx0 : X → N0 be the mapping defined by

x →

1 if x = x0 , 0 if x = x0 . (X)

It is easily proved that ΔX := { δx0 | x0 ∈ X } is a free set of generators for N0 . Proposition 1.11 (Universal property of free monoids). Let M be a free commutative monoid, X a free set of generators for M , and ε : X → M the embedding of X into M . Then, for every monoid N and every mapping f : X → N , there exists a unique monoid morphism f˜: M → N making the diagram /N > }} } } ε }}  }} f M X

f

commute, that is, such that f˜ ◦ ε = f . The previous universal property characterizes free monoids, and with the standard technique that proves the uniqueness up to isomorphism of the solution of universal problems, one proves that if M is a free commutative monoid with (X) free set X of generators, then M ∼ = N0 . Also, every commutative monoid M is a homomorphic image of a free monoid, that is, M is isomorphic to a monoid of (X) the form N0 /∼ for a suitable set X and a suitable congruence ∼ on the monoid (X) (M) and its free set ΔM N0 . (Given a monoid M , consider the free monoid N0 of generators, and apply the universal property to the inverse f of the bijection (M) M → ΔM , x0 ∈ M → δx0 , getting a unique monoid morphism f˜: N0 → M. ˜ This morphism is onto. Now let ∼ be the kernel of f .) (X)

When X is finite of cardinality n, the free monoid N0 is isomorphic to the monoid Nn0 of all n-tuples of nonnegative integers with addition defined componentwise: (x1 , . . . , xn ) + (y1 , . . . , yn ) = (x1 + y1 , . . . , xn + yn ). Similarly to how we have done in the previous paragraph, ones proves that every finitely generated commutative monoid is a homomorphic image of Nn0 .

14

Chapter 1. Monoids, Krull Monoids, Large Monoids

Recall that an element x in a commutative monoid M is irreducible (or an atom) if x ∈ / U (M ) and x = y + z with y, z ∈ M implies y ∈ U (M ) or z ∈ U (M ). Let A(M ) be the set of all atoms of the monoid M . A monoid M is atomic if every element of M \ U (M ) is a sum of finitely many atoms of M , that is, the union A(M ) ∪ U (M ) is a set of generators of M . If M is reduced and cancellative, atoms are the minimal elements of M \ {0} with respect to the algebraic preorder ≤, that is, for every atom x ∈ M one has that y ≤ x with y ∈ M implies that either y = 0 or y = x. Free monoids are atomic. Abelian groups are atomic monoids. In a free commutative monoid, the atoms are exactly the elements of a free set of generators. The free set of generators is unique in a free monoid. Any automorphism of an atomic monoid permutes the atoms. Hence there is an embedding of the group of all automorphisms of an atomic monoid M into the group of all permutations of the set of atoms of M . This embedding is an isomorphism when the monoid is free, so that the group of all automorphisms of a free commutative monoid M is isomorphic to the group of all permutations of the free set of generators of M . The proof we give here of the following result is due to F. Ced´o. For different proofs, see [Clifford and Preston 1967, Theorem 9.28] or [Freyd]. Theorem 1.12 (R´edei’s theorem [R´edei]). Every congruence on a finitely generated commutative monoid is finitely generated. Proof. Let M be a commutative monoid with a finite set {a1 , a2 , . . . , an } of generators, and let ∼ be a congruence on M . We must show that ∼ is generated by a finite set of pairs of elements of M . Let k be the field with two elements, R := k[x1 , x2 , . . . , xn ] the ring of polynomials in n indeterminates with coefficients in k, and M the set of all monomials of R. Here, by “monomial” we mean nonzero monomial, necessarily monic. Thus M is the free commutative monoid freely generated by the set {x1 , x2 , . . . , xn }. There is a unique monoid morphism ϕ : M → M that maps each xi to ai . Let I be the ideal of R generated by all elements f −g ∈ R with f, g ∈ M and ϕ(f ) ∼ ϕ(g). Since R is a Noetherian ring, its ideal I is finitely generated, so that there is a finite set { (fj , gj ) | j = 1, 2, . . . , t } of pairs of elements fj , gj ∈ M, with ϕ(fj ) ∼ ϕ(gj ) for every j, such that I is generated by the set { fj − gj | j = 1, 2, . . . , t }. In order to prove that ∼ is finitely generated, it suffices to prove that it is generated by the set G := { (ϕ(fj ), ϕ(gj )) | j = 1, 2, . . . , t }. Clearly, if ≡ denotes the congruence on M generated by the set G, one has that ≡ ⊆ ∼. Thus it suffices to prove that if f, g ∈ M and ϕ(f ) ∼ ϕ(g), then ϕ(f ) ≡ ϕ(g). This is trivial when f = g, so that we can assume f = g. Let f, g be distinct elements of M with ϕ(f ) ∼ ϕ(g). Then f − g ∈ I, so that there exist polynomials hj ∈ R, j = 1, 2, . . . , t, with f −g =

t j=1

hj (fj − gj ).

(1.2)

For every polynomial h ∈ R, let (h) be the number of monomials that appear in writing h as a sum of distinct elements of M. The proof that ϕ(f ) ≡ ϕ(g)

1.1. Commutative Monoids

15

t will be by induction on L := j=1 (hj ). If L = 0, then every hj is zero, so that f = g, and we are done. Assume that this is true for L − 1. The monomial f must also appear on the right of the identity (1.2), so that there exist a polynomial hj0 (j0 = 1, 2, . . . , t) and a monomial p that appears in writing hj0 as a sum of distinct monomials such that f is one of the two monomials that appears in writing p(fj0 − gj0 ) as a sum of distinct monomials, that is, f is one of the two monomials pfj0 and pgj0 . Assume, for instance, that f is equal to pfj0 . Then from (1.2), we t get that pgj0 − g = −p(f j0 − gj0 ) + f − g = −p(fj0 − gj0 ) + j=1 hj (fj − gj ) = (−p + hj0 )(fj0 − gj0 ) + j=j0 hj (fj − gj ). Now pgj0 , g ∈ M, pgj0 − g ∈ I, and

(−p + hj0 ) = (hj0 ) − 1. Thus, by the inductive hypothesis, ϕ(pgj0 ) ≡ ϕ(g). Now f = pfj0 , so that ϕ(f ) ≡ ϕ(pfj0 ). Since the pairs in G generate ≡, we know that ϕ(fj0 ) ≡ ϕ(gj0 ), so that ϕ(pfj0 ) ≡ ϕ(pgj0 ). Thus ϕ(f ) ≡ ϕ(pfj0 ) ≡ ϕ(pgj0 ) ≡ ϕ(g), as we wanted to prove. Similarly for the case f = pgj0 .  In particular, all congruences on the monoids Nn0 are finitely generated. Another important property of the monoids Nn0 concerns antichains. A subset W of a partially ordered set (X, ≤) is an antichain if the elements of W are pairwise incomparable, that is, if w, w ∈ W and w ≤ w imply w = w . For the monoids N0n , the preorder ≤ is a partial order (i.e., it is also antisymmetric) and coincides with the componentwise order: (x1 , . . . , xn ) ≤ (y1 , . . . , yn ) if and only if xi ≤ yi for every i = 1, . . . , n. For every m ≥ 0, the set Wm := {(0, m), (1, m−1), (2, m−2), . . . , (m−1, 1), (m, 0)} is an antichain of cardinality m + 1 in the monoid (N20 , ≤). Hence there is not a finite upper bound on the cardinality of the antichains of (N02 , ≤). Nevertheless: Theorem 1.13 ([Clifford and Preston 1967, Theorem 9.18]). Let n be a nonnegative integer. Every antichain in the partially ordered set (N0n , ≤) is finite. Proof. Induction on n. The case n = 1 is trivial. Suppose that the theorem has been proved for n − 1. Let A be an antichain in (Nn0 , ≤). Let πi : Nn0 → N0 , i = 1, 2, . . . , n, be the n projections. Set li := min(πi (A)) and Ai := A ∩ πi−1 (li ). Then Ai ⊆ A is an antichain in Nn0 , and the ith coordinate of all the elements of Ai . By the inductive hypothesis, every Ai is a finite set, clearly nonempty. is equal to li n Ai is a finite nonempty subset of A. Set mi := max(pii (M )) and Thus M := i=1 m := (m1 , . . . , mn ) ∈ N0n . Then, for every x ∈ M , πi (x) ≤ mi = πi (m) for all i, so that x ≤ m for every x ∈ M . Now li ≤ mi . Let [li , mi ] denote the set of all integers t with li ≤ t ≤ mi , and set Ni := A ∩ πi−1 ([li , mi ]). By the inductive hypothesis, n every Ni is also a finite set. Thus N := i=1 Ni is also a finite set. In order to conclude, it suffices to show that A ⊆ N . By contradiction, suppose that there exists an element a ∈ A, a ∈ / N . Then / [li , mi ]. By the definition of li , it follows that a∈ / Ni for every i, so that πi (a) ∈ πi (a) > mi , for every i. Hence πi (m) = mi < πi (a), so that m < a. It follows that x < a for every x ∈ M . This is a contradiction, because A is an antichain. 

16

Chapter 1. Monoids, Krull Monoids, Large Monoids

A monoid M is directly finite if for every x, y ∈ M , x = x + y implies y = 0. If M is any directly finite reduced monoid, then the algebraic preorder on M is a partial order. Conversely, if M is a monoid on which the algebraic preorder is a partial order, then M is reduced, but not necessarily directly finite, as the following example shows. Example 1.14. Let M = {0, 1} be the multiplicative monoid of the field with two elements. Clearly, M is not directly finite, because x = xy does not imply y = 1. The monoid M with the algebraic preorder is the linearly ordered set with 1 < 0. Finally, M is reduced because U (M ) = {1}.

1.2 Preordered Groups, Positive Cones If G is an abelian group, a translation-invariant preorder ≤ on G is completely determined by the set of elements x ∈ G with x ≥ 0, because for any x, y ∈ G, we have that x ≤ y if and only if y − x ≥ 0. (To see this, notice that x ≤ y implies 0 = x + (−x) ≤ y + (−x), and conversely y − x ≥ 0, that is, 0 ≤ y − x implies x = 0 + x ≤ (y − x) + x = y.) More precisely: Lemma 1.15. There is a one-to-one correspondence between the set of all submonoids of an abelian group G and the set of all translation-invariant preorders on G. This correspondence associates to every translation-invariant preorder ≤ on G the positive cone G+ := { x ∈ G | 0 ≤ x }. Conversely, if M is a submonoid of G, the corresponding preorder ≤M on G is defined, for every x, y ∈ G, by x ≤ y if y − x ∈ M . A preordered abelian group (G, +, ≤) is an abelian group (G, +) with a translation-invariant preorder ≤ on G. Equivalently, a preordered abelian group can be defined as a pair (G, C), where G is an abelian group and C is a submonoid of G. Preordered abelian groups form a category in which the morphisms f : (G, +, ≤) → (H, +, ≤) are the group morphisms f : G → H for which x ≤ y implies f (x) ≤ f (y) for every x, y ∈ G (equivalently, such that f (G+ ) ⊆ H + ). A partially ordered abelian group is a preordered abelian group (G, +, ≤) in which ≤ is a partial order, that is, the preorder ≤ is an antisymmetric relation. A submonoid of an abelian group G is sometimes called a cone in G. A reduced submonoid of an abelian group G is sometimes called a strict cone in G. Thus a strict cone is a submonoid C with the property that x ∈ C and −x ∈ C imply x = 0. It is easily seen that for a preordered abelian group (G, +, ≤), ≤ is a partial order if and only if the positive cone G+ of G is a reduced submonoid of G. Thus the one-to-one correspondence of the previous lemma induces a one-to-one correspondence between the set of all reduced submonoids of the abelian group G and the set of all translation-invariant partial orders on G.

1.2. Preordered Groups, Positive Cones

17

For any preorder ≤ on a set S, the equivalence relation associated to ≤ is defined, for all x, y ∈ S, by x ∼ y if x ≤ y and y ≤ x (Exercise 1.1). In the case of a preordered abelian group, we have the following. Proposition 1.16. Let G be a preordered abelian group. Set H := { x ∈ G | x ≤ 0 and 0 ≤ x }. Then: (a) H is a subgroup of G. (b) Define a relation  on G/H by x + H  y + H if x ≤ y, for every x, y ∈ G. This definition is independent of the choice of the representatives x and y of x + H and y + H, that is, the relation  on G/H is well defined. (c) The relation  defined in (b) is a partial order on G/H, and G/H, with this partial order, turns out to be a partially ordered group. Conversely, if G is an abelian group, H is a subgroup of G, and  is a translation-invariant partial order on G/H, then the relation ≤ on G, defined by x ≤ y if x + H  y + H, is a translation-invariant preorder on G. Thus there is a canonical one-to-one correspondence between the set of all translation-invariant preorders on G and the set of all pairs (H, ) with H a subgroup of G and  a translation-invariant partial order on G/H. Proof. We show only that H is a subgroup of G and leave the rest to the reader. Clearly, 0 ∈ H. If x, y ∈ H, then x ≤ 0 and y ≤ 0, so that x + y ≤ 0 + y = y ≤ 0. Similarly 0 ≤ x and 0 ≤ y, so that 0 ≤ x = x + 0 ≤ x + y. This proves that H is a submonoid of G. Finally, adding −x to both x ≤ 0 and 0 ≤ x, one gets that  0 ≤ −x and −x ≤ 0, and thus −x ∈ H. Notice that in Proposition 1.16, if G+ is the positive cone of the preordered group G, then the positive cone of the corresponding partial group G/H is G+/H = (G+ )red . For any commutative monoid M , endow the Grothendieck group G(M ) of M with the structure of a preordered group given by G(M )+ = {[m] | m ∈ M }, where [m] is the image of m ∈ M under the canonical map ψM : M → G(M ). Every monoid homomorphism ϕ : M → N induces a homomorphism of preordered groups G(ϕ) : G(M ) → G(N ). Hence G is a functor of CMon into the category of preordered abelian groups. For every monoid homomorphism ϕ : M → N , there is a commutative diagram ϕ /N M ψM

 G(M )

ψN

G(ϕ)

 / G(N )

It follows that if F is the forgetful functor of the category of preordered abelian groups into the category CMon that sends a preordered abelian group (G, +, ≤) to the commutative monoid (G, +), then ψ is a natural transformation of the identity functor CMon → CMon into the composite functor F G : CMon → CMon.

18

Chapter 1. Monoids, Krull Monoids, Large Monoids

1.3 The Monoid V (C), Discrete Valuations, Krull Monoids We will denote by Ob C the class of objects of any category C. Recall that a terminal object in a category C is an object T of C with the property that for every A ∈ Ob(C), there is a unique morphism A → T in C. Similarly, I is called an initial object of C if for every A ∈ Ob(C) there is exactly one morphism I → A. Finally, an object Z of C is called a null object (or a zero object) if it is both initial and terminal. Thus an object I is initial if and only if HomC (I, A) has cardinality 1 for every object A, and T is terminal if and only if HomC (A, T ) has cardinality 1 for every A. Obviously, an object is an initial object in a category C if and only if it is a terminal object in the dual category C op . If a category has an initial object, then all its initial objects are isomorphic. Similarly for terminal objects and zero objects. For instance, in the category Set, whose objects are all sets and morphisms are all mappings between them, the unique initial object is the empty set, the terminal objects are the sets of cardinality one, and there is no zero object. In the category Ab of abelian groups, the initial objects coincide with the terminal objects and are the groups of order 1. In the category Ring of rings with identity and ring morphisms preserving identities, the initial objects are the rings isomorphic to Z, but there are no terminal objects. In the category Grp the trivial group with one element is a zero object. Let C be a category and let Z be a zero object of C. Then there exist exactly one morphism A → Z and exactly one morphism Z → B for every pair A, B of objects. Their composite morphism A → B is called the zero morphism of A into B. In fact, it is easily seen that in a category C with a zero object of C, there is a unique zero morphism A → B for every pair A, B of objects of C. (One must prove that if Z, Z  are two zero objects, then the composite morphism A → Z → B is equal to the composite morphism A → Z  → B.) The unique zero morphism A → B will be denoted by ζA,B . Let C be a category. For every object A of C, let Iso(A) denote the isomorphism class of A, that is, the class of all objects of C isomorphic to A. The class Iso(A) is a subclass of the class Ob(C) of all objects of C, and the isomorphism classes Iso(A) form a partition of Ob(C). Let V (C) denote a skeleton of C, that is, a class of representatives of the objects of C modulo isomorphism. Notice that V (C) exists by the axiom of choice for classes. For every object A in C, there is a unique object A in V (C) isomorphic to A. Thus there is a mapping C → V (C), A → A, that associates to every object A of C the unique object A in V (C) isomorphic to A. Assume that a product A × B exists in C for every pair A, B of objects of C. Define an addition + in V (C) by A + B := A × B for every A, B ∈ V (C). In the next lemma, we show that in this way we get a monoid that is large, in the sense that it is a class and not a set when the category C is not skeletally small.

1.3. The Monoid V (C), Discrete Valuations, Krull Monoids

19

Notice that if C is an arbitrary category, so that the product A × B does not necessarily exist for any pair A, B of objects of C, then the skeleton V (C) turns out to be a class in which the operation induced by the product is only partially defined, that is, it is a mapping + : S → V (C) for a subclass S of V (C) × V (C). Lemma 1.17. Let C be a category with a terminal object and in which a product A × B exists for every pair A, B of objects of C. Then V (C) is a large reduced commutative monoid. Proof. It is easily seen that addition is commutative and associative. In order to prove that the terminal object T turns out to be the zero element of the monoid, we ∼ A. Let 1A : A → A be the must prove that if A is an object of V (C), then A × T = identity morphism and ζA : A → T the unique morphism. Then, for every object B, the mapping HomC (B, A) → HomC (B, A) × HomC (B, T ), f → (1A f, ζA f ) = ∼ A × T. (f, ζB ) is clearly a bijection. Thus A = The monoid V (C) is reduced, because if A, B are elements of V (C) and A × B = T , then there exist morphisms g : T → A and h : T → B such that for every object C, the mapping HomC (C, T ) → HomC (C, A) × HomC (C, B), f → (gf, hf ), is a bijection. Since HomC (C, T ) has one element for every C, it follows that both HomC (C, A) and HomC (C, B) have exactly one element each for every C, so that A and B are terminal objects; hence A = T and B = T .  Moving on to the dual category, one sees that if C is a category with an initial object and in which a coproduct A B exists for every pair A, B of objects of C, any skeleton V (C) of C isa large reduced monoid with respect to the addition + defined by A + B := A B for every A, B ∈ V (C). All the categories we will meet in the rest of this book will be preadditive categories. For the terminology and basic results about preadditive categories, see Chapter 4. In a preadditive category, an object is terminal if and only if it is initial (Lemma 4.3), and coproduct is canonically isomorphic to product. Hence, in this case, the construction with product and the dual construction with coproduct coincide. Otherwise, the two constructions can be “essentially different,” as the following example shows. Let C be the full subcategory of the category of sets whose objects are finite sets. In this case, it is clear that V (C) is in one-to-one correspondence with N0 , because every set is determined up to isomorphism by its cardinality. In C, the terminal objects are the sets of cardinality 1 and product is the Cartesian product. Thus, with respect to product, the monoid V (C) turns out to be isomorphic to the multiplicative monoid N0 . But the initial object of C is the empty set, and coproduct is the disjoint union. Therefore, with respect to coproduct, the monoid V (C) is isomorphic to the additive monoid N0 . Let M be a monoid. A discrete valuation of a monoid M is a nonzero monoid homomorphism v : M → N0 . Here N0 is the additive monoid of nonnegative integers. Every discrete valuation M → N0 induces a nonzero group homomorphism G(M ) → Z that maps ψM (M ) into N0 . Conversely, every nonzero group homomorphism f : G(M ) → Z with f (ψM (M )) ⊆ N0 induces a discrete valuation

20

Chapter 1. Monoids, Krull Monoids, Large Monoids

M → N0 . Thus discrete valuations can also be seen as the nonzero group homomorphisms G(M ) → Z that map ψM (M ) into N0 , i.e., nonzero homomorphisms of preordered groups, where G(M ) is the preordered group whose positive cone is the image ψM (M ) of M in G(M ). A monoid homomorphism f : M → M  is called a divisor homomorphism if for every x, y ∈ M , f (x) ≤ f (y) implies x ≤ y. Here ≤ denotes the algebraic preorder. A monoid M is a Krull monoid if there exists a divisor homomorphism of M into a free commutative monoid. Equivalently, a monoid M is a Krull monoid if and only if there exists a set { vi | i ∈ I } of monoid homomorphisms vi : M → N0 such that (1) if x, y ∈ M and vi (x) ≤ vi (y) for every i ∈ I, then x ≤ y; (2) for every x ∈ M , the set { i ∈ I | vi (x) = 0 } is finite. Our main applications of Krull monoids will be to the reduced monoid V (C). We leave to the reader the proof of the following elementary lemma. Lemma 1.18. A commutative monoid M is a Krull monoid if and only if the reduced monoid Mred is a Krull monoid. Reduced Krull monoids are characterized among Krull monoids in the next elementary lemma. Lemma 1.19. Let f : M → F be a divisor homomorphism of a commutative monoid M into a free commutative monoid F . The following conditions are equivalent: (a) (b) (c) (d)

The The The The

monoid M is reduced and cancellative. monoid M is reduced and directly finite. monoid M is reduced. homomorphism f is injective.

Proof. The implications (a) ⇒ (b) ⇒ (c) are trivial. (c) ⇒ (d) Assume M reduced. If x, y ∈ M and f (x) = f (y), then x ≤ y, so that there exists z ∈ M with x+z = y. Then f (x) = f (y) = f (x+z) = f (x)+f (z), from which f (z) = 0. Thus z ≤ 0, that is, there exists t ∈ M with z + t = 0. Since M is reduced, we get that z = 0, and x = y. (d) ⇒ (a) Every submonoid of a free monoid is both reduced and cancellative.  Thus when the reduced monoid V (C) is a Krull monoid, it is necessarily a cancellative monoid. The case of cancellative Krull monoids is characterized in the following proposition. Recall that if M is a commutative cancellative monoid, the canonical homomorphism ψM : M → G(M ) is injective, and its image ψM (M ) is the positive cone G(M )+ of G(M ). Thus ψM (M ) = G(M )+ is isomorphic to M via ψM . In the next proposition, we will identify the elements of M with their images in G(M ). Proposition 1.20. Let M be an additive, cancellative, commutative monoid with Grothendieck group G(M ). The following conditions are equivalent:

1.3. The Monoid V (C), Discrete Valuations, Krull Monoids

21

(a) M is a Krull monoid. (b) There exists a set { vi | i ∈ I } of nonzero group morphisms vi : G(M ) → Z such that (1) M = { x ∈ G(M ) | vi (x) ≥ 0 for every i ∈ I } and (2) for every x ∈ G(M ), the set { i ∈ I | vi (x) = 0 } is finite. (c) There exist an abelian group G, a set I, and a subgroup H of the free abelian (I) group Z(I) such that M ∼ = G ⊕ (H ∩ N0 ). Proof. (a) ⇒ (b) Since M is a Krull monoid, there exists a set { vi | i ∈ I } of discrete valuations of M satisfying the conditions in the definition of Krull monoid. Extend each vi to a nonzero group morphism vi : G(M ) → Z. We will prove that (1) and (2) of statement (b) hold for the group morphisms vi . Clearly, if x ∈ M , then vi (x) = vi (x) ≥ 0. Conversely, assume that x ∈ G(M ) is such that vi (x) ≥ 0 for all i. Then x = y − z with y, z ∈ M , so that vi (y) ≥ vi (z). Thus vi (y) ≥ vi (z) for every i, whence y ≥ z in M . It follows that y = z + w for some w ∈ M . Hence y = x + z and y = z + w in the abelian group G(M ). Thus x = w ∈ M . This shows that (1) holds. The proof of (2) is immediate. nonzero group morphisms satisfying (b). (b) ⇒ (c) Let { vi | i ∈ I } be a set of The image of the product mapping v := i∈I vi : G(M ) → ZI is contained in the subgroup Z(I) of the group ZI , so that v can be viewed as a group homomorphism v : G(M ) → Z(I) . Since every subgroup of a free abelian group is a free abelian group, it follows that G(M ) is the direct sum of ker v and a suitable complement C, that is, G(M ) = ker v ⊕ C. Moreover, v induces by restriction an isomorphism between C and the subgroup H := v(C) = v(G(M )) of ZI . Set G := ker v, so (I) (I) that M = v −1 (N0 ) = G ⊕ (v −1 (N0 ) ∩ C). It remains to show that v induces (I) (I) by restriction an isomorphism between v −1 (N0 ) ∩ C and H ∩ N0 . Clearly, v (I) (I) induces by restriction a monomorphism of v −1 (N0 ) ∩ C into H ∩ N0 . As far as (I) surjectivity is concerned, every element of H ∩ N0 has a unique inverse image in (I) C, which clearly belongs to v −1 (N0 ) ∩ C. (c) ⇒ (a) Assume that (c) holds. Let f be the composite mapping of the ∼ G ⊕ (H ∩ N(I) ) onto H ∩ N(I) and the inclusion canonical projection of M = 0 0 (I) (I) H ∩ N0 → F := N0 . If x, y ∈ M and f (x) ≤ f (y) in F , there exists z ∈ F with f (x) + z = f (y). Since f (x) and f (y) belong to H, which is a subgroup of Z(I) , it follows that z ∈ H. Since H ∩ F is the image of f , there exists z  ∈ M with f (z  ) = z. Then f (x) + f (z  ) = f (y) implies that x + z  and y differ by an element  of G, that is, an invertible element of M . Hence x ≤ y in M . In the next proposition, we have collected some further properties of reduced Krull monoids. By Lemma 1.19, a reduced Krull monoid is necessarily cancellative. Notice that if M is a reduced atomic monoid, the set of all atoms of M is the least set of generators of M , i.e., it is contained in any other set of generators of M .

22

Chapter 1. Monoids, Krull Monoids, Large Monoids

Proposition 1.21. Let M be a reduced Krull monoid. Then: (a) The algebraic preorder on M is a partial order on M . (b) For every x, y ∈ M with x ≤ y, the interval [x, y] := { z ∈ M | x ≤ z ≤ y } has only finitely many elements. (c) The number of decompositions of any nonzero element of M as a sum of nonzero elements of M is finite. (d) The monoid M is atomic. (e) An element x of the monoid M is an atom if and only if x is a minimal nonzero element of M (that is, a minimal element in the partially ordered subset M \ {0} of M with respect to the partial order induced by the algebraic partial order on M ). (f) Every element of M can be expressed in only finitely many ways as a finite sum of atoms. Proof. Statements (a), (b), (c), and (e) follow immediately from Lemma 1.19. To (I) prove (d), fix a divisor homomorphism f : M → N0 of M into a free commu(I) (I) tative monoid N0 . Let σ : N0 → N0 be the monoid homomorphism defined by σ(ni )i∈I = i∈I ni . Show that every element x of M is a sum of atoms of M by induction on σf (x).  Statement (f) is a consequence of (c). Corollary 1.22. A reduced Krull monoid M is a finitely generated monoid if and only if there is a divisor homomorphism f : M → Nn0 for some n ≥ 0. Proof. If there is a divisor homomorphism f : M → N0n , the algebraic preorder on M is the partial order induced by the componentwise order on N0n . The set A of all atoms of M is a set of generators for M and is an antichain in M . Hence A is an antichain in N0n . By Theorem 1.13, the set A is finite. For the converse, assume M is a reduced Krull monoid. Then there exists a (I) (I) divisor homomorphism f : M → N0 for some set I. Let πi : N0 → N0 be the canonical projection onto the ith summand. If M is finitely generated, there is a (J) finite subset J of I such that πi f = 0 for every i ∈ I \J. Then (πj f )j∈J : M → N0  is a divisor homomorphism. Example 1.23. Let r, s > 1 be integers and let M (r, s) be the submonoid of N0r generated by A = {(1, . . . , 1), e1 = (s, 0, . . . , 0), e2 = (0, s, 0, . . . , 0), . . . , er = (0, . . . , 0, s)}. Then M (r, s) is a Krull monoid and A is the set of atoms of M (r, s). Proof. In order to prove that M (r, s) is a Krull monoid, it suffices to show that if H is the subgroup of the free abelian group Zr generated by A, then M (r, s) = H ∩Nr0 (Proposition 1.20). The inclusion M (r, s) ⊆ H ∩ N0r is trivial.

1.3. The Monoid V (C), Discrete Valuations, Krull Monoids

23

For H ∩ N0r ⊆ M (r, s), let h = (n1 , . . . , nr ) be an element of H ∩ Nr0 . Let π : Z → (Z/sZ)r−1 be the group homomorphism defined by r

π(m1 , m2 , . . . , mr ) = (m1 − m2 + sZ, m2 − m3 + sZ, . . . , mr−1 − mr + sZ) for every (m1 , m2 , . . . , mr ) ∈ Zr . Notice that A ⊆ ker π, so that H ⊆ ker π. For every i = 1, . . . , r, divide ni by s, getting nonnegative integers qi , ti such that ni = qi s + ti and 0 ≤ ti < s. Then h = q1 e1 + · · · + qr er + (t1 , . . . , tr ). It follows that (t1 , . . . , tr ) ∈ ker π. Thus ti ≡ ti+1 (mod s) for every i = 1, . . . , r − 1. This and 0 ≤ ti < s imply that t1 = t2 = · · · = tr . It follows that h = q1 e1 + · · · + qr er + t1 (1, . . . , 1) belongs to M (r, s). This proves that the submonoid M (r, s) of N0 r is a Krull monoid. Since r, s > 1, the set A is an antichain. Since A generates M (r, s), A is the  set of atoms of M (r, s). In view of Proposition 1.20(c), Krull monoids have a very regular geometric structure. In the language of Minkowski’s geometry of numbers, if we represent a subgroup G of Zt in the n-dimensional real space, we get a “lattice,” that is, a structure with a very regular geometric pattern. (Here we are using the word lattice with a meaning completely different from the meaning employed until now in this chapter, for instance on page 4.) If M is a reduced Krull monoid, then M ∼ = Nt0 ∩ G is represented by the intersection of the sublattice G of Zt with the positive cone N0t . In an atomic monoid M , every element of M can be written as a sum of atoms, but this can be done in different ways, unless M is a free commutative monoid. That is, the uniqueness of the representation of an element as a sum of atoms fails in monoids that are not free. The failure of this uniqueness is minimal when the monoid M is a reduced Krull monoid, because it is due only to the presence of the border of N0t ∩ G. Hence, when M is a reduced Krull monoid that is not free, uniqueness of the representation of an element as a sum of atoms fails, but the monoid still has a very regular geometric pattern. In order to better explain what we mean by a “regular geometric pattern,” consider the four lattices in Figure 1.3. These four lattices give an idea of what we mean by regular geometric pattern. They represent four submonoids of N20 for which the inclusion into N02 is a divisor homomorphism. The first monoid (top left) is N20 itself, the free monoid with free set of generators {(1, 0), (0, 1)}. The second monoid (top right) is (Z(2, 0) + Z(1, −1)) ∩ N20 . Its atoms, which form its least set of generators, are (2, 0), (1, 1), and (0, 2). This monoid is not free, because (2, 0) + (0, 2) = (1, 1) + (1, 1) is a relation between the atoms. The third monoid (bottom left) is (Z(2, 0) + Z(0, 2)) ∩ N20 = N0 (2, 0) + N0 (0, 2), a free monoid with free set of generators {(2, 0), (0, 2)}. The fourth monoid (bottom right) is (Z(1, 2) + Z(2, −3)) ∩ N02 . Its atoms are (7, 0), (4, 1), (1, 2), and (0, 7). It is not a free monoid, because (7, 0) + (0, 7) = (4, 1) + 3(1, 2). A prime ideal P of a commutative monoid M is said to be a prime ideal of height one if it is minimal in the set of all nonempty prime ideals of M . We will see

24

Chapter 1. Monoids, Krull Monoids, Large Monoids

Figure 1.3: Four submonoids of N20 that are Krull monoids. Notice their regular geometric pattern. in the next proposition and at the end of Section 1.5 that prime ideals of height one abound in monoids with order-units and in Krull monoids. Immediately after the proof of Proposition 1.24, we give an example of a monoid very close to being a Krull monoid (it is cancellative, reduced, and its Grothendieck group is a free abelian group) that does not have prime ideals of height one. Proposition 1.24. In a commutative monoid with order-unit, every nonempty prime ideal contains a prime ideal of height one. Proof. If u is an order-unit in the commutative monoid M , every nonempty prime ideal of M contains u. Let P be a nonempty prime ideal of M . Apply Zorn’s lemma to the set of all nonempty prime ideals of M contained in P , partially ordered by inverse inclusion. 

1.3. The Monoid V (C), Discrete Valuations, Krull Monoids

25

The monoid with one element is a trivial example of a reduced cancellative commutative monoid without prime ideals of height one. A less trivial example is the following. Fix a limit ordinal α. Let G be the free abelian group of all functions α → Z that are zero almost everywhere, so that G is a free monoid with free set of generators the cardinal α itself. For every nonzero element g ∈ G, set supp(g) := { μ | μ < α, g(μ) = 0 }. Let M be the submonoid of G whose nonzero elements are all g ∈ G, g = 0, with g(λ) > 0, where λ is the greatest element of the finite set supp(g). It is easily checked that the prime ideals of the monoid M form a descending chain Pα = ∅ ⊂ · · · ⊂ P1 ⊂ P0 = M \ {0}, indexed in the set of all ordinals λ ≤ α, where Pλ = { g ∈ M | there exists an ordinal μ < α with g(μ) = 0 and μ ≥ λ }. The monoid M is cancellative, reduced, and its Grothendieck group is the free abelian group G, but M does not have prime ideals of height one. A valuation v : M → N0 of a commutative monoid M is essential if for every x, y ∈ M with v(x) ≤ v(y), there exists s ∈ M with x ≤ y + s and v(s) = 0. (Again, we use the term “essential valuation” in analogy with essential valuations of commutative rings.) (A)

Example 1.25. Let M := N0 be the free commutative monoid with free set A of (A) generators. Assume that |A| ≥ 2. The canonical projections πa : N0 → N0 are essential valuations. that is not essential. Here is an example of a valuation of the free monoid M Let v : M → N0 be the augmentation defined by v(s) := a∈A πa (s). Clearly, v(s) = 0 if and only if s = 0, and there are elements x and y in M with the same augmentation that are not comparable. Hence v(x) = v(y), but there does not exist an element s ∈ M with v(s) = 0 and x ≤ y + s. If v : M → N0 is a valuation of M , the index of v is e(v) := gcd(v(M )), that is, the greatest common divisor of all the elements of v(M ). Two valuations v, v  : M → N are said to be equivalent if e(v)−1 v = e(v  )−1 v  . Clearly, v is essential if and only if e(v)−1 v is essential. A monoid M is a discrete valuation monoid if Mred is isomorphic to N0 . If M is a discrete valuation monoid, then (1) there is a unique isomorphism θ : Mred → N0 ; (2) the mapping vM : M → N0 , defined by vM (x) = θ(x + U (M )), is an essential surjective valuation; and (3) for any other valuation v : M → N0 of M , one has that v = e(v)vM . Lemma 1.26. Let v : M → N0 be a valuation of a monoid M . Set Pv := { x ∈ M | v(x) > 0 }. Then (a) Pv is a nonempty prime ideal of M , and v induces a surjective homomorphism v : (MPv )red → v(M ), defined by v(x − s + U (MPv )) = v(x) for every x ∈ M and s ∈ M \ Pv .

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Chapter 1. Monoids, Krull Monoids, Large Monoids

(b) The following conditions are equivalent: (1) The valuation v is essential. (2) v(M ) = e(v)N0 , and v is an isomorphism. (3) The localization MPv of M at Pv is a discrete valuation monoid. (c) If v is essential, then Pv is a prime ideal of height one. Proof. (a) It is immediately seen that Pv is a nonempty prime ideal of M . If x, x ∈ M and s, s ∈ M \ Pv are such that x − s + U (MPv ) = x − s + U (MPv ), then there are t, t ∈ M \ Pv such that x + t = x + t . Thus v(x) = v(x ). This shows that v is a well-defined mapping. It is now clear that the induced mapping v : (MPv )red → v(M ) is a surjective monoid homomorphism. (b) (1) ⇒ (2) Set e0 := min(v(M ) \ {0}), so that e0 = v(x0 ) for a suitable x0 ∈ M . Clearly, v(M ) ⊇ e0 N0 . We will prove that v(M ) = e0 N0 . Suppose that x ∈ M . Then v(x) = e0 k + r with k, r ∈ N0 and r < e0 . Since v(kx0 ) = e0 k ≤ v(x) and v is essential, there exists s ∈ M such that kx0 ≤ x + s and v(s) = 0. Thus x+s = kx0 +y for some y ∈ M , from which v(y) = r. Thus r = 0 and v(x) ∈ e0 N0 . In order to prove that v : (MPv )red → v(M ) is injective, let x, x ∈ M and  s, s ∈ M \ Pv be such that v(x − s + U (MPv )) = v(x − s + U (MPv )). Then v(x) = v(x ). Since v is essential, there exist elements y ∈ M and t ∈ M \ Pv such that x+y = x +t. Thus v(y) = 0. It follows that y ∈ M \Pv and x−s+U (MPv ) = x − s + U (MPv ). (2) ⇒ (3) Since v is an isomorphism, one has that (MPv )red ∼ = v(M ) = e(v)N0 ∼ = N0 . Thus MPv is a discrete valuation monoid. (3) ⇒ (1) Let ϕ : M → (MPv )red be the canonical morphism, which is an onto mapping. The valuation v : M → N0 factors through (MPv )red , i.e., we have a commutative diagram M ϕ

/ N0 : vv v v v vv vv v v

 (MPv )red

Now (MPv )red ∼ = N0 , so that there is a unique isomorphism θ : (MPv )red → N0 . The endomorphisms of N0 are given by multiplication by elements of N0 . It follows that v = tθϕ for a suitable t ∈ N. Now let x, y ∈ M be such that v(x) ≤ v(y). Then there exists an element a ∈ M such that v(x)+ v(a) = v(y). Then ϕ(x+ a) = ϕ(y). Thus there exist elements s, s ∈ M \ Pv with x + a + s = y + s . Consequently, x ≤ y + s and v(s ) = 0. ∼ N0 has (c) By (b), MPv is a discrete valuation monoid. Hence (MPv )red = exactly two prime ideals (Example 1.4). By Propositions 1.5 and 1.6, Pv is a prime ideal of height one. 

1.4. Essential Morphisms

27

1.4 Essential Morphisms We now generalize the definition of essential valuation given in Section 18 to arbitrary morphisms between monoids. Let M, N be commutative additive monoids. We say that a monoid morphism ϕ : M → N is essential if for every x, y ∈ M with ϕ(x) ≤ ϕ(y), there exist elements z, t ∈ M with ϕ(z) = 0 and x + t = y + z. Let us determine all surjective essential monoid homomorphisms of a monoid M into a directly finite reduced monoid N . This is equivalent to determining all congruences ∼ on the monoid M such that the canonical projection M → M/∼ is an essential monoid homomorphism and M/∼ is a directly finite reduced monoid. Recall that a convex subgroup H of a preordered group G is a subgroup H of G with the property that for every a ∈ G and every b ∈ H, 0 ≤ a ≤ b implies a ∈ H. A directed subgroup H of G is a subgroup such that for every h ∈ H, there are elements h , h ∈ G+ ∩ H such that h = h − h . Proposition 1.27 ([Ara and Facchini, Proposition 6.2]). For every commutative monoid M with Grothendieck group G(M ), let L(G(M )) be the set of all directed convex subgroups of the preordered group G(M ), and let Spec (M ) be the set of all prime ideals P ∈ Spec(M ) with (MP )red a directly finite monoid. Then there is an inclusion-reversing one-to-one correspondence Spec (M ) → L(G(M )). The proof is easy, but rather long, and we omit it. The interested reader can find it in [Ara and Facchini]. Recall that for any prime ideal P of M , the canonical morphism M → (MP )red is a surjective mapping. Its kernel is a congruence on M , which we denote by ∼P . It is defined, for every x, x ∈ M , by x ∼P x if there exist s, s ∈ M \ P with x + s = x + s . Proposition 1.28. For every commutative monoid M , the assignment P ∈ Spec (M ) → ∼P gives a one-to-one correspondence of the set Spec (M ) onto the set of all the congruences ∼ on M such that M/∼ is directly finite and reduced and the canonical projection M → M/∼ is an essential monoid homomorphism. Proof. Let us prove that the correspondence is well defined. If P ∈ Spec (M ), ∼ (MP )red is directly finite and reduced. It remains to prove the monoid M/∼P = that the canonical projection π : M → M/∼P is an essential morphism. Let x, y be elements of M with π(x) ≤ π(y). Then there exists t ∈ M with x + t ∼P y. It follows that there are t , z ∈ M \ P with x + t + t = y + z. This shows that π

28

Chapter 1. Monoids, Krull Monoids, Large Monoids

is an essential monoid morphism and that the correspondence in the statement is well defined. We will now prove that the correspondence is injective. Suppose that P = P  are two primes belonging to Spec (M ). Without loss of generality, we can suppose P ⊆ P  , so that there exists p ∈ P \ P  . Then p ∼P  0 and p ∼P 0. This shows that ∼P = ∼P  . In order to prove that the correspondence is surjective, fix a congruence ∼ on M such that the canonical projection M → M/∼ is essential and the factor monoid M/ ∼ is directly finite and reduced. Set P := { x ∈ M | x ∼ 0 }. It is easy to see that P is a prime ideal. In order to prove that the correspondence is surjective, it suffices to show that ∼P = ∼. If x, y ∈ M and x ∼P y, then there exist z, t ∈ M \ P such that x + z = y + t. Then z ∼ 0 and t ∼ 0. But ∼ and + are compatible, so that x ∼ x + z = y + t ∼ y. For the converse, suppose x ∼ y, so that π(x) = π(y). Since π is essential, there exist z, t ∈ M with π(z) = 0 and x + t = y + z. It follows that π(x) + π(t) = π(y) = π(x). Now M/∼ is directly / P , hence x ∼P y. This shows that finite, so that π(t) = 0. It follows that z, t ∈ ∼P = ∼, as desired. Finally, the factor monoid (MP )red ∼ = M/∼P = M/∼ is directly finite, so that P ∈ Spec (M ). 

1.5 Further Results on Krull Monoids* Since we will not use the content of this section in the rest of the book, the results we will present now will be stated without proof. We have decided to include them because we think they are very interesting. Krull monoids are the analogue for commutative monoids of what Krull domains are in commutative algebra. Let us briefly recall the first notions about Krull domains. They are mainly due to Wolfgang Krull [Krull 1932b]. A detailed presentation with all the proofs can be found, for instance, in [Bourbaki, Chapter 7]. Recall that a discrete valuation on a field K is a mapping v : K → Z ∪ {+∞} satisfying the following properties: (1) v(xy) = v(x) + v(y) for every x, y ∈ K; (2) v(x + y) ≥ min{v(x), v(y)} for every x, y ∈ K; (3) v(1) = 0, v(0) = +∞. We explicitly exclude the trivial valuation defined by v(x) = 0 for every x ∈ K, x = 0. A discrete valuation domain (DVR) is an integral domain R for which there exists a (nontrivial) discrete valuation v on the field of fractions K of R with R = {x ∈ K | v(x) ≥ 0}. An integral domain R with field of fractions K is said to be a Krull domain if there exists a set { vi | i ∈ I } of discrete valuations vi : K → Z ∪ {+∞} with

1.5. Further Results on Krull Monoids*

29

the following two properties: (1) R = { x ∈ K | vi (x) ≥ 0 for every i ∈ I}; (2) for every x ∈ K the set { i ∈ I | vi (x) = 0 } is finite. A principal fractional ideal of an arbitrary integral domain R with field of fractions K is a cyclic R-submodule of K, that is, a submodule of K of the form kR := { kr | r ∈ R }. A divisorial fractional ideal of R is a nonzero intersection of a nonempty set of principal fractional ideals. The set D(R) of all divisorial fractional ideals of an integral domain R is a commutative monoid with respect to the operation ∗ defined, for every I, J ∈ D(R), by I ∗ J :=“the intersection of all the principal fractional ideals containing IJ.” Let Prin(R) be the set of all nonzero principal fractional ideals of R. Then Prin(R) is a subgroup of D(R), and the divisor class semigroup of R is the factor group Cl(R) := D(R)/Prin(R). An integral domain R is an integrally closed domain if for every element x of the field of fractions K of R, whenever R[x] is a finitely generated R-submodule of K, then necessarily x ∈ R. The domain R is completely integrally closed if every element x of K such that R[x] is contained in a finitely generated R-submodule of K belongs to R. Equivalently, if for every x ∈ K and every d ∈ R such that d = 0 and dxn ∈ R for all n ≥ 0, one has that x ∈ R. Clearly, every completely integrally closed domain is integrally closed. Conversely, every integrally closed Noetherian domain is completely integrally closed. Every valuation domain is integrally closed, but a valuation domain is completely integrally closed if and only if it has rank 1. It is possible to prove that the semigroup D(R) is a group if and only if R is completely integrally closed [Bourbaki, Theorem 1, page 479]. An integral domain R is a Krull domain if and only if R is completely integrally closed and has the ascending chain condition on divisorial ideals. A Krull domain R is a locally finite intersection of its localizations at the prime ideals of height 1, which are discrete valuation domains. It was [Chouinard] who realized that the same arguments could be adapted from fields to torsion-free abelian groups. In the rest of this section, all monoids M are commutative cancellative monoids and have a Grothendieck group G(M ) that is a torsion-free group. Notice that if M is a commutative monoid, then M is a Krull monoid if and only if Mred is a Krull monoid, and valuations and prime ideals behave well in the passage from M to Mred . Hence, for our study, we can suppose without loss of generality that our Krull monoids are reduced, hence cancellative (Lemma 1.19). In this case our monoids are isomorphic to a submonoid of a free monoid (Lemma 1.19(d) or Proposition 1.20(c)), so that their Grothendieck groups are free, hence torsion-free. Therefore the reduction to cancellative monoids with torsion-free Grothendieck group is very natural in this setting. We have already defined discrete valuations of a monoid. In particular, a discrete valuation of an abelian group G is a nonzero homomorphism v : G → Z. ∼ Z ⊕ ker v. The subset { x ∈ G | v(x) ≥ 0 } turns out to Clearly, in this case, G = be a submonoid of G, called the valuation submonoid of v. It is clearly isomorphic to N0 ⊕ ker v. (We will always deal with discrete valuations in this book; hence we

30

Chapter 1. Monoids, Krull Monoids, Large Monoids

will call discrete valuations simply valuations. A more general notion of valuation can be obtained by replacing the linearly ordered abelian additive group Z with an arbitrary linearly ordered abelian additive group, but this more general notion will not be treated here.) For a cancellative monoid M with G(M ) torsion-free, it is possible to define, as in the case of integral domains, principal fractional ideals (they are the subsets of G(M ) of the form k + M := { k + x | x ∈ M } for some k ∈ G(M )) and divisorial fractional ideals of M (they are the nonempty intersections of a nonempty set of principal fractional ideals), and if D(M ) denotes the set of all divisorial fractional ideals of M , then D(M ) turns out to be a commutative monoid with respect to the operation ∗ defined, for every I, J ∈ D(M ), by I ∗ J := “the intersection of all the principal fractional ideals that contain I + J.” Let Prin(M ) denote the subgroup of D(M ) consisting of all the principal fractional ideals. The divisor class semigroup is Cl(M ) := D(M )/Prin(M ). A monoid M is integrally closed if x ∈ G(M ), a ∈ M , n ∈ N0 , and nx = a imply x ∈ M . The monoid M is completely integrally closed if x ∈ G(M ), a ∈ M , and a + nx ∈ M for all n ∈ N0 imply x ∈ M . Every completely integrally closed monoid is integrally closed. Conversely, every integrally closed monoid with the ascending chain condition on ideals is completely integrally closed. The monoids D(M ) and Cl(M ) are groups if and only if M is completely integrally closed. An additive, commutative, cancellative monoid M with torsion-free Grothendieck group G(M ) is a Krull monoid if and only if M is completely integrally closed and satisfies the ascending chain condition on divisorial ideals in M . The following wonderful result is due to Ulrich Krause [Krause, Proposition]. Theorem 1.29. A commutative integral domain R is a Krull domain if and only if its multiplicative monoid R∗ := R \ {0} is a Krull monoid. Let us set aside the previous brief presentation of the analogy and the relations between the theory of Krull domains and fields with the theory of Krull monoids and torsion-free abelian groups, and let us go back to consider the more general case of monoids that are not necessarily cancellative but with G(M ) torsion-free. A divisor homomorphism ϕ : M → F of M into a free monoid F is a divisor theory if for every u ∈ F there exist finitely many elements x1 , . . . , xm ∈ M such that u = ϕ(x1 ) ∧ · · · ∧ ϕ(xm ). Here ∧ refers to the algebraic preorder on F . Notice that a free monoid F is a lattice with respect to its algebraic preorder ≤. Let M be a Krull monoid, so that there exists a divisor homomorphism (I) v = (vi )i∈I : M → N0 . First of all, we can cancel the valuations vi that are zero. All the remaining vi ’s are valuations, and after cancellation we get a divisor homomorphism of M into a free monoid. Then we can take only one valuation up to equivalence, and again, we still have a divisor homomorphism of M into a free monoid. Therefore, without loss of generality, we can suppose that the

1.6. Some Further Notions About Commutative Monoids* (I)

divisor homomorphism v = (vi )i∈I : M → N0 inequivalent.

31

is such that the valuations vi are

Proposition 1.30 ([Halter-Koch 1991, Satz 1, Satz 2, and Korollar] and [Facchini and Halter-Koch, Proposition 4.3]). Let M be a cancellative Krull monoid, so that (I) there exists a divisor homomorphism v = (vi )i∈I : M → N0 where the valuations vi are inequivalent. Let J be the subset of I consisting of all indices j ∈ I with vj essential. Then: (a) The mapping v ∗ = (e(vj )−1 vj )j∈J : M → N0 is a divisor theory. (b) v is a divisor theory if and only if I = J and e(vi ) = 1 for every i ∈ I. (c) If v : M → N0 is an arbitrary essential valuation of M , then there exists an index j ∈ J with v equivalent to vj . (J)

For any two divisor theories v : M → F and v  : M → F  of M , there is a unique isomorphism Φ : F → F  such that Φ ◦ v = v  . Every nonempty prime ideal of a cancellative Krull monoid M contains a prime ideal of height one [Facchini 2006a, Lemma 3.2]. We will denote the set of all prime ideals of height one of a commutative monoid M by X (1) (M ). If M is a cancellative Krull monoid, the mapping v → Pv is a one-to-one correspondence between the set of all essential valuations of M up to equivalence and the set X (1) (M ). Theorem 1.31 ([Facchini 2006a, Theorem 3.4]). The following conditions are equivalent for a cancellative monoid M : (a) M is a Krull monoid. (b) The localization MP is a discrete valuation monoid for every prime ideal is contained in at most finitely many prime P ∈ X (1) (M ), every x ∈ M  ideals of height one, and M = P ∈X (1) (M) MP .

1.6 Some Further Notions About Commutative Monoids* In any category with a zero object it is possible to define the concept of subobject and kernel (the equalizer of a morphism and the zero morphism). Sometimes there is also a notion of pure monomorphism (pure kernel). See Section 11.10. For instance, in the category Ab of abelian groups, subobjects and kernels coincide, and a subobject (subgroup) H of an abelian group G is pure if nG ∩ H = nH for every positive integer n. When the factor group G/H of G modulo a subgroup H is torsion-free, then H is a pure subgroup of G. Moreover, if G is torsion-free, then H is pure in G if and only if G/H is torsion-free. In the subcategory of torsion-free abelian groups, kernels are exactly pure subgroups. Let us see a concept about commutative monoids that has a behavior somewhat similar to the concept of

32

Chapter 1. Monoids, Krull Monoids, Large Monoids

pure subgroup in the theory of abelian groups. As in the rest of this chapter, in this section all monoids will be commutative additive monoids. Let us recall some terminology already previously introduced in this chapter. We saw in Section 1.1 that monomorphisms in the category CMon of commutative monoids are exactly the monoid morphisms that are injective mappings. Then we defined in Section 1.3 divisor homomorphisms as those monoid homomorphisms f : M → M  between two commutative monoids M, M  for which f (x) ≤ f (y) implies x ≤ y for every x, y ∈ M (here ≤ denotes the algebraic preorder on M and M  ), and Krull monoids as those commutative monoids M for which there exists a divisor homomorphism of M into a free commutative monoid. Then we proved in Lemma 1.19 that a commutative monoid is a reduced cancellative Krull monoid if and only if there is an injective divisor homomorphism of M into a free commutative monoid. We will call any divisor homomorphism that is an injective mapping a divisor monomorphism. Fix a submonoid K of a commutative monoid M . For every m ∈ M , it is possible to consider the “coset” m + K := { m + k | k ∈ K }. In the next lemma, we will consider the relation ρK on M defined, for every m, m ∈ M , by mρK m if both m and m belong to the same coset m + K for some m ∈ M . This relation ρK is not an equivalence relation on M because the cosets m + K, m ∈ M , do not form a partition of M . They form a partition of M if and only if (m + K) ∩ (m + K) = ∅ implies m + K = m + K for every m, m ∈ M . Lemma 1.32. Let M be commutative monoid and K a submonoid of M . The congruence ∼K on M generated by all the relations k ∼K k  with k, k  ∈ K (that is, the smallest congruence in which all the elements of K are pairwise congruent) is the transitive closure of the relation ρK on M defined, for every m, m ∈ M , by m ∼K m if there exist m ∈ M and a, b ∈ K with m = m + a and m = m + b. Proof. [Clifford and Preston 1961, Theorem 1.8] The relation ρK is the smallest relation on M that is reflexive, symmetric, compatible with the addition on the monoid M , and contains all the relations k ∼K k  with k, k  ∈ K. Therefore its  transitive closure is the congruence on M generated by all these relations. The category CMon is not preadditive, but has a zero object, the trivial monoid with one element. Hence it is possible to consider the kernel of any morphism f : M → N . (Here kernel has a completely different meaning from the meaning it had in Section 1.1, where kernel was a congruence.) The kernel of any morphism f : M → N in CMon is the equalizer of f and the zero morphism M → N , and is clearly the embedding ε of f −1 (0N ) into M . Lemma 1.33. Let K be a submonoid of a monoid M and let ε : K → M be the embedding. Then: (a) The monomorphism ε is a kernel in CMon if and only if for every m ∈ M , K ∩ (m + K) = ∅ implies m ∈ K.

1.6. Some Further Notions About Commutative Monoids*

33

(b) If the monomorphism ε is a kernel in CMon, then ε is a divisor monomorphism. (c) If M is a cancellative monoid, then ε is a kernel in CMon if and only if ε : K → M is a divisor monomorphism. Proof. (a) Let ∼K be the congruence on M generated by all the relations k ∼K k  with k, k  ∈ K. It is clear that ε is a kernel of a morphism in CMon if and only if it is the kernel of the canonical projection π : M → M/∼K , that is, if and only if K = π −1 (0M/∼K ), i.e., if and only if K = [0M ]∼K . The inclusion K ⊆ [0M ]∼K always holds trivially. Thus ε is a kernel if and only if K ⊇ [0M ]∼K . Now K ⊇ [0M ]∼K if and only if for every k ∈ K and every m ∈ M , k ∼K m implies m ∈ K. But ∼K is the transitive closure of ρK (notation as in Lemma 1.32), so that K ⊇ [0M ]∼K if and only if for every k ∈ K and every m ∈ M , kρK m implies m ∈ K, equivalently, if and only if for every k, a, b ∈ K and m, m ∈ M , k = m + a and m = m + b imply m ∈ K. This condition can be rewritten as “for every m ∈ M , if K ∩ (m + K) = ∅, then m + b ∈ K for every b ∈ K.” That is, “for every m ∈ M , if K ∩ (m + K) = ∅, then m + K ⊆ K.” Finally, m + K ⊆ K if and only if m ∈ K. (b) Let ε be the kernel of a monoid morphism f : M → N . Suppose that a, b ∈ K = f −1 (0N ) and a ≤ b in M . Then there exists m ∈ M with a + m = b. Then f (a) + f (m) = f (b), so that f (m) = 0, i.e., m ∈ K. This proves that ε is a divisor monomorphism. (c) Suppose M cancellative. Let ε be a divisor monomorphism. In view of (a) and (b), it suffices to show that K ∩ (m + K) = ∅ implies m ∈ K for every fixed m ∈ M . Now if k = k  + m with k, k  ∈ K and m ∈ M , then k  ≤ k in M , so k  ≤ k in K. Thus k = k  + a for some a ∈ K. Since M is cancellative, it follows  that m = a ∈ k. Statement (b) cannot be inverted if M is not cancellative, that is, (c) does not necessarily hold when M is not cancellative. For instance, let M be the multiplicative monoid Z/3Z and K its submonoid {0, 1}. By (a), ε is not a kernel in CMon, because m = 2 ∈ M \ K, but K ∩ (m + K) = {2} = ∅. It is easily seen that ε is a divisor monomorphism. Thus the notion of being a kernel in CMon is stronger, in general, than the notion of being a divisor monomorphism. Now let cCMon be the full subcategory of CMon whose objects are all cancellative commutative monoids. The same proof that in Section 1.1 showed us that monomorphisms in CMon are exactly the monoid morphisms that are injective mappings lets us see that monomorphisms in cCMon are also the monoid morphisms that are injective mappings. Monomorphisms in cCMon are not all divisor homomorphisms. As an example of a monoid monomorphism that is not a divisor homomorphism consider the additive monoid N0 , its submonoid N0 \ {1} and the embedding ε : N0 \ {1} → N0 . As in the category CMon, the kernel of any morphism f : M → N in cCMon is clearly the embedding ε of f −1 (0N ) into M .

34

Chapter 1. Monoids, Krull Monoids, Large Monoids

Proposition 1.34. Let cCMon be the full subcategory of CMon whose objects are all cancellative commutative monoids. Let K be a submonoid of a cancellative monoid M and let ε : K → M be the embedding. The following conditions are equivalent: (a) ε is a kernel in cCMon. (b) ε : K → M is a divisor monomorphism. (c) There exists a subgroup H of the Grothendieck group G(M ) such that K = M ∩ H. Proof. (a) ⇔ (b) Since M is cancellative, the kernel of any morphism f : M → N in cCMon coincides with the kernel of f : M → N in CMon, so that (a) holds by Lemma 1.33(c). (a) ⇒ (c) Let ε : K → M be a kernel of f : M → N in cCMon. Then f induces a preordered abelian group morphism G(f ) : G(M ) → G(N ). By the commutativity of the diagram M

f

ψM

 G(M )

/N ψN

G(f )

 / G(N )

(last paragraph of Section 1.2), in which the vertical arrows are injective, we get that K = f −1 (0N ) = (ψN f )−1 (0G(N ) ) −1 = (G(f )ψM )−1 (0G(N ) ) = ψM (ker(G(f ))) = M ∩ ker(G(f )).

Hence it suffices to take H := ker(G(f )). (c) ⇒ (a) If H is a subgroup of G(M ) and K = M ∩ H, then ε is the kernel of the composite morphism of ψM : M → G(M ) and the canonical projection G(M ) → G(M )/H.  We now further narrow our attention to the full subcategory rKMon of CMon of all reduced Krull monoids. All the objects of rKMon are cancellative monoids, and their algebraic preorders are partial orders. More precisely, every object M in rKMon is a subobject of a free commutative monoid, so that G(M ) is a free abelian group. Corollary 1.35. The kernel of any morphism f : M → N in rKMon coincides with the kernel of f in the category CMon. In particular, K := f −1 (0N ) is a reduced Krull monoid, the embedding ε : K → M is a divisor homomorphism, and there exists a pure subgroup H of the free abelian group G(M ) such that K = M ∩ H.

1.6. Some Further Notions About Commutative Monoids*

35

Proof. Consider the kernel ε : K → M of f : M → N in CMon (equivalently, in cCMon). Since ε is a divisor homomorphism (Proposition 1.34(b)) and M is a reduced Krull monoid, there is an injective divisor homomorphism of M into a free monoid F , so that there is an injective divisor homomorphism of K into F . This proves that K is a reduced Krull monoid. Therefore the kernel of f : M → N in rKMon coincides with the kernel of f in CMon. Finally, as we have seen in the proof of Proposition 1.34((a) ⇒ (c)), it is possible to take H := ker(G(f )). Since G(f ) is a group morphism of the free abelian group G(M ) into the free abelian group G(N ), the kernel H of G(f ) is a pure subgroup of G(M ).  If we have two objects M and N in rKMon and we fix two arbitrary monomor(X) (Y ) phisms ε : M → N0 and N → N0 , then any morphism f : M → N induces a commutative diagram  ε / N(X) M 0 f

  N

/ N(Y ) . 0

Now every object of rKMon is cancellative, hence it embeds in its Grothendieck group. Passing to the Grothendieck groups, we get the diagram G(M )

  G(ε) / (X) Z0

G(f )

  G(N )

/ Z(Y )   0

A

" / Q(Y ) . 0

(Y )

Here the dotted arrow exists because Q0 is an injective Z-module. Moreover, (X) (Y ) this arrow Z0 → Q0 is given by left multiplication by a column-finite Y × X (X) matrix A with entries in Q. As usual, we view the elements of Z0 as column (Y ) matrices indexed in X. Similarly for the elements of Q0 . Then the kernel of (X) the dotted arrow consists of all column matrices C ∈ Z0 such that AC = 0. (X) Restricting to G(M ) first, and then to M , one sees that the image ε(K) in N0 (X) of the kernel K := f −1 (0N ) of f : M → N via the monomorphism ε : M → N0 (X) consists of all column matrices C ∈ N0 with AC = 0 and C ∈ ε(M ). In the next proposition, we consider the special case in which the reduced Krull monoid M is a free commutative monoid. Proposition 1.36. The following conditions are equivalent for a submonoid K of a (X) free monoid N0 : (X)

(a) The embedding ε : K → N0 in the category rKMon.

(X)

is the kernel of some morphism f : N0

(Y )

→ N0

36

Chapter 1. Monoids, Krull Monoids, Large Monoids (X)

(X)

(b) The embedding ε : K → N0 is the kernel of some morphism f : N0 → N in the category rKMon, for some object N of rKMon. (X) (c) K = { x ∈ N0 | Ax = 0 } for some column finite matrix A with entries in N0 . Proof. (a) ⇒ (b) is trivial. (X)

(X)

(b) ⇒ (c) Let ε : K → N0 be the kernel of a morphism f : N0 → N (Y ) in the category rKMon. Since N embeds in a free commutative monoid N0 , (Y ) (X) (X) ε : K → N0 is also the kernel of a morphism g : N0 → N0 . Passing to the (Y ) (X) Grothendieck groups, g extends to the group morphism G(g) : Z0 → Z0 . This group morphism corresponds to left multiplication by a matrix A with entries in (Y ) (X) Z. But G(g) maps N0 into N0 , so that the matrix A has its entries in N0 . Now (c) follows easily. (X)

(c) ⇒ (a) If K = { x ∈ N0 | Ax = 0 } for some column-finite Y × X (X) matrix A with entries in N0 , then the embedding ε : K → N0 is the kernel of (X) (Y )  left multiplication by A, which is a morphism N0 → N0 . We will denote by Q≥0 the set of all nonnegative rational numbers: Q≥0 := { q | q ∈ Q, q ≥ 0 }. Proposition 1.37. The following conditions are equivalent for a submonoid K of a finitely generated free monoid Nn0 : (a) The embedding ε : K → Nn0 is the kernel in rKMon. (b) The embedding ε : K → Nn0 is the kernel rKMon, for some object N of rKMon. (c) K = { x ∈ N0n | Ax = 0 } for some m × n (d) K = { x ∈ Nn0 | Bx = 0 } for some m × n

(Y )

of some morphism f : Nn0 → N0

of some morphism f : Nn0 → N in matrix A with entries in N0 . matrix B with entries in Q≥0 .

Proof. (a) ⇒ (b) and (c) ⇒ (d) are trivial. (b) ⇒ (c) Let ε : K → Nn0 be the kernel of a morphism f : Nn0 → N in the (Y ) category rKMon. Since N embeds in a free commutative monoid N0 , ε : K → N0n ) (Y is also the kernel of a morphism g : Nn0 → N0 . Passing to the Grothendieck (Y ) groups, g extends to the group morphism G(g) : Z0n → Z0 . This group morphism  corresponds to left multiplication by a matrix A with entries in Z. But G(g) maps (Y ) N0n into N0 , so that the matrix A has all its entries in N0 . Now A has only finitely many nonzero entries, because m is finite. Hence we can replace A with the matrix A obtained from A by cancelling all zero rows of A , because A C = 0 if and only if AC = 0. Thus (c) follows easily. (d) ⇒ (a) Let B  = aB be the m × n matrix obtained by multiplying all the entries of B by a suitable positive integer a such that all the entries of B 

1.6. Some Further Notions About Commutative Monoids*

37

become nonnegative integers. Then K = { x ∈ Nn0 | B  x = 0 } is the kernel of the  morphism N0n → N0m given by left multiplication by B  . Most of the discussion above can be generalized from kernels of morphisms to equalizers of pairs of parallel morphisms. The equalizer of any two morphisms f, g : M → N in the categories CMon and cCMon is the embedding ε : K → M , where K := { m ∈ M | f (m) = g(m) }. It is easily seen that if ε : K → M is the equalizer of any two morphisms f, g : M → N in cCMon, then ε : K → M is a divisor monomorphism. Thus the equalizer of any two morphisms f, g : M → N in rKMon coincides with the equalizer of f and g in the category CMon. Also, a morphism ε : K → M in cCMon is the equalizer of some pair of morphisms f, g : M → N in cCMon if and only if it is the kernel of some morphism h : M → N  in cCMon. In particular, K := f −1 (0N ) is a reduced Krull monoid and there exists a pure subgroup H of the free abelian group G(M ) such that K = M ∩ H. Proposition 1.38. The following conditions are equivalent for a submonoid K of a (X) free monoid N0 : (X)

(a) The embedding ε : K → N0 is the equalizer of a pair of morphisms (Y ) (X) f, g : N0 → N0 in the category rKMon. (X)

(b) The embedding ε : K → N0 is the equalizer of a pair of morphisms (X) f, g : N0 → N in the category rKMon, for some object N of rKMon. (X)

(c) K = { x ∈ N0 entries in Z.

| Ax = 0 } for some column-finite matrix A with (X)

Proof. (b) ⇒ (c) Let ε : K → N0 be the equalizer of a pair of morphisms (X) (Y ) f, g : N0 → N in the category rKMon. Without loss of generality, N = N0 . Passing to the Grothendieck groups, f and g extend to group morphisms (X)

G(f ), G(g) : Z0

(Y )

→ Z0

(X)

and K = N0 (X)

∩ ker(G(f ) − G(g)).

(Y )

The group morphism G(f )− G(g) : Z0 → Z0 corresponds to left multiplication by a matrix A with entries in Z. Now (c) follows easily. (X)

(c) ⇒ (a) If K = { x ∈ N0 | Ax = 0 } for some column-finite Y × X matrix A with entries in Z, then A can be written as A = B − C for suitable column-finite Y × X matrices B, C with entries in N0 . Then K is the equalizer of (X) (Y ) f, g : N0 → N0 , where f is left multiplication by B and g is left multiplication by C.  Similarly: Proposition 1.39. The following conditions are equivalent for a submonoid K of a finitely generated free monoid Nn0 : (a) The embedding ε : K → Nn0 is the equalizer of a pair of morphisms f, g : Nn0 → (Y ) N0 in the category rKMon.

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(b) The embedding ε : K → Nn0 is the equalizer of a pair of morphisms f, g : N0n → N in rKMon, for some object N of rKMon. (c) K = { x ∈ Nn0 | Ax = 0 } for some m × n matrix A with entries in Z. (d) K = { x ∈ N0n | Bx = 0 } for some m × n matrix B with entries in Q. (e) K = W ∩ N0n for some vector Q-subspace W of the vector space Qn . (X)

Notice that every divisor monomorphism ε : K → N0 is the kernel of some morphism in cCMon (Proposition 1.34), but not every divisor monomorphism (X) ε : K → N0 is the kernel of some morphism in rKMon. For instance, the inclusion 2N0 → N0 is a morphism in rKMon and is a divisor monomorphism, but is not the kernel of some morphism in rKMon. Otherwise, if we suppose that 2N0 → N0 is the kernel of a morphism f : N0 → N in rKMon, then 2N0 is contained in the kernel of the group morphism G(f ) : Z → G(N ), where G(N ) is a free abelian group, so that the kernel of G(f ) is a pure subgroup of Z that contains 2N0 . Thus G(f ) = 0, hence f = 0, so that the kernel of f : N0 → N is the identity N0 → N0 , contradiction. Let M be a commutative monoid, U (M ) its group of units, and π : M → Mred = M/U (M ), π : x → x+U (M ), the canonical projection (cf. Proposition 1.6). Two elements x, y ∈ M are associates if π(x) = π(y). A commutative monoid M is factorial if it is an atomic cancellative monoid such that if x1 , . . . , xn , y1 , . . . , ym are atoms of M and x1 +· · ·+xn = y1 +· · ·+ym , then n = m and there exists a permutation σ of {1, 2, . . . , n} such that xi is associated to yσ(i) for every i = 1, 2, . . . , n. Equivalently, M is a cancellative monoid with Mred free. A commutative monoid M is half-factorial if it is an atomic cancellative monoid such that if x1 , . . . , xn , y1 , . . . , ym are atoms of M and x1 + · · · + xn = y1 + · · · + ym , then n = m. In the next result, we will give a characterization of half-factorial monoids. Let M be a commutative atomic cancellative monoid. Let A(M ) be the set of all atoms of M , so that M is generated by A(M ) ∪ U (M ). Let G(M ) denote the Grothendieck group of M . For simplicity of notation, we will assume M ⊆ G(M ). Let V be the Q-vector space G(M ) ⊗Z Q. Applying the functor G(M ) ⊗Z − to the inclusion Z → Q, we get a group morphism ι : G(M ) → V . Since Q is the ring of fractions of Z with respect to its multiplicatively closed subset S consisting of all its nonzero elements, V is simply the localization of the Z-module G(M ) at S, and the group morphism ι : G(M ) → V is the canonical mapping of G(M ) into its localization. Thus ker(ι) = { g ∈ G(M ) | ng = 0 for some n ∈ S } is the torsion subgroup t(G(M )) of G(M ). A hyperplane in V is a subset H of V for which there exist a nonzero Q-linear mapping f : V → Q and an element q ∈ Q with H = f −1 (q). Proposition 1.40. For a commutative atomic cancellative monoid M , the following conditions are equivalent:

1.7. Appendix to Chapter 1: Sets and Classes*

39

(a) M is half-factorial. (b) There exists a monoid morphism v : M → N0 with ker(v) = U (M ) and v(A(M )) ⊆ {c} for some nonzero c ∈ N0 . (c) There exists a hyperplane H in V such that 0 ∈ / H and ι(A(M )) ⊆ H. Proof. The proposition is trivially true if M is an abelian group. Thus we can suppose M is not an abelian group, in which case A(M ) is nonempty. (a) ⇒ (b) If M is half-factorial and v : M → N0 associates 0 to the elements of U (M ) and the length of (any) factorization of m to any element m ∈ M \U (M ), then v is a monoid morphism with the properties in (b). (b) ⇒ (c) Let v : M → N0 be a monoid morphism with ker(v) = U (M ) and v(A(M )) ⊆ {c} for some nonzero c ∈ N0 . The monoid morphism v extends to a group morphism G(v) : G(M ) → Z, and tensoring with Q, we get a Q-linear mapping f := G(v) ⊗Z Q : G(M ) ⊗Z Q → Q. Then v(A(M )) = {c} implies that f = 0, so that H := f −1 (c) is a hyperplane in G(M ) ⊗Z Q = V and 0 ∈ / H. Finally ι(A(M )) ⊆ H, because f (ι(A(M ))) = v(A(M )) ⊆ {c}. (c) ⇒ (a) Suppose that ι(A(M )) ⊆ H for some hyperplane H in V with / H. Then there exist a nonzero Q-linear mapping f : V → Q and an element 0∈ q ∈ Q with H = f −1 (q). Thus q = 0 and f (ι(A(M ))) ⊆ {q}. It follows that if x1 , . . . , xn , y1 , . . . , ym are atoms of M and x1 + · · · + xn = y1 + · · · + ym , then f (ι(x1 )) + · · · + f (ι(xn )) = f (ι(y1 )) + · · · + f (ι(ym )), that is, qn = qm. Hence n = m. 

1.7 Appendix to Chapter 1: Sets and Classes* In this monograph we deal with categories, their skeletons, and equivalences between two categories. This immediately makes us run into foundational problems. The aim of this section is definitely not that of introducing a new foundation of mathematics or criticizing the existing theories or explicating them in detail. The aim of this section is simply that of giving the reader some confidence not to run into paradoxes when dealing with classes. I began to be interested in this sort of questions when I realized that often in category theory it is necessary to make a class of choices. To be more explicit, let us consider the following two examples. As a first example, recall that a skeleton V (C) of a category C is a full subcategory of C with the property that for every object A of C there exists a unique object B of V (C) isomorphic to A. A functor F : C → D is fully faithful if it is full and faithful, and is essentially surjective if for every D ∈ Ob D there exists C ∈ Ob C such ∼ D. As a second example of a class of choices, recall the two possible that F (C) = equivalent definitions of equivalence between two categories:

40

Chapter 1. Monoids, Krull Monoids, Large Monoids

Lemma 1.41. For a covariant functor F : C → D, the following conditions are equivalent: (a) There exists a covariant functor G : D → C such that G ◦ F is naturally isomorphic to the identity functor 1C and F ◦ G is naturally isomorphic to the identity functor 1D . (b) The functor F is fully faithful and essentially surjective. A functor F satisfying the equivalent conditions of this lemma is called a category equivalence. Two categories C and D are said to be equivalent if there exists an equivalence F : C → D. As soon as we try to prove the existence of a skeleton of any category C or the equivalence of conditions (a) and (b) in Lemma 1.41, we are forced to apply the axiom of choice for classes. That is, we must make not a set of choices, but a class of choices. In order to prove that every category has a skeleton, we must fix one object in each isomorphism class of C. In order to prove that (b) implies (a) in Lemma 1.41, we must fix for every object D of D an object G(D) in E ∼ D. We know that the axiom of choice for sets is equivalent with F (G(D)) = to the fact that every set can be well ordered, to Zorn’s lemma about partially ordered sets, and to the fact that every onto mapping between any two sets has a right inverse. The axiom of choice for sets is an important and fundamental tool in mathematics, and how can we make a class of choices without taking care of stressing the application of the axiom of choice for classes? This is what we will do in this section. The Zermelo–Fraenkel theory. The most popular and accepted axiomatic set theory is ZFC, the Zermelo–Fraenkel set theory with the axiom of choice. It has a single primitive ontological notion, the notion of set. That is, it treats only sets (and not classes): all individuals in the universe of discourse are sets. Sets are denoted by lowercase letters. The only binary relations are equality and set membership, denoted by ∈. Thus the formula x ∈ y indicates that x and y are sets and that x belongs to y (or x is an element of y, or x is a member of y). We can only use the logical symbols ¬, ∧, ∨, →, ↔, ∀, ∃, = (equality), parentheses, lowercase letters (variable symbols) and the symbol ∈. Clearly, one must follow the rules studied in any course of mathematical logic to get well-formed formulas. Here is a list of the axioms of ZFC. We essentially follow the presentation of [Kunen]. Alternative forms of the axioms can be found, for instance, in [Jech]. Notice that there are various formulations with different axioms. For instance, one may adjoin an axiom that explicitly states that at least one set exists, and this, in the following list, follows from axiom 7. Notice that the axioms are formulas, to which we have added some comments for clarity. 1. Axiom of extensionality. Two sets are equal if they have the same elements, that is, a set is determined by its elements: ∀x∀y(∀z(z ∈ x ↔ z ∈ y) → x = y).

1.7. Appendix to Chapter 1: Sets and Classes*

41

2. Axiom of regularity. Every nonempty set x contains an element y such that x and y are disjoint sets. ∀x(∃a(a ∈ x) → ∃y(y ∈ x ∧ ¬∃z(z ∈ y ∧ z ∈ x))). 3. Axiom schema of specification (also called the axiom schema of separation). If z is a set, and φ is a property that the elements x of z can have or not have, then there exists a subset y of z containing the elements x in z that satisfy the property φ: ∀z∀w1 . . . ∀wn ∃y∀x(x ∈ y ↔ (x ∈ z ∧ φ)). Here φ is a formula in the language of ZFC in the variables x, y, z, w1 , . . . , wn with free variables among x, z, w1 , . . . , wn and y not free in φ. 4. Axiom of pairing. If x and y are sets, then there exists a set whose elements are exactly x and y: ∀x∀y∃z∀w(w ∈ z ↔ w = x ∨ w = y). 5. Union axiom. For any set x there is a set whose elements are exactly the elements of the elements of x: ∀x ∃y ∀z(z ∈ y ↔ ∃w(z ∈ w ∧ w ∈ x)). 6. Axiom schema of collection. If φ is a formula in the language of ZFC with free variables among x, y, z, w1 , . . . , wn and with a nonfree variable w, then ∀z ∀w1 . . . ∀wn (∀x(x ∈ z → ∃!yφ) → ∃w∀x(x ∈ z → ∃y(y ∈ w ∧ φ))). Here ∃!y means “there exists a unique y such that. . . .” The axiom essentially says that if f : z → z  is a function, then the image of f is a subset of a set. A function f : z → z  is a triple (f, z, z  ) of sets, where f ⊆ z × z  and for every x ∈ z there exists a unique y ∈ z  with (x, y) ∈ f . 7. Axiom of infinity. The axiom essentially states that there exists a set with infinitely many members: ∃x (∅ ∈ x ∧ ∀y(y ∈ x → y ∪ {y} ∈ x)) . 8. Axiom of power set. For any set x, there is a set y whose elements are exactly the subsets of x: ∀x∃y∀z(z ∈ y ↔ (∀q(q ∈ z → q ∈ x))). 9. Axiom of choice. For any set x, every equivalence relation on x has a set of representatives. The axiom of choice AC is independent of the other eight axioms of ZF (the set theory ZFC without choice), and the continuum hypothesis 2ℵ0 = ℵ1 is independent of ZFC.

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Chapter 1. Monoids, Krull Monoids, Large Monoids

With ZFC we can handle sets, but in category theory we need classes as well. In a category, for instance in the category of sets, objects form a class, not a set. To define a functor between two categories, we need a function from the class of objects of a category to the class of objects of the other category. In order to prove the equivalence of the various standard definitions of category equivalence between two categories or the existence of a skeleton of a category, we need the axiom of choice for classes. In category theory, there is an absolute need of classes, and they cannot be handled in ZFC. The problem is that ZFC does not make it possible to handle classes, hence it is not convenient for dealing with category theory. The theory ZFC can be enriched with further (independent) axioms. For instance, recall that if P is a partially ordered set, a subset A of P is said to be cofinal in P if for every x ∈ P there exists y ∈ A with x ≤ y. A mapping f : X → P is said to be cofinal if f (X) is a cofinal subset of P . The least cardinal of cofinal subsets of P is called the cofinality of P , and is denoted by cf (P ). Thus the cofinality of P is the least ordinal α such that there is a cofinal mapping f : α → P . For any ordinal α, we must clearly have cf (α) ≤ α, because the identity mapping is cofinal. In particular, this holds for cardinals, so any cardinal κ, viewed as an ordinal, either satisfies cf (κ) = κ, in which case it is said to be regular, or it satisfies cf (κ) < κ, in which case κ is called singular. Thus a cardinal number is regular if and only if all its cofinal subsets have cardinality κ. As far as finite cardinals are concerned, 0 and 1 are regular cardinals, and all cardinals ≥ 2 are singular. It can be proved that an infinite cardinal is regular if and only if for  every family { Ai | i ∈ I } of sets, |I| < κ and |Ai | < κ for all i ∈ I implies  i∈I Ai  < κ. A cardinal number λ is weakly inaccessible if it is > ℵ0 , regular, and for every κ < λ one has that κ+ < λ. A cardinal number λ is strongly inaccessible if it is > ℵ0 , regular, and for every κ < λ one has that 2κ < λ. Every strongly inaccessible cardinal is weakly inaccessible. The existence of weakly inaccessible cardinals and strongly inaccessible cardinals cannot be proved in ZFC. More precisely, consistency of ZFC implies both consistency of ZFC+“there are no strongly inaccessible cardinals” and consistency of ZFC+“there are no weakly inaccessible cardinals.” There are several other axioms independent of ZFC, like the continuum hypothesis (there are no sets of cardinality strictly between ℵ0 and 2ℵ0 ), Martin’s axiom, V = L, and the Diamond principle. The introduction of these axioms does not allow a treatment of classes either. Universes. An alternative approach is due to Grothendieck. We fix a set U , called a universe, in which we perform all our constructions, and call U -small sets the sets that are members of U and U -large sets the sets that do not belong to U . Let us give the exact definition. A universe is a set U satisfying the following properties: (a) X ∈ Y ∈ U → X ∈ U . (b) X, Y ∈ U → {X, Y } ∈ U .

1.7. Appendix to Chapter 1: Sets and Classes*

(c) (d) (e) (f) (g)

43

X, Y ∈ U → X × Y ∈ U . X ∈ U → P(X) ∈ U .

X ∈ U → Y ∈X Y ∈ U . The set ω of natural numbers is an element of U . If X ∈ U and f : X → U is a mapping, then { f (Y ) | Y ∈ X } ∈ U .

The axioms of ZFC do not guarantee the existence of a universe. Following Grothendieck, we adjoin a further axiom to the axioms of ZFC: Axiom of universes: Every set is a member of a universe. Given any universe U , if the axioms of ZFC are satisfied by the class of all sets with the relation ∈, then they are also satisfied by the set of all sets belonging to U with the relation ∈ between them. Hence we can argue remaining in the universe U , which we suppose fixed once for all. In the universe, we find all what we need, and if we do not find it, we can always adjoin it to the universe thanks to the axiom of universes. In other words, we decide to work in a set that we possibly expand. The cardinal number of a universe U is a strongly inaccessible cardinal. Hence the axiom of universes implies the existence of arbitrarily large strongly inaccessible cardinals. The idea is interesting, but some mathematicians are not yet satisfied, since the solution of the problem is not elegant. (The method of “postulating” what we want has many advantages; they are the same as the advantages of theft over honest toil [Russell, p. 71].) How is it possible to deal with the category Set of all sets in our universe, in this universe in expansion? For the notion of class, we must introduce NBG. odel theory. The von Neumann–Bernays–G¨ The von Neumann–Bernays–G¨ odel set theory (NBG) is a conservative extension of ZFC [Smullyan and Fitting]. The ontology of NBG includes proper classes. The members of both sets and proper classes are sets. Proper classes cannot be members. “Conservative extension” means that a statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC, that is, any theorem in NBG which speaks only about sets is a theorem in ZFC. In NBG, quantified variables in the defining formula can range only over sets. Let us try to be more precise. The characteristic of NBG is the distinction between proper classes and sets. NBG is a two-sorted theory, that is, two types of variables are used in NBG. Lowercase letters will denote variables ranging over sets, and uppercase letters will denote variables ranging over classes. The atomic sentences a ∈ b and a ∈ A are defined for a, b sets and A a class, but A ∈ a and A ∈ B are not defined for any two classes A, B. Equality can have the form a = b or A = B. The notation a = A stands for ∀x(x ∈ a ↔ x ∈ A), but is an abuse of notation. Alternatively, NBG can also be presented as a one-sorted theory of classes, with sets being those classes that are members of at least one other class. That is, NBG can be presented as a system having only one type of

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Chapter 1. Monoids, Krull Monoids, Large Monoids

variables (class variables) with a unary relation M(A) (M stands for the German word “Menge,” which means “set”), and M(A) indicates that A is a set. Thus M(A) ↔ ∃B(A ∈ B). Notice that NBG admits the class V of all sets, but it does not admit the class of all classes or the set of all sets. Here is a list of the axioms of NBG. Notice that the first five coincide with five axioms of ZFC and deal only with sets, not classes. 1. Axiom of extensionality. Two sets are equal if they have the same elements: ∀a∀b(∀z(z ∈ a ↔ z ∈ b) → a = b). 2. Axiom of pairing. If x and y are sets, then there exists a set whose elements are exactly x and y. 3. Union axiom. For any set x there is a set whose elements are exactly the elements of the elements of x. 4. Axiom of power set. For any set x, there is a set y whose elements are exactly the subsets of x. 5. Axiom of infinity. ∃x (∅ ∈ x ∧ ∀y(y ∈ x → y ∪ {y} ∈ x)) . The remaining axioms are primarily concerned with classes rather than sets. 6. Axiom of extensionality for classes. Two classes are equal if they have the same elements: A = B ↔ ∀x(x ∈ A ↔ x ∈ B). 7. Axiom of regularity for classes. Every nonempty class A contains an element disjoint from A: ∃x(x ∈ A) → ∃y(y ∈ A ∧ ¬∃z(z ∈ y ∧ z ∈ A)). Finally, the last two axioms are peculiar to NBG: 8. Axiom of limitation of size. For any class A, there exists a set a such that a = A if and only if there is no bijection between A and the class V of all sets. This is really a powerful axiom. From this axiom, every proper class is equipotent to the class V of all sets. Moreover, the axiom of choice for classes holds, because the class of ordinals is not a set, so that there is a bijection between the ordinals and any class that can be well ordered. Equivalently, if A is any class and ∼ is an equivalence relation on A, a class of representatives exists. In particular, every category has a skeleton. 9. Class comprehension schema. For any formula φ containing no quantifiers over classes (it may contain class and set parameters), there exists a class A such that ∀x(x ∈ A ↔ φ(x)).

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45

It can be proved that NBG can be finitely axiomatized. What is important for us is that in NBG, which is a conservative extension of ZFC, we can deal with classes and have the axiom of choice for classes. Thus every category has a skeleton, we have a class of representatives for any equivalence relation on any class, and we can define an equivalence between two categories either as a functor with a quasi-inverse or as a faithful, full, essentially surjective functor. If C is any additive category, any skeleton V (C) of C is a reduced additive large monoid with respect to the product on V (C) induced by the product between objects of C. Here “large” means that V (C) can be a class, and not necessarily a set. Notice that the direct product of a class of classes does not exist in NBG, because a mapping from a class to a class is a class; hence it cannot be an element of a class. The Morse–Kelley set theory. Morse–Kelley (MK) set theory is a first-order axiomatic set theory closely related to NBG [Morse]. MK allows the bound variables in the schematic formula appearing in the axiom schema of class comprehension to range over proper classes as well as sets. NBG restricts these bound variables to sets alone. With the exception of class comprehension, the axioms of MK are the same as those for NBG. The union of a class of classes and the disjoint union of a class of classes exist in MK. The theory NFUP . This is a very interesting theory, which has Solomon Feferman [Feferman 1974, Feferman 2006] as one of its supporters. In the acronym NFUP , NF stands for “new foundations for mathematical logic,” U means “urelements” (there can be more than one class that has no members), and P indicates the “pairing axiom”: (X1 , X2 ) = (Y1 , Y2 ) → X1 = Y1 ∧ X2 = Y2 . NFUP is a one-sorted theory (only one type of variables, called class variables and denoted by uppercase letters). The only binary relations are equality = and class membership ∈. Moreover, NFUP has one binary operation symbol (−, −) for “pairing.” The terms in the language are obtained from the variables by closing under the pairing operations: (1) the variables are terms; and (2) if s, t are terms, then (s, t) is a term. That is, the terms are generated from the variables by closing under the pairing operation. A formula ϕ in this theory is said to be stratified if it is possible to assign a type to each term t occurring in the formula ϕ, where the type is a natural number, in such a way that (1) each variable of ϕ is assigned the same type at all its occurrences, (2) each term t of ϕ is assigned the same type as all the variables occurring in t, (3) for each subformula of ϕ of the form t = s, the terms t and s are assigned the same type, (4) for each subformula of ϕ of the form t ∈ s, if the term t is assigned type n, then the term s is assigned type n + 1.

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Chapter 1. Monoids, Krull Monoids, Large Monoids

Examples: X ∈ Y and Y ∈ X are stratified formulas for X and Y distinct variables. X ∈ X and (X ∈ Y ) ∧ (Y ∈ X) are not stratified formulas. Remark 1.42. In 1921 Kuratowski introduced the idea of defining pairs as (X, Y ) = {{X}, {X, Y }}. This is not a stratified formula. This is the reason why in NFUP pairs are defined using a binary operation symbol. The three axioms of NFUP are extensionality, the stratified comprehension axiom schema and the pairing axiom P. See [Feferman 2006, Section 3] for the exact definitions and details. Here extensionality is weakened to be applied only to non-urelements: ∃X(X ∈ A) ∧ ∀X(X ∈ A ↔ X ∈ B) → A = B The stratified comprehension axiom schema consists of all formulas of the form ∃A∀X(X ∈ A ↔ ϕ), where ϕ is a stratified formula and the variable A does not occur in ϕ. NFUP allows the formation of (1) the Cartesian product of an indexed class of classes, (2) the union of any class of classes, (3) the construction of the category of all functors between any two given categories, and (4) the construction of the category Cat of all categories. The theory NFUP has been proved to be consistent.

1.8 Notes on Chapter 1 The great difference between groups and monoids is that groups have a much more regular structure than monoids. This is due to the fact that in a group G there is the symmetry G → G defined by g → g −1 . This mapping is an antiautomorphism of G, and yields all the bijections (symmetries) G → G given by left multiplications by g. These bijections produce a great regularity in the structure of a group G that monoids don’t have. Groups should always be seen as (universal) algebras of G, ·, 1, −1  with a binary, a nullary, and a unary operation. In topological groups these three operations are supposed to be continuous. Also, when one defines bialgebras, one assume that they are algebras and coalgebras, with compatible operations. But this is not sufficient to define a symmetric enough structure. Thus one defines Hopf algebras, requiring the existence of the antipode, which plays the role of the symmetry g → g −1 in groups. In this book, Section 1.1 is essentially folklore. Proposition 1.8 has been taken from [Facchini and Fern´andez-Alonso, Section 6]. The content of Section 1.2 has been taken from [Goodearl 1986] and [Goodearl 1991]. The spectrum Spec(M ) of a commutative monoid M is a commutative monoid. The monoid structure in Spec(M ) is given by the union ∪ of prime ideals,

1.8. Notes on Chapter 1

47

and the zero is the empty ideal. The spectrum Spec(M ) is also equipped with a topology, where a basis of open sets is given by the sets / P }, D(a) := { P ∈ Spec(M ) | a ∈

a ∈ M.

For any monoid morphism f : M → N , there is a continuous map f ∗ : Spec(N ) → Spec(M ),

Q → f −1 (Q)

(notice that the inverse image of a prime ideal via a monoid morphism is a prime ideal). The operation ∪ on Spec(M ) and the topology of Spec(M ) are compatible, i.e., the mapping ∪ : Spec(M ) × Spec(M ) → Spec(M ) is continuous [Pirashvili]. Thus Spec(M ) is a topological monoid. For any commutative monoid M , there is a natural isomorphism of topological monoids Spec(M ) ∼ = HomCMon (M, {0, 1}) [Pirashvili]. Here the monoid {0, 1} is endowed with multiplication and is a topological monoid with respect to the topology in which the open subsets are ∅, {1}, and {0, 1}. For instance, the topological monoids Spec(N) and {0, 1} are isomorphic. The topology  on HomCMon (M, {0, 1}) is the topology induced by the product topology on m∈M {0, 1}. One has that Spec(M ) ∼ = Spec(M/), where  is the least congruence on M such that every element in the quotient monoid M/ is idempotent (page 302). More precisely, there is a relation between monoids and semilattices. A joinsemilattice (or upper semilattice) is a partially ordered set in which every nonempty finite subset has a least upper bound. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set in which every nonempty finite subset has a greatest lower bound. A semilattice with 1 is a meet-semilattice with a greatest element 1. A morphism of semilattices with 1 is a mapping that respects the greatest lower bound of two elements and the greatest elements 1. If M is a commutative additive monoid in which 2x = x for all x ∈ M , one defines y ≤ x if x + y = y. In this way, M becomes a semilattice with 1. Conversely, if L is a semilattice with 1, then L is a monoid with respect to the operation ∧. The category of monoids satisfying the identity 2x = x turns out to be equivalent to the category of semilattices with 1. There is a notion of tensor product of commutative monoids, and one finds the isomorphism M/  ∼ = M ⊗ {0, 1}. For all finitely generated semilattices L with 1, there is an isomorphism ev : L → Hom(Hom(L, {0, 1}), {0, 1}), defined by ev(x))(f ) = f (x) for every x ∈ L and f ∈ Hom(L, {0, 1}). For any monoid M , the canonical homomorphism q : M → M/ induces an isomorphism of topological monoids q ∗ : Spec(M/) → Spec(M ). See [Pirashvili]. The advantage in moving on from a category C to a skeleton V (C) is that all the objects that in the category C are defined only up to (natural) isomorphisms become unique in the category V (C). For instance, in V (C) there is at most one

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terminal object, at most one initial object, at most one zero object (which becomes like the identity in an algebraic structure), and any two objects have at most one product and at most one coproduct, so that if C is additive, then V (C) is a commutative additive, possibly large, monoid. All solutions of universal properties are unique in the skeleton V (C). If C is a monoidal category, then V (C) is a multiplicative, possibly large, monoid. Reduced localization with respect to a prime ideal has found several applications in the study of the monoids V (C), for instance in [Facchini and HalterKoch, Facchini 2006a, Facchini and Pˇr´ıhoda 2008a, Ara and Facchini], and, implicitly, in [Schofield, Theorem 1.5]. In the literature, the terminology concerning semigroups and monoids depends a lot on the author. For instance, the submonoids M of Nn0 (or Qn≥0 ) such n that the embedding ε : M → N0n (or ε : M → Q≥0 ) is a divisor homomorphism, n that is, such that x + y = z with x in N0 and y, z ∈ N0n implies x ∈ M , appear in [Hochster] under the name full semigroups and in [Leuschke and Wiegand] under the name full subsemigroups of N0n . Cf. positive normal affine semigroups [Wiegand (Y ) 2001, Wiegand and Wiegand]. Equalizers of a pair of morphisms f, g : Nn0 → N0 in the category rKMon (Proposition 1.39) are called expanded subsemigroups of Nn0 in [Leuschke and Wiegand]. For a further study of kernels in categories of monoids and related notions, see [Facchini and Rodaro]. Theorem 1.13 is essentially [Clifford and Preston 1967, Theorem 9.18]. It was [Chouinard] who realized the analogy between Krull domains and Krull monoids and introduced the divisor class group of a Krull monoid. Proposition 1.20 is due to him [Chouinard, Proposition 1]. For the definition of essential valuations in commutative monoids and Proposition 1.30, we have followed [Halter-Koch 1991]. Lemma 1.26 is taken from [Facchini and Halter-Koch]. Proposition 1.40 is taken from [Kainrath and Lettl, Proposition 1].

Chapter 2

Basic Concepts on Rings and Modules All rings we will consider will be associative rings with identity, and all modules will be unital modules. They will be right modules unless otherwise specified. Since we want all modules MR to have an endomorphism ring End(MR ) (the zero module as well), the identity of the ring may be equal to zero, in which case the ring trivially reduces to zero. Ring homomorphisms send the identity of the domain to the identity of the codomain, and subrings must have the same identity of the ring. Whenever we will say “ideal,” we will mean a “two-sided ideal,” though we will sometimes say “a two-sided ideal” in order to emphasize it. For any ring R, Mod-R will denote the category of all right R-modules, and R -Mod will be the category of all left R-modules. For any set I, we will denote by |I| the cardinality of I.

2.1 Semisimple Rings and Modules In this chapter, we introduce some basic results that will be used freely in the rest of the book. All these results are elementary and well known, and we suppose that the reader is well acquainted with them. In this first section, we will not give proofs of the statements, which the reader can find in the book of [Anderson and Fuller]. A module M is simple if 0 and M are the only submodules of M and M = 0. A module M is a semisimple module if it is a direct sum of simple submodules. It is possible to see that a module M is semisimple if and only if every submodule of M is a direct summand of M . The class of all semisimple R-modules is closed under submodules, homomorphic images, and arbitrary direct sums. For a module M , the radical of M is the intersection of all maximal submodules of M . We will denote it by rad(M ). If M has no maximal submodules, we set rad(M ) := M . It follows that an element x of a right R-module M belongs to rad(M ) if and only if ϕ(x) = 0 for every simple right R-module S and every homomorphism ϕ : M → S. It is easy to see that rad(M/rad(M )) = 0 for any © Springer Nature Switzerland AG 2019 A. Facchini, Semilocal Categories and Modules with Semilocal Endomorphism Rings, Progress in Mathematics 331, https://doi.org/10.1007/978-3-030-23284-9_2

49

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Chapter 2. Basic Concepts on Rings and Modules

module M . If M, N are right R-modules, then rad(M ⊕ N ) = rad(M ) ⊕ rad(N ).

(2.1)

In particular, rad(eR) = e rad(R) for every idempotent element e of R. For a ring R, one has that rad(RR ) = rad(R R), that is, the radical of R over itself as a right module is the same as over a left module. Thus, this right/left radical is a two-sided ideal of R, called the Jacobson radical of R. It is denoted by J(R). The Jacobson radical J(R) coincides with the annihilator in R of the class of all simple right R-modules. It is possible to prove that J(R) is the set of all x ∈ R such that 1 − xr has a right inverse for every r ∈ R, or equivalently, the set of all x ∈ R such that 1 − rx has a left inverse for every r ∈ R, or the set of all x ∈ R such that 1 − rxr has a two-sided inverse for every r, r ∈ R. If e ∈ R is idempotent, then J(eRe) = eJ(R)e. If I is an ideal of a ring R and I ⊆ J(R), then J(R/I) = J(R)/I. Lemma 2.1. If MR is a right module over a ring R, then M J(R) ⊆ rad(MR ). A ring R is local if it has a unique maximal right ideal, that is, a unique maximal element in the set of all proper right ideals partially ordered by set inclusion. A ring R is local if and only if it has a unique maximal left ideal, if and only if the two-sided ideal J(R) is both a maximal right ideal and a maximal left ideal, if and only if R/J(R) is a division ring. Theorem 2.2 (Nakayama’s lemma). If M is a finitely generated module and N is a submodule of M such that N + M J(R) = M , then N = M . Every module with a local endomorphism ring is necessarily indecomposable [Anderson and Fuller, p. 144].

2.2 Free Rings and Free Algebras For any module MR , we will denote by MRn the direct sum of n copies of MR , and (I) for every set I, we will denote by MR the direct sum of |I| copies of MR and by I MR the direct product of |I| copies of MR . For instance, if R is a ring and I is (I) a set, RR is a free right R-module with free set of generators I. Similarly, R R(I) is a free left R-module with free set of generators I. The cardinality of a free set of generators of a free module will be called the rank of the free module. If I is (I) n finite of cardinality n, we will use the notation RR and R Rn for RR and R R(I) , respectively. Let k be a commutative ring. Recall that a k-algebra is a ring R with a ring homomorphism fR : k → R of k to R whose image fR (k) is contained in the center Z(R) of R. A k-algebra homomorphism ϕ : R → S, where R and S are k-algebras via the ring homomorphisms fR : k → R and fS : k → S with fR (k) ⊆ Z(R) and fS (k) ⊆ Z(S) respectively, is a ring homomorphism such that fS = ϕfR . If k is a

2.2. Free Rings and Free Algebras

51

commutative ring and X is a set, we will denote by k[X] the ring of polynomials in the set X of commuting indeterminates with coefficients in k. It is a commutative overring of k, so that it is a k-algebra via the natural embedding fk[X] : k → k[X]. The ring k[X] contains the set X and has the following universal property: for every commutative k-algebra S and every mapping ψ : X → S, there is a unique k-algebra homomorphism k[X] → S whose restriction to X is equal to ψ. Equivalently, this can be stated as follows. Let S X denote the set of all mappings of the set X to S, and Algk (k[X], S) the set of all k-algebra homomorphisms from k[X] to S. Then the mapping Algk (k[X], S) → S X that associates to each ϕ ∈ Algk (k[X], S) the restriction ϕ|X of ϕ to X is a bijection. If X = {x1 , . . . , xn } is a finite set, we will write k[x1 , . . . , xn ] instead of k[{x1 , . . . , xn }]. The ring k[X] is a graded ring over the monoid N0 of all nonnegative integers. The component of degree 0 is a subring of k[X] isomorphic to k. For instance, if X = {x, y} has two elements, then the component of degree 0 consists of all polynomials of degree ≤ 0, the component of degree 1 is a free k-module of rank two freely generated by the set {x, y}, the component of degree 2 is a free k-module of rank three freely generated by the set {x2 , xy, y 2 }, and so on, the component of degree n is a free k-module of rank n + 1 freely generated by the set {xn , xn−1 y, . . . , y n }. If k is a commutative ring and X is a set, we will denote by kX the free k-algebra in the set X of noncommuting indeterminates. It is a free k-module with basis consisting of all words xi1 xi2 . . . xit of length t ≥ 0, where xi1 , xi2 , . . . , xit range over the set X. Multiplication is defined on the words by juxtaposition: (xi1 xi2 . . . xit )(xj1 xj2 . . . xjs ) = xi1 xi2 . . . xit xj1 xj2 . . . xjs for every xi1 , xi2 , . . . , xit , xj1 , xj2 , . . . , xjs in X, and then extended to kX by k-bilinearity. It is clear that kX is a k-algebra via the natural embedding fkX : k → kX, i.e., kX can be viewed as an overring of k, and has the following universal property: for every k-algebra S and every mapping ψ : X → S there is a unique k-algebra homomorphism kX → S whose restriction to X is equal to ψ. Equivalently, the mapping Algk (kX, S) → S X that associates to each ϕ ∈ Algk (kX, S) its restriction to X is a bijection. If X = {x1 , . . . , xn } is a finite set, we will write kx1 , . . . , xn  instead of k{x1 , . . . , xn }. The ring kX is also a graded ring over the monoid N0 . For instance, if X = {x, y} has two elements, then the component of degree 0 is a subring of kx, y isomorphic to k, the component of degree 1 is a free k-module of rank two freely generated by the set {x, y}, the component of degree 2 is a free k-module of rank four freely generated by the set {x2 , xy, yx, y 2 }, and so on, the component of degree n being a free k-module of rank 2n . Notice that the elements of X do not commute, but that they commute with the elements of k, so that k = Z(kX) when |X| ≥ 2. Let us consider some variations of the rings of commutative or noncommutative polynomials with coefficients in a commutative ring k considered above. Let R be a fixed ring. An R-ring is a ring S with a ring homomorphism fS : R → S.

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A homomorphism of R-rings is a ring homomorphism ω : S → S  such that ωfS = fS  . If R is a ring and X is a set, we will denote by R{X} the free Rring over X. Here, again, the elements of X are noncommuting indeterminates. The ring R{X} is the R-ring containing X that is the solution of the following universal problem: for every R-ring S and every mapping ψ : X → S, there is a unique R-ring homomorphism R{X} → S whose restriction to X is equal to ψ. If RingR (R{X}, S) denotes the set of all R-ring homomorphisms from R{X} to S, we have a one-to-one correspondence between the sets RingR (R{X}, S) and S X . As a right R-module, R{X} is the free right R-module having as a free set of generators the set whose elements are 1 and all the formal expressions of the type r1 xi1 r2 xi2 . . . rn xin , where n ≥ 1, r1 , r2 , . . . , rn ∈ R, and xi1 , xi2 , . . . , xin ∈ X. The multiplication on R{X} is defined by   (r1 xi1 . . . rn xin r)(r1 xj1 . . . rm xj m r ) = r1 xi1 . . . rn xin (rr1 )xj1 . . . rm xj m r

and then extended by Z-bilinearity. The ring R{X} is a graded ring over the monoid N0 of all nonnegative integers, and its component of degree 0 is isomorphic to R, so that the structure of an R-ring on R{X} is given by an isomorphism between R and the component of degree 0 of R{X}. As a second variation, consider the case of a fixed commutative ring k and a fixed k-algebra R, so that we have a ring homomorphism fR : k → R with fR (k) ⊆ Z(R). An R-ringk is a k-algebra S with a k-algebra homomorphism fS : R → S [Bergman 1974b]. An R-ringk homomorphism ϕ : R → S is a kalgebra homomorphism such that fS = ϕfR . If R is a k-algebra and X is a set, let R{X}k be the free R-ringk over X. It is the R-ringk containing X as a subset that is the solution of the following universal problem: for every R-ringk S and every mapping ψ : X → S, there is a unique R-ringk homomorphism R{X}k → S whose restriction to X is equal to ψ. It is easily seen that R{X}k is the factor ring of R{X} modulo the two-sided ideal of R{X} generated by all the elements of the form λx − xλ with x ∈ X and λ ∈ k. As a third variation, we mention the possibility of adjoining central indeterminates. If R is a ring and X is a set, let R[X] be the R-ring containing X as a subset that is the solution of the following universal problem: for every R-ring S with ring homomorphism fS : R → S, and every mapping ψ : X → Z(S), there is a unique R-ring homomorphism R[X] → S whose restriction to X is equal to ψ. It is easily verified that R[X] ∼ = R ⊗Z Z[X], where Z[X] is the ring of polynomials with integral coefficients.

2.3 Ranks of Free Modules Proposition 2.3. Let R be a ring and let MR be a right R-module with an infinite set I of generators. If no proper subset of I generates MR , then every set of generators of MR has cardinality ≥ |I|.

2.3. Ranks of Free Modules

53

Proof. Let J be another set of generators of the module MR . For every a ∈ J, there is a finite subset I a of I such that a can be written as a linear combination I of elements of . Thus a subset of I that generates MR . By hypothesis, a a∈J Ia is

is finite, then If I = I. J a∈J a

a∈J Ia = I is finite, a contradiction. Therefore  J must be infinite, so that |I| = | a∈J Ia | ≤ ℵ0 |J| = |J|. Corollary 2.4. The rank of a free module MR of infinite rank is uniquely determined by MR . In the case of free modules of finite rank, the situation is different. For a fixed ring R, define an equivalence relation ∼ on N0 by setting, for all n, m ∈ N0 , n ∼ m m n ∼ n ∼ m if RR = RR . Using the duality functor HomR (−, R) one easily sees that RR = RR m n ∼ if and only if R R = R R , so that the equivalence relation ∼ depends only on R and not on the side considered. It is easily proved that ∼ is a congruence on the additive monoid N0 . Therefore either ∼ is the equality = or ∼ is a congruence ∼k,n , where k ≥ 0 and n ≥ 1 are integers (page 6). We will call ∼ the Leavitt congruence associated to R [Leavitt 1962]. If ∼ is the equality = on N0 , that is, m n ∼ RR implies n = m for all n, m ∈ N0 , that the ring R is said to have the = RR invariant basis property, or the invariant basis number (IBN). For example, in linear algebra one proves that right vector spaces have bases of unique cardinality, so that division rings have IBN. If f : R → S is a ring homomorphism, then S can be viewed as an (R, S)-bimodule, and for every (finitely generated) free right R-module AR , A ⊗R S turns out to be a (finitely generated) free right S-module of the same rank. The proof of part (a) in the statement of the next proposition follows immediately. Proposition 2.5. (a) If R and S are rings such that there exists a ring homomorphism R → S and S has IBN, then R has IBN as well. (b) Every local ring has IBN. (c) Every nonzero commutative ring has IBN. For (b), apply (a) to the canonical projection R → R/J(R), where J(R) is the Jacobson radical of R. Similarly for (c) with any maximal ideal of R instead of J(R). Example 2.6. We will now give an example of a (nonzero) ring R with RR ∼ = RR ⊕ RR , so that in particular the ring R does not have IBN. Let F be a field. Let VF be a vector space over F of infinite dimension, so that VF ⊕ VF ∼ = VF . Let R := End(VF ) be the endomorphism ring of VF , so that R VF is an R-F -bimodule. Thus there is a covariant functor Hom(R VF , −) : Mod-F → Mod-R. Applying this ∼ VF , we get a right Rfunctor to the right F -module isomorphism VF ⊕ VF = ∼ module isomorphism Hom(R VF , VF ) ⊕ Hom(R VF , VF ) = Hom(R VF , VF ), that is, an isomorphism RR ⊕ RR ∼ = RR . This is an isomorphism between two free right R-modules of rank 2 and 1 respectively. Therefore R is not an IBN ring. Notice

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∼ Rn for every n ≥ 1. Thus Rn = ∼ Rm for that for this ring R, we have that RR = R R R every n, m ≥ 1. Before stating the next example, let us recall what the two-sided ideal generated by a subset is. Let R be a ring and X a subset of R. Any intersection of two-sided ideals of R is a two-sided ideal of R. In particular, the intersection of all two-sided ideals of R containing X (there are always such ideals, for instance R itself) is a two-sided ideal containing X. It is the smallest two-sided ideal containing X, called the two-sided ideal of R generated by X, and denoted by (X). If X = ∅, n ri xi ri | n ≥ 1, ri , ri ∈ R xi ∈ X }. For instance, it is easily seen that (X) = { i=1 if X is the set with only one element a ∈ R, then theprincipal (two-sided) ideal of ideal generated by the set X = {a}, is the R generated by a, that is, the two-sided n set of all elements of the form i=1 ri ari , where n ≥ 1 and ri , ri ∈ R. This set can be properly larger than the set of all elements of the form rar with r, r ∈ R. For instance, if  Ris the ring of all 2 × 2 matrices with entries in a field F and a is the matrix 01 00 , then R is a simple ring, so that the principal ideal generated by a is not a proper ideal, but it is easily seen that the set of all elements of R of the form rar with r, r ∈ R is a proper subset of R. Example 2.7 (Leavitt algebra). Fix m, n ≥ 1. We want to give another example of n ∼ m a ring R such that RR = RR . Let F be a field, and let Z := { xij , yji | i = 1, . . . , n, j = 1, . . . , m } be a set of 2mn noncommuting indeterminates. Let P := F { xij , yji | i = 1, . . . , n, j = 1, . . . , m } be the free F -algebra over Z. Consider the matrices X := (xij )i,j ∈ Mn×m (P ) and Y := (yji )j,i ∈ Mm×n (P ) with entries in P . Let I be the two-sided ideal of P generated by the n2 + m2 entries of the two matrices XY − 1n and Y X − 1m . Let us prove that I is a proper ideal. Consider a vector space VF over F of infinite dimension, so that, as we have seen in Example 2.6, if T is its endomorphism ring End(VF ), then TTn ∼ = TTm . Hence there exist two matrices A := (tij )i,j ∈ Mn×m (T )  and B := (tji )j,i ∈ Mm×n (T ) such that AB = 1n and BA = 1m . By the universal property of free F -algebras, there is a unique F -algebra homomorphism ϕ : P →  End(VF ) such that xij → tij and yji → tji for every i and j. Then I is contained in / I. This shows that I is a proper ideal the kernel of ϕ and ϕ(1) = 1 = 0, so that 1 ∈ of P . It follows that R := P/I is a F -algebra with identity 1 = 0. If f ∈ P , set f := f + I ∈ P/I. Then X := (xij )i,j ∈ Mn×m (R) and Y := (yji )j,i ∈ Mm×n (R). Since XY = 1n , Y X = 1m , the mappings m n n m X : RR → RR , Y : RR → RR n ∼ m are inverses of each other, so that RR = RR . The F -algebra R is called the Leavitt algebra of type (n, m). The ring R = Rn,m is the F -algebra with a universal

2.4. Projective Modules and Radicals

55

m m n n → RR isomorphism i : RR in the following sense. Let i : RR → RR be the right R-module isomorphism given by left multiplication by the n × m matrix (xij + I). Then, for every F -algebra S and every isomorphism j : SSm → SSn , there is a unique F -algebra homomorphism f : R → S such that j = i ⊗R S [Bergman 1974b, p. 35].

2.4 Projective Modules and Radicals We have already defined the commutative monoid V (C). It is the algebraic object that describes direct-sum decompositions of objects in an additive category C. Our main example of a commutative monoid will be the monoid V (C) when C is the full subcategory of Mod-R whose objects are all finitely generated projective right modules over a ring R. We will now recall the definition and the main properties of projective modules, which we assume that the reader knows. For a detailed presentation with all the proofs, see [Anderson and Fuller]. Recall that the epimorphisms of modules over a given ring R, that is, the epimorphisms in Mod-R, are exactly the surjective homomorphisms. Let R be a ring. A right R-module PR is projective if for every epimorphism f : MR → NR and every homomorphism g : PR → NR , there exists a morphism h : PR → MR with f ◦ h = g. For instance, every free module is projective, every direct summand of a projective module is projective, and every direct sum of projective modules is projective. Proposition 2.8. The following conditions are equivalent for a right R-module PR : (a) The module PR is projective. (b) If MR , NR are right R-modules and f : MR → NR is an epimorphism, then the induced mapping Hom(PR , f ) : Hom(PR , MR ) → Hom(PR , NR ) is an epimorphism. (c) The functor Hom(PR , −) is exact, that is, for every exact sequence f

g

MR − → MR − → MR of right R-modules, the sequence Hom(PR ,f )

Hom(PR ,g)

Hom(PR , MR ) −−−−−−−→ Hom(PR , MR ) −−−−−−−→ Hom(PR , MR ) is exact. (d) If MR is a right R-module, then every epimorphism f : MR → PR splits, that is, ker f is a direct summand of MR . (e) The module PR is isomorphic to a direct summand of a free module. If PR is a right R-module and n ≥ 0 is an integer, then PR is a projective module with a set of generators of cardinality ≤ n if and only if it is isomorphic n to a direct summand of RR . For an infinite cardinal ℵ instead of an integer n ≥ 0,

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56

we have the following proposition, whose nontrivial part (a) ⇒ (c) is known as Eilenberg’s trick. Proposition 2.9. The following conditions are equivalent for a module PR and an infinite cardinal ℵ: (a) PR is a projective module with a set of generators of cardinality ≤ ℵ. (ℵ)

(b) PR is isomorphic to a direct summand of the free module RR . (ℵ) ∼ (ℵ) (c) PR ⊕ R = R . R

R

Proof. (a) ⇒ (b) and (c) ⇒ (a) are easy. (ℵ) (b) ⇒ (c) Since PR ⊕ QR ∼ = RR , one has that (ℵ) (ℵ) ∼ ∼ (PR ⊕ QR ) ⊕ (PR ⊕ QR ) ⊕ (PR ⊕ QR ) ⊕ · · · RR = (RR )(ℵ0 ) = (ℵ) ∼ PR ⊕ (QR ⊕ PR ) ⊕ (QR ⊕ PR ) ⊕ (QR ⊕ PR ) ⊕ · · · ∼ = PR ⊕ R . = R



The notion dual to that of projective module is the notion of injective module. A module ER is injective if it satisfies the equivalent conditions of the following Proposition 2.10. With the notation Ab we indicate the category of all abelian groups. It is isomorphic to the category Mod-Z. Proposition 2.10. The following conditions are equivalent for an R-module ER : (a) The functor Hom(−, ER ) : Mod-R → Ab is exact, that is, for every exact sequence MR → MR → MR of right R-modules, the sequence of abelian groups Hom(MR , ER ) → Hom(MR , ER ) → Hom(MR , ER ) is exact. (b) For every monomorphism MR → MR of right R-modules, Hom(MR , ER ) → Hom(MR , ER ) is an epimorphism of abelian groups. (c) For every submodule MR of a right R-module MR , every morphism MR → ER can be extended to a morphism MR → ER . (d) For every monomorphism f : MR → MR and every homomorphism g : MR → ER , there exists a morphism h : MR → ER with h ◦ f = g. If S is a ring and n ≥ 1 is an integer, Mn (S) will denote the ring of n × n matrices with entries in S. Theorem 2.11. The following conditions are equivalent for a ring R: (a) (b) (c) (d) (e)

The right R-module RR is a semisimple module. The right R-module RR is the direct sum of a finite set of simple right ideals. Every right R-module is semisimple. Every right R-module is projective. Every right R-module is injective.

2.4. Projective Modules and Radicals

57

(f) The ring R is right Artinian and J(R) = 0. (g) (Artin–Wedderburn Theorem) There exist an integer t ≥ 1, division rings ∼ t Mn (Di ). D1 , . . . , Dt , and positive integers n1 , . . . , nt such that R = i i=1 A ring R satisfying any of these equivalent conditions is said to be a semisimple Artinian ring. We prefer not to call these rings semisimple rings, which would be shorter, because we want simple rings to be semisimple, and simple non-Artinian rings do not satisfy the equivalent conditions of Theorem 2.11. A submodule N of a module MR is superfluous (or small, or inessential ) in MR if for every submodule L of MR , N +L = MR implies L = MR . To denote that N is superfluous in MR we will write N ≤s MR . Notice that the zero submodule is a superfluous submodule of any module MR , also when MR = 0. An epimorphism f : M → M  is superfluous if its kernel is a superfluous submodule of M . We leave to the reader the proof of the following three easy results: Lemma 2.12. If f : M → M  is an R-module morphism and N ≤s M , then f (N ) ≤s M  . Notice that by Lemma 2.12, if N is any submodule of MR , all modules of the form M  /N with M  ≤s M and M  ⊇ N are superfluous in M/N , but the converse is not true. Lemma 2.13. Let N be a superfluous submodule of a right module MR . Then the superfluous submodules of M/N are exactly the modules of the form M  /N with M  ≤s M and M  ⊇ N . Proposition 2.14. An epimorphism g : M → N is superfluous if and only if for every module L and every homomorphism h : L → M , if gh is onto, then h is onto. Lemma 2.15. For any module MR , the submodule rad(MR ) is the sum of all superfluous submodules of MR . Proof. In order to prove that rad(MR ) contains the sum of all superfluous submodules, fix a superfluous submodule N of MR and a maximal submodule M  of MR . We must prove that N ⊆ M  . Assume the contrary. Then N + M  = MR because M  is maximal, so that M  = MR because N is superfluous. Contradiction. For the inverse inclusion, it suffices to show that if x ∈ rad(MR ), then xR is superfluous in MR . Equivalently, we will prove that if x ∈ MR and xR is not / rad(MR ). Let x be an element of MR with xR superfluous in MR , then x ∈ nonsuperfluous in MR , so that there exists a proper submodule L of MR with / S }. The / L. Set F := { S | S ≤ MR , S ⊇ L, x ∈ xR + L = MR . Then x ∈ family F is nonempty, because it contains L. Zorn’s lemma shows that there is a maximal element S in F . Then S is a maximal submodule of MR , because if S  is / F , so that x ∈ S  , whence a submodule of MR that contains S properly, then S  ∈  S ⊇ xR + L = MR . Since x does not belong to the maximal submodule S of MR , it follows that x ∈ / rad(MR ), as desired. 

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Nakayama’s lemma (Theorem 2.2) says that if M is a finitely generated module, then M J(R) is superfluous in M . By Lemma 2.15, M J(R) ⊆ rad(MR ) for every finitely generated R-module MR . Notice that this inclusion can be proper. For instance, let R be the ring Z of integers and M = Z/4Z. Then M is a finitely generated module, rad(M ) has two elements, J(Z) = 0, and M J(Z) has one element. The next result describes the radical of projective modules. Proposition 2.16. If PR is a projective right module over a ring R, then PR J(R) = rad(PR ). Proof. The projective module PR is isomorphic to a direct summand of a free module. Hence, we can suppose that PR is a direct summand of a free module (X) (X) (X) RR : PR ⊕ PR = RR . By (2.1) in Section 2.1, rad(PR ) ⊕ rad(PR ) = rad(RR ) = (X) (X) (X)   = RR J(R) = (PR ⊕ PR )J(R) = P J(R) ⊕ P J(R). Inter= J(R) rad(RR ) (X) (X) secting this submodule of RR with the submodule PR of RR , which is a direct (X) summand of RR , we get that (rad(PR )⊕rad(PR ))∩PR = (P J(R)⊕P  J(R))∩PR , that is, rad(PR ) = P J(R).  For a proof of the following important theorem, due to Irving Kaplansky, see [Facchini 1998, Corollary 2.48]. Theorem 2.17 ([Kaplansky]). Every projective module is a direct sum of countably generated projective modules. Now we will present a very powerful result due to Pˇr´ıhoda (Theorem 2.20). If PR , QR are projective modules and πP : PR → PR /PR J(R),

πQ : QR → QR /QR J(R)

are the canonical projections, then for every homomorphism f : PR /PR J(R) → QR /QR J(R) there exists a homomorphism f : PR → QR with πQ f = f πP . In this notation, we will say that f lifts f . Lemma 2.18. Let PR and QR be two countably generated projective modules and f : PR → QR a homomorphism. Assume that the induced morphism f : PR /PR J(R) → QR /QR J(R) is an isomorphism. Let X be a finite subset of PR . Then there exists a homo−1 morphism g : QR → PR that lifts f and such that gf (x) = x for every x ∈ X. (ℵ ) (ℵ ) Proof. By the Eilenberg trick (Proposition 2.9), PR ⊕ RR 0 ∼ = RR 0 and QR ⊕ (ℵ ) (ℵ ) (ℵ ) RR 0 ∼ = RR 0 . Fix a free set of generators Y = {e1 , e2 , . . . } of PR ⊕ RR 0 . Let −1 g0 : QR → PR be a homomorphism that lifts f : QR /QR J(R) → PR /PR J(R).

2.4. Projective Modules and Radicals

59 (ℵ )

Consider the endomorphism h = (g0 f )⊕1R(ℵ0 ) of PR ⊕RR 0 . Notice that e−h(e) ∈ (ℵ ) (ℵ ) (PR ⊕ RR 0 )J(R) for every e ∈ PR ⊕ RR 0 . Let A = (aij )ij be the matrix of the (ℵ ) endomorphism h of the free module PR ⊕ RR 0 with respect to the free set of generators Y , so that A is a column-finite matrix with entries in R, and the columnfinite matrix 1 − A = (δij − aij )ij has all its entries in J(R). Let n be an index such every element of X can be written as a linear combination of {e1 , . . . , en }. Let m ≥ n be an index such that every element of {h(e1 ), . . . , h(en )} can be written as a linear combination of {e1 , . . . , em }. Let B be the square m × m submatrix of A corresponding to the first m rows and the first m columns of A. Then B ∈ Mm (R) and 1m −B, where 1m is the m×m identity matrix, belongs to the Jacobson radical J(Mm (R)) = Mm (J(R)) of the ring Mm (R). Thus B is an invertible element of the ring Mm (R) and its inverse B −1 ∈ Mm (R) is also such that B −1 ≡ 1m modulo Mm (J(R)). Let h be the endomorphism of the free countably generated module (ℵ ) PR ⊕ RR 0 whose matrix with respect to the free set Y of generators is the matrix obtained by replacing the top left m × m corner in the identity ℵ0 × ℵ0 matrix (ℵ ) with B −1 . Clearly, h is an automorphism of PR ⊕ RR 0 . (ℵ ) (ℵ ) The isomorphism PR ⊕ R 0 ∼ = R 0 yields a monomorphism εP : PR → (ℵ )

R

(ℵ )

R

RR 0 and an epimorphism πP : RR 0 → PR with πP εP the identity of PR . Set g := πP h εP g0 : QR → PR . Modulo the Jacobson radical, h is congruent to the (ℵ ) identity of the free module RR 0 , so that g is congruent to g0 , that is, g lifts −1 f . Finally, for every i = 1, . . . , m, we have that gf (ei ) = πP h εP g0 f (ei ) = πP h h(ei ⊕ 0) = πP (ei ⊕ 0) = ei . Thus gf (x) = x for every x ∈ X.  The next lemma will be generalized in Theorem 2.20 from countably generated projective modules to arbitrary projective modules. Lemma 2.19. Let PR and QR be countably generated projective modules and let f : PR /PR J(R) → QR /QR J(R) be an isomorphism. Then there exists an isomorphism f : PR → QR that lifts f . Proof. Let {p1 , p2 , . . . }, {q1 , q2 , . . . } be sets of generators for PR , QR , respectively. Let g : QR /QR J(R) → PR /PR J(R) be an inverse of f . We will define, for every n ≥ 0, a quadruple (Xn , Yn , fn , gn ), where for every n ≥ 0, Xn is a finite subset of PR containing p1 , . . . , pn , Yn is a finite subset of QR containing q1 , . . . , qn , fn : PR → QR and gn : QR → PR are homomorphisms that lift f and g respectively, gn fn (x) = x for every x ∈ Xn , and fn gn−1 (y) = y for every y ∈ Yn−1 and n ≥ 1. For n = 0, let (X0 , Y0 , f0 , g0 ) be a quadruple with X0 and Y0 the empty set and f0 : PR → QR and g0 : QR → PR arbitrary homomorphisms that lift f and g respectively. Suppose n ≥ 1 and that the quadruple (Xn−1 , Yn−1 , fn−1 , gn−1 ) with the required properties has been defined. Set Xn := gn−1 (Yn−1 ) ∪ {p1 , . . . , pn }. Let fn : PR → QR be a homomorphism that lifts f and such that fn gn−1 (y) = y

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for every y ∈ Yn−1 (Lemma 2.18). Set Yn := fn (Xn ) ∪ {q1 , . . . , qn }. Applying Lemma 2.18 again, let gn : QR → PR be a homomorphism that lifts g and with gn fn (x) = x for every x ∈ Xn . This completes the construction by induction. Notice that Xn−1 = gn−1 fn−1 (Xn−1 ) ⊆ gn−1 (Yn−1 ) ⊆ Xn . Moreover, fn−1 (x) = fn (x) for every x ∈ Xn−1 , because x ∈ Xn−1 implies fn−1 (x) ∈ Yn−1 , so that fn gn−1 (fn−1 (x)) = fn−1 (x), that is, fn (x) = fn−1 (x). Thus we can take the inductive limit, getting a well-defined homomorphism f : PR → QR such that f (x) = fn (x) for every x ∈ Xn . Since fn lifts f for every n, it is clear that f also lifts f . It remains to show that f is an isomorphism. If p is in the kernel of f , then there exists an n ≥ 1 with p in the submodule of PR generated by Xn . Then p = gn fn (p) = gn f (p) = 0. This shows that f is a monomorphism. It is also an epimorphism, because, for every n ≥ 1, qn = fn+1 gn (qn ) ∈ fn+1 (Xn+1 ) = f (Xn+1 ) ⊆ f (PR ). This concludes the proof.  Theorem 2.20 ([Pˇr´ıhoda 2007, Theorem 2.3]). Let PR , QR be projective modules and f : PR /PR J(R) → QR /QR J(R) an isomorphism. Then there exists an isomorphism f : PR → QR that lifts f . Proof. By Theorem 2.17, there are direct-sum decompositions PR = ⊕i∈I Ai and QR = ⊕j∈J Bj in which all the modules Ai and Bj are countably generated. It follows that PR /PR J(R) ∼ = ⊕i∈I Ai /Ai J(R) and QR /QR J(R) ∼ = ⊕j∈J Bj /Bj J(R). Thus the module QR /QR J(R) has two direct-sum decompositions QR /QR J(R) = ⊕i∈I Ai = ⊕j∈J Bj , in which Ai /Ai J(R) ∼ = Ai via an isomorphism induced by the isomorphism f : PR /PR J(R) → QR /QR J(R) and Bj /Bj J(R) ∼ = Bj via the isomorphism induced by the embedding Bj → QR . We claim that if i0 ∈ I, then there exist countable subsets I  ⊆ I and J  ⊆ J such that i0 ∈ I  and ⊕i∈I  Ai = ⊕j∈J  Bj . In order to prove the claim, define by induction, for every integer n ≥ 0, countable sets In and Jn as follows. Set I0 := {i0 } and J0 := ∅. Suppose In and Jn have been defined. Then (⊕i∈In Ai ) + (⊕j∈Jn Bj ) is a countably generated R-module, so that there exists a countable   subset Jn+1 of J such that (⊕i∈In Ai )+(⊕j∈Jn Bj ) ⊆ ⊕j∈Jn+1 Bj . Since (⊕i∈In Ai )+    (⊕j∈Jn+1 Bj ) is also countably generated, there exists a countable subset In+1 of

      I with (⊕i∈In Ai ) + (⊕j∈Jn+1 Bj ) ⊆ ⊕i∈In+1 Ai . Clearly, the sets I = n≥0 In and

J  = n≥0 Jn satisfy the claim. Now define a chain of subsets K0 ⊆ K1 ⊆ · · · ⊆ Kλ ⊆ · · · of I and a chain of subsets L0 ⊆ L1 ⊆ · · · ⊆ Lλ ⊆ · · · of J for each ordinal λ by transfinite

induction as follows. Set K := ∅ and L := ∅. If λ is a limit ordinal, set K = 0 0 λ μ 1, the abelian group Zn with the lexicographic order is not order-isomorphic to the Grothendieck group K0 (R) of any semilocal ring R.

3.5 Modules with Semilocal Endomorphism Rings We have seen at the end of the previous section that being semilocal is a finiteness condition on a ring. Now let R be an arbitrary ring, MR a right R-module, and suppose that the endomorphism ring End(MR ) is semilocal. In particular, MR = 0. Having a semilocal endomorphism ring is a finiteness condition on the module MR . This follows immediately from the equivalence between the category add(MR ) and proj- End(MR ) (Theorem 2.35). Thus we immediately get that for a module MR with End(MR ) semilocal: (a) The monoid V (add(MR )) is a finitely generated cancellative Krull monoid. (b) The module MR has only finitely many direct summands up to isomorphism. (c) The number of direct-sum decompositions of MR is finite up to isomorphism. In particular, the number of direct-sum decompositions of MR as a direct sum of indecomposable modules is finite up to isomorphism. (d) The module MR is a direct sum of finitely many indecomposable modules. (e) The module MR is not a direct sum of infinitely many nonzero modules. ∼ MR ⊕ NR for some right (f) The module MR is directly finite, that is, if MR = R-module NR , then NR = 0. From (a), we get that modules with a semilocal endomorphism ring have a very regular geometric behavior as far as their direct summands are concerned; cf. Section 1.3.

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Proposition 3.35 ([Facchini and Herbera 2006, Proposition 2.7]). Let R → S be a ring morphism, so that every S-module can be viewed as an R-module. If MS is an S-module and MR has a semilocal endomorphism ring, then MS has a semilocal endomorphism ring. Proof. Clearly, every S-module endomorphism of MS is an R-module endomorphism, so that End(MS ) can be viewed as a subring of End(MR ). The embedding End(MS ) → End(MR ) is clearly a local morphism, because an endomorphism of MS is an automorphism of MR if and only if it is a bijection, if and only if it is an automorphism of MS . The conclusion now follows from Corollary 3.28.  In the next proposition, we have collected three properties of modules with semilocal endomorphism rings. Their proofs can be found in [Facchini 1998, Corollary 4.6, Propositions 4.8 and 4.9]. Proposition 3.36. Let MR be a module and assume End(MR ) semilocal. ∼ (a) (Cancellation property) If AR and BR are arbitrary modules and MR ⊕AR = ∼ MR ⊕ BR , then AR = BR . n for some (b) (nth root property) If AR is an arbitrary module and MRn ∼ = AR ∼ integer n ≥ 1, then MR = AR . (c) If codim(End(MR )) = d, then MR has at most 2d direct summands up to isomorphism. If MR is a right module over a ring R, we can define δ(MR ) := codim(End(MR )), the dual Goldie dimension of the endomorphism ring of MR [Facchini and Herbera 2004]. (For MR = 0, we set δ(MR ) := 0.) Thus δ(MR ) is either a nonnegative integer or ∞. It is a nonnegative integer if and only if the endomorphism ring of MR is semilocal. Notice that a module MR has a local endomorphism ring if and only if δ(MR ) = 1. Proposition 3.37 ([Sarath and Varadarajan, Theorem 2.5]). Let PR be a finitely generated projective right R-module and E := End(PR ). Then codim(PR ) = codim(EE ) = δ(PR ). Proof. Since PR is a finitely generated projective right R-module, we know that PR J(R) = rad(PR ) (Proposition 2.16), and that this is a superfluous submodule of PR by Nakayama’s lemma, Theorem 2.2. Moreover, E/J(E) ∼ = End(PR /PR J(R)) by Corollary 2.25. Assume codim(PR ) finite, codim(PR ) = n say. We can apply Theorem 3.10, getting that PR /PR J(R) is a semisimple module of composition length n, so that ∼ E/J(E) is a semisimple Artinian ring its endomorphism ring End(PR /PR J(R)) = of composition length n. This proves that E is a semilocal ring of dual Goldie dimension codim(EE ) = n (Proposition 3.12).

3.6. Notes on Chapter 3

109

It remains to prove that if codim(PR ) is infinite, then codim(EE ) is also infinite. Assume on the contrary codim(PR ) infinite and n := codim(EE ) finite. By Proposition 3.12, E is semilocal and E/J(E) is semisimple Artinian of composition length n. Set R := R/J(R) and P  := P/P J(R). Then P  is a finitely generated projective R -module with an endomorphism ring that is semisimple Artinian of composition length n. Thus there are pairwise orthogonal idempotent endomorphisms e1 , . . . , en of P  such that P  = e1 P  ⊕ · · · ⊕ en P  , and each ei P  has an endomorphism ring that is a division ring. Thus ei P  satisfies condition (c) of Lemma 3.9, so that ei P  is the projective cover of a simple R -module. Now J(R ) = 0 implies that every finitely generated projective right R -module has radical zero (first paragraph of this proof), hence has no nonzero superfluous submodule. Thus each ei P  is a simple R -module. Hence P  is a semisimple module of finite length n as both an R -module and an R-module. Theorem 3.10 allows us to conclude that PR has finite dual Goldie dimension codim(PR ) = n, a  contradiction. Proposition 3.38. If MR , NR are modules over an arbitrary ring R, then δ(MR ⊕ NR ) = δ(MR ) + δ(NR ). In particular, MR ⊕ NR has a semilocal endomorphism ring if and only if both MR and NR have a semilocal endomorphism ring. Proof. Let E be the endomorphism ring of MR ⊕NR and let e ∈ E be the idempotent endomorphism with image MR and kernel NR . Apply the previous proposition to the finitely generated projective right E-modules eE and (1 − e)E, whose endomorphism rings are isomorphic to eEe and (1 − e)E(1 − e) respectively, getting that codim(eEE ) = codim(eEe) = codim(End(MR )) = δ(MR ) and, similarly, codim((1 − e)EE ) = δ(NR ). By Proposition 3.8(e), it follows that δ(MR ⊕ NR ) = codim(EE ) = codim(eEE ) + codim((1 − e)EE ) = δ(MR ) + δ(NR ).  We have thus proved that the class of the modules with semilocal endomorphism rings is closed under direct summands and finite direct sums. If R is a semilocal ring, δ induces a monoid valuation V (R) → N0 .

3.6 Notes on Chapter 3 By Proposition 3.12, the dual Goldie dimensions codim(RR ) and codim(R R) are always equal. More precisely, a ring R is semilocal if and only if the right module RR has finite dual Goldie dimension codim(RR ), if and only if the left module R R has finite dual Goldie dimension codim(R R). Now codim(RR ) = codim(R R) for any ring R, and this common dual Goldie dimension can be either any positive integer or infinite. For the Goldie dimensions dim(RR ) and dim(R R), the situation is completely different. For instance, for a (noncommutative) integral domain R, either dim(RR ) = 1 or dim(RR ) = ∞. The reason for this is that if dim(RR ) > 1, then there exist nonzero elements a, b ∈ R such that aR ∩ bR = 0. Then R contains the infinite direct sum n≥0 an bR, so that dim(RR ) = ∞. An integral

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domain R with dim(RR ) = 1 is called a right Ore domain. Right Ore domains are exactly the rings that have a classical right ring of quotients that is a division ring. Thus a right Ore domain R that is not left Ore is an example of a ring for which dim(RR ) = dim(R R). Such an example of a right Ore domain that is not left Ore is the skew polynomial ring k[x; σ], where k is a field and σ : k → k is an endomorphism of k that is not an automorphism. See Section 11.6 and [Berrick and Keating, Section 3.2]. Couniform projective modules have been studied by [Mares] and [Ware]. Some further elementary properties of semilocal rings can be found in [Anderson and Fuller, pp. 170–172]. The last example in Section 3.4 is taken from [Facchini and Herbera 2000b, p. 188]. We have already remarked at the end of Section 3.4 that every semilocal ring has only finitely many pairwise nonisomorphic indecomposable finitely generated projective right modules. It was proved by [Pˇr´ıhoda 2007, Corollary 2.6] that every semilocal ring has at most countably many pairwise nonisomorphic indecomposable projective right modules. Several examples of modules with a semilocal endomorphism ring appear in [Herbera and Shamsuddin]. Endomorphism rings of Artinian modules have been the object of a number of studies; see, for instance, [Camps and Facchini], where Pr¨ ufer rings that are endomorphism rings of Artinian modules are characterized, and [Camps, Facchini, and Puninski], where serial rings that are endomorphism rings of Artinian modules are studied. The nth root property for projective modules has been an object of deep study. For an interesting discussion of the nth root property for projective modules over von Neumann regular rings, see the Introduction in [Goodearl 1995]. In [Facchini and Herbera 2004, Theorem 2.3], it is proved that if a module MR is isomorphic to a direct summand of a direct sum of finitely many modules Ai and δ(MR ) = n, then MR is isomorphic to a direct summand of a direct sum of at most n of the modules Ai . This shows that δ(MR ) := codim(End(MR )) can be used as a measure of the size of the module MR . Notice that δ(MR ) ≤ dim(MR ) + codim(MR ) for any module MR ([Herbera and Shamsuddin], [Facchini 1998, Theorem 4.3]).

Chapter 4

Additive Categories The aim of this chapter is to point out some aspects of additive categories that usually are not sufficiently stressed in a first course in category theory.

4.1 Preadditive Categories A category C is a preadditive category (or a Z-category) if the sets HomC (A, B) can be given an additive abelian group structure in such a way that the composition ◦ becomes Z-bilinear. That is, for every A, B ∈ Ob(C), HomC (A, B) is an abelian group, and for every A, B, C ∈ Ob(C), f, f  : A → B, and g, g  : B → C, g ◦ (f + f  ) = g ◦ f + g ◦ f 

and (g + g  ) ◦ f = g ◦ f + g  ◦ f.

In the next example, we show that the abelian group structure on the Homsets of a preadditive category making the composition Z-bilinear can be given in different ways. Example 4.1. Let R and S be two rings that are isomorphic as multiplicative monoids, but not as rings. For example, the commutative rings of polynomials R = Z2 [x] and S = Z2 [x, y] in one and two indeterminates with coefficients in the field Z2 with two elements are not isomorphic as rings; for instance, they have different Krull dimension. Let us prove that R and S are isomorphic as monoids. The unique invertible element of R is 1R . Since R is a unique factorization domain, every nonzero element of R can be written in a unique way as a product of irreducible monic polynomials. Thus the multiplicative monoid of R is isomorphic (P ) to {0}∪N0 , where P is the set of all irreducible polynomials in the indeterminate (Q) x. Similarly, S is isomorphic to {0} ∪ N0 , where Q is the set of all irreducible polynomials in the indeterminates x and y. But P and Q are both countable sets, so ∼ S as monoids. Define a category C with a unique object ∗, EndC (∗) = R, that R = and multiplication in R as composition. The category C is a preadditive category, in which there is a first additive structure on EndC (∗) = R given by the additive © Springer Nature Switzerland AG 2019 A. Facchini, Semilocal Categories and Modules with Semilocal Endomorphism Rings, Progress in Mathematics 331, https://doi.org/10.1007/978-3-030-23284-9_4

111

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Chapter 4. Additive Categories

structure of R. But there is also a second additive structure on EndC (∗) given by the additive structure of S. More precisely, the second additive structure is defined as follows. Since R and S are isomorphic as multiplicative monoids, there is a monoid isomorphism ϕ : R → S. The bijection ϕ cannot be an isomorphism of additive abelian groups, since otherwise, we would have R ∼ = S as rings, which is not the case. Thus on the set EndC (∗) we can give two different ring structures, with the same multiplication, but with different additions. Hence EndC (∗) has two different abelian group structures making the composition in C Z-bilinear. If C and D are preadditive categories, a functor F : C → D is an additive functor if F (f +f ) = F (f )+F (f  ) for every f, f  : A → B in C. Equivalently, if the mapping FA,B : HomC (A, B) → HomD (F (A), F (B)), defined by FA,B (f ) = F (f ) for every morphisms f : A → B in C, is a group morphism for all A, B ∈ Ob(C). Similarly for contravariant functors. Notice that in this definition of an additive functor, we intend fixed once for all the abelian group structure on the Hom-sets. For example, if C is the preadditive category of the previous Example 4.1, with one object ∗ with endomorphism ring R and D is the preadditive category with one object ∗ with endomorphism ring S, then the categories C and D are isomorpic, so that there is a category isomorphism F : C → D, but the isomorphism F is not an additive functor. If C is a preadditive category, then for any two objects A, B of Ob(C), the additive abelian group HomC (A, B) has a zero with respect to the group addition. We will call it the zero morphism of A into B and denote it by ζA,B . We had already defined on p. 18 a zero morphism between any two objects in an arbitrary category C with a zero object. Hence it is convenient to verify immediately the following: Lemma 4.2. If C is a preadditive category with a zero object Z and A, B ∈ Ob(C), then the zero morphism A → B defined on p. 18 as the composite morphism of the unique morphism A → Z and the unique morphism Z → B coincides with the zero element ζA,B of the abelian group HomC (A, B). Proof. The composition of morphisms in C is a Z-bilinear mapping Hom(Z, B) × Hom(A, Z) → Hom(A, B), so (0, 0) → 0◦0 = 0. Here the first two morphisms 0 are the unique elements of Hom(Z, B) and Hom(A, Z) respectively, so that the fifth 0 is the zero element ζA,B of the group Hom(A, B) by Z-bilinearity, and 0 ◦ 0 is the composite morphism of the unique morphism A → Z and the unique morphism Z → B.  It follows that though the abelian group structure on a preadditive category C is not necessarily unique, the zero morphism in HomC (A, B) is uniquely determined by the category C. Notice that the dual category of a preadditive category is a preadditive category.

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In the next lemma we show that in a preadditive category, zero objects, initial objects and terminal objects coincide. Lemma 4.3. The following conditions are equivalent for an object Z of a preadditive category C: (a) Z is an initial object. (b) Z is a terminal object. (c) 1Z = ζZ,Z . (d) There is only one morphism Z → Z, that is, HomC (Z, Z) is the trivial group with one element. Proof. (a) ⇒ (c) If Z is an initial object, there is a unique morphism Z → Z, so that in particular 1Z = ζZ,Z . (c) ⇒ (d) For every f ∈ HomC (Z, Z), one has f = f ◦ 1Z = f ◦ ζZ,Z = ζZ,Z by Z-bilinearity. (d) ⇒ (a) For every object A, HomC (Z, A) is a group, hence has at least one element, so that there is a morphism Z → A. Let us show that (d) implies that this morphism is unique. From (d), it follows that 1Z = ζZ,Z . Hence, for every f : Z → A, we have f = f ◦ 1Z = f ◦ ζZ,Z = ζZ,A . Thus | HomC (Z, A)| = 1 for every A, that is, Z is an initial object. Now one proves that (b) is logically equivalent to the other conditions by applying the logical equivalences (a) ⇔ (c) ⇔ (d) to the dual category of C.  From now on, the zero morphism ζA,B : A → B will be denoted simply by 0, for any pair A, B of objects. We leave as an exercise to the reader to show that additive functors send zero morphisms to zero morphisms and zero objects to zero objects. Example 4.4. Initial objects do not coincide with terminal objects in the category Set of sets, and so Set is not preadditive. For the same reason, the category Top of topological spaces and continuous mappings and the category Ring are not preadditive either. The categories Ab, Mod-R and R -Mod are preadditive. As we have already seen in Example 4.1, given a ring R, there is a preadditive category C with exactly one object whose endomorphism ring is R. For every right module MR , the functors HomC (−, MR ) : Mod-R → Ab and HomC (MR , −) : Mod-R → Ab are additive functors. The first is contravariant, the second covariant. There are categories that cannot be embedded as a full subcategory of a category with a zero object. For instance, if C is a category with a unique object ∗, with EndC (∗) the multiplicative monoid N of positive integers, and composition ◦ the usual multiplication between positive integers, then there is not a category C  with a zero object and containing C as a full subcategory, because in such a category C  there is the morphism ζ : ∗ → ∗ with the property that ζ ◦ f = ζ for every f ∈ EndC (∗), while there is not a z ∈ N such that zn = z for every n ∈ N. We leave to the reader to show that any preadditive category can be embedded as a full subcategory in a preadditive category with a zero object.

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4.2 Products, Coproducts, Biproducts of Objects Definition 4.5. Let C be a category and let A, B be objects of C. A product    A B, πA , πB of A and B in C consists of an object A πA : A



B→A



and

B of C and morphisms πB : A



B→B

such that for any pairof morphisms f : C → A, g : C → B, there is a unique morphism h : C → A B with πA ◦ h = f and πB ◦ h = g. Equivalently, the mapping    B → HomC (C, A) × HomC (C, B), h → (πA ◦ h, πB ◦ h) HomC C, A is a bijection for every C ∈ Ob(C). A product of two objects A and B is the solution of a universal problem. Hence it does not necessarily exist for any two objects A, B in a category C, but when it exists, it is always unique up to isomorphism in C. Definition 4.6. Let C be a category and let A, B be objects of C. A coproduct    A B, εA , εB   of A and B consists of an object A B of C and mappings εA : A → A B and  εB : B → A B such that for any  pair of morphisms f : A → C, g : B → C, there is a unique morphism h : A B → C such that h ◦ εA = f and h ◦ εB = g. Equivalently,    B, C → HomC (A, C) × HomC (B, C), h → (h ◦ εA , h ◦ εB ) HomC A is a bijection for every C ∈ Ob(C).  When a coproduct A B of two objects A and B in a category C exists, it is unique up to isomorphism. Example 4.7. Consider the category Set. The product of X, Y ∈ Ob(Set) is (X × Y, πX , πY ), where X × Y is the Cartesian product and πX , πY are the canonical projections. The coproduct is (X ∪˙ Y, εX , εY ), where X ∪˙ Y is the disjoint union and εX , εY are the inclusions. Example 4.8. In the category Ab, the product of the abelian groups G and H is (G ⊕ H, πG , πH ), the coproduct is (G ⊕ H, εG , εH ).

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Example 4.9. In the category Grp of all groups, the product of two groups G and H is their direct product (Cartesian product) G×H with the canonical projections πG : G × H → G and πH : G × H → H. The coproduct is (G ∗ H, εG , εH ), where G ∗ H is the free product of G and H, which is the group defined as follows. Let G ∪˙ H be the disjoint union of the sets G and H, and let F be the free group on the set G ∪˙ H. Let ε : G ∪˙ H → F be the canonical mapping with the universal property of free groups. Set G ∗ H := F/N , where N is the normal subgroup of F generated by the set { ε(g)ε(g  )ε(gg  )−1 , ε(h)ε(h )ε(hh )−1 | g, g  ∈ G, h, h ∈ H }. Then G ∗ H, called the free product of G and H, is a group generated by the set { ε(x)N | x ∈ G ∪˙ H }. Define εG : G → G ∗ H by εG (g) = ε(g)N for every g ∈ G and εH : H → G ∗ H by εH (h) = ε(h)N for every h ∈ H. It is easily seen that these two mappings εG , εH are group morphisms. For instance, if g, g  ∈ G, then εG (g)εG (g  ) = εG (gg  ), i.e., ε(g)N ε(g  )N = ε(gg  )N , because ε(g)ε(g  )ε(gg  )−1 ∈ N . It remains to show that the free product with the two morphisms εG , εH satisfies the universal property of coproduct. Let L be a group and let ϕG : G → L, ϕH : H → L be two group morphisms. Let ω : F → L be the unique group morphism such that ω(ε(g)) = ϕG (g) for every g ∈ G and ω(ε(h)) = ϕH (h) for every h ∈ H. The generators of N are mapped to one by ω. Hence ω induces  : F/N → L such that ω a unique group morphism ω  (ε(g)N ) = ϕG (g) for every  (ε(h)N ) = ϕH (h) for every h ∈ H. Thus ω g ∈ G and ω  εG = ϕG and ω  εH = ϕH . In order to show that ω  is the unique morphism with this property, assume that ω  : F → L is also a group morphism such that ω  εG = ϕG and ω  εH = ϕH . Then ω  (ε(g)N ) = ϕG (g) for every g ∈ G and ω  (ε(h)N ) = ϕH (h) for every h ∈ H. Hence ω  and ω  coincide on a set of generators of F/N . Therefore they coincide on F/N . Example 4.10. In the categories R -Mod and Mod-R, the product and the coproduct of two modules is their direct sum. Example 4.11. In the category Ring, the product of two objects R, S is their direct product (Cartesian product) R × S. The coproduct is (R ∗ S, εR , εS ), where R ∗ S is the free product of the rings R and S. It is defined as follows. Let R ∪˙ S be the disjoint union of the sets R and S, and let F be the free ring on the set R ∪˙ S, that is, the ring of all noncommutative polynomials with coefficients in Z in the set of indeterminates R ∪˙ S. Let ε : R ∪˙ S → F be the canonical mapping. Set R ∗ S := F/I, where I is the two-sided ideal of F generated by the set { ε(r) + ε(r ) − ε(r + r ), ε(r)ε(r ) − ε(rr ), ε(s) + ε(s ) − ε(s + s ), ε(s)ε(s ) − ε(ss ), | r, r ∈ R, s, s ∈ S }. Then R ∗ S is the free product of R and S. As a ring, it is generated by the set { ε(x) + I | x ∈ R ∪˙ S }. Define εR : R → R ∗ S, εS : S → R ∗ S by εR (r) = ε(r) + I for every r ∈ R and εS (s) = ε(s) + I for every s ∈ S. Example 4.12. Let k be a commutative ring and k -Alg the category of all kalgebras (p. 50). The product of (R, fR ) and (S, fS ) in k -Alg is their direct product (R × S, fR×S ), where fR×S (λ) = (fR (λ), fS (λ)). The coproduct of (R, fR )

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and (S, fS ) in k -Alg is their suitably defined free product R ∗k S. This generalizes Example 4.11. If we consider the full subcategory k -CAlg of k -Alg of all commutative k-algebras, whose objects are the pairs (R, fR ) with R a commutative ring and fR : k → R a ring homomorphism, the situation changes as follows. The product of (R, fR ) and (S, fS ) is always their direct product, but the coproduct of (R, fR ) and (S, fS ) is their tensor product (R ⊗k S, fR⊗S ), where fR⊗k S (λ) = fR (λ) ⊗k fS (λ). More precisely, for any two (not necessarily commutative) k-algebras (R, fR )

and

(S, fS ),

it is possible to define their tensor product (R ⊗k S, fR ⊗ fS ), and there are two canonical morphisms eR : R → R ⊗k S and eS : S → R ⊗k S, defined by eR (r) = r ⊗ 1 and eS (s) = 1 ⊗ s, and two canonical morphisms εR : R → R ∗k S and εS : S → R ∗k S. These morphisms εR , εS induce a bijection F : Homk -Alg (R ∗k S, T ) → Homk -Alg (R, T ) × Homk -Alg (S, T ), h → (h ◦ εR , h ◦ εS ), for every k-algebra T . There is a surjective morphism ϕ : R ∗k S → R ⊗k S, and eR = ϕ ◦ εR , eS = ϕ ◦ εS . Via this morphism ϕ, the bijection F restricts to a bijection F  : Homk -Alg (R ⊗k S, T ) → → { (a, b) ∈ Homk -Alg (R, T ) × Homk -Alg (S, T ) | [a(R), b(S)] = 0 }, defined by

F  : h ∈ Homk -Alg (R ⊗k S, T ) → (h ◦ eR , h ◦ eS ).

For instance, if R = k[x] and S = k[y] are polynomial rings in one inde∼ k[x, y], R ∗k S ∼ terminate, then R ⊗k S = = kx, y, ϕ : kx, y → k[x, y] is the canonical mapping, in the bijection F we have sets corresponding to the Cartesian product T × T , in the bijection F  we have sets corresponding to the subset { (t, t ) ∈ T × T | tt = t t } of T × T  . The notions of product and coproduct of two objects easily extend to the notions of product and coproduct of any set of objects of the category. Let A1 , A2 be objects of a preadditive category C. A biproduct of A1 and A2 is a 5-tuple (B, π1 , π2 , ε1 , ε2 ), where B ∈ Ob(C) and π1 : B → A1 , π2 : B → A2 , ε1 : A1 → B, ε2 : A2 → B are morphisms such that π1 ◦ ε1 = 1A1 , π2 ◦ ε2 = 1A2 , ε1 ◦ π1 + ε2 ◦ π2 = 1B . Lemma 4.13. If (B, π1 , π2 , ε1 , ε2 ) is a biproduct of two objects A1 and A2 in a preadditive category C, then π2 ◦ ε1 = 0

and

π1 ◦ ε2 = 0.

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Proof. One has that π2 ◦ ε1 = π2 ◦ 1B ◦ ε1 = π2 ◦ (ε1 ◦ π1 + ε2 ◦ π2 ) ◦ ε1 = π2 ◦ ε1 ◦ π1 ◦ ε1 + π2 ◦ ε2 ◦ π2 ◦ ε1 = π2 ◦ ε1 + π2 ◦ ε1 , from which π2 ◦ ε1 = 0. Similarly, π1 ◦ ε2 = 0.  Clearly, the notion of biproduct of two objects also extends to the notion of biproduct of any finite set of objects of the category. In the next lemma, we show that products and biproduct in a preadditive category are essentially the same thing. Lemma 4.14. Let C be a preadditive category, and let A1 , A2 ∈ Ob(C). (i) If (B, π1 , π2 , ε1 , ε2 ) is a biproduct of A1 and A2 , then (B, π1 , π2 ) is a product of A1 and A2 . (ii) If (B, π1 , π2 ) is a product of A1 and A2 , then there exist morphisms ε1 : A1 → B and ε2 : A2 → B such that (B, π1 , π2 , ε1 , ε2 ) is a biproduct of A1 and A2 . Proof. (i) Let (B, π1 , π2 , ε1 , ε2 ) be a biproduct of A1 and A2 . We must prove that for every fixed C ∈ Ob(C), the mapping HomC (C, B) → HomC (C, A1 ) × HomC (C, A2 ), h → (π1 ◦ h, π2 ◦ h) is a bijection. It is injective, because if h, h : C → B are such that πi ◦h = πi ◦h for both i = 1, 2, then h = 1B ◦ h = (ε1 ◦ π1 + ε2 ◦ π2 ) ◦ h = ε1 ◦ π1 ◦ h + ε2 ◦ π2 ◦ h = h . It is surjective, because if f : C → A1 and g : C → A2 , then π1 ◦ (ε1 ◦ f + ε2 ◦ g) = f and π2 ◦ (ε1 ◦ f + ε2 ◦ g) = g. (ii) Let (B, π1 , π2 ) be a product of A1 and A2 . By the universal property of product, there exists a unique morphism ε1 : A1 → B such that π1 ◦ ε1 = 1A1 and π2 ◦ ε1 = 0. Similarly, there exists a unique morphism ε2 : A2 → B such that π2 ◦ ε2 = 1A2 , π1 ◦ ε2 = 0. In order to show that (B, π1 , π2 , ε1 , ε2 ) is a biproduct of A1 and A2 , it remains to show that ε1 ◦ π1 + ε2 ◦ π2 = 1B . For this, by the universal property of product, it suffices to show that the two morphisms ε1 ◦ π1 + ε2 ◦ π2 and 1B of B into B are such that πi ◦ (ε1 ◦ π1 + ε2 ◦ π2 ) = πi ◦ 1B for both i = 1, 2. Now πi ◦ (ε1 ◦ π1 + ε2 ◦ π2 ) = πi ◦ εi ◦ πi = 1Ai ◦ πi = πi = πi ◦ 1B , as we wanted to prove.  One easily sees that (B, π1 , π2 , ε1 , ε2 ) is a biproduct of A1 and A2 in C if and only if (B, ε1 , ε2 , π1 , π2 ) is a biproduct of A1 and A2 in the dual category C 0 . Moreover, applying Lemma 4.14 to the dual category, one finds that if (B, π1 , π2 , ε1 , ε2 ) is a biproduct of A1 and A2 in C, then (B, ε1 , ε2 ) is a coproduct of A1 and A2 , and every coproduct (B, ε1 , ε2 ) of A1 , A2 in C can be completed to a biproduct (B, π1 , π2 , ε1 , ε2 ). Thus, in a preadditive category, two objects A1 and A2 have a biproduct π1 π2 / A2 A1 o /Bo ε2

ε1

if and only if they have a product A1 o

π1

B

π2

/ A2

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if and only if they have a coproduct A1

ε1

/Bo

ε2

A2 .

In this case, we will denote by (A1 ⊕ A2 , π1 , π2 , ε1 , ε2 ) such a biproduct. We will also often omit the morphisms and simply write A1 ⊕ A2 instead of the 5-tuple. Instead of biproduct, we will use the term direct sum as well. Finally, note that if (B, ε1 , ε2 , π1 , π2 ) is a biproduct of A1 and A2 in the preadditive category C, then (B, π1 , π2 ) is a product of A1 and A2 , and (B, ε1 , ε2 ) is a coproduct of A1 and A2 , so that the biproduct (B, ε1 , ε2 , π1 , π2 ) depends only on C, and not on the abelian group structure on the Hom-sets, though in the formulas defining the biproduct the sign of addition appears.

4.3 Additive Categories A preadditive category C is said to be an additive category if it has a zero object and have a biproduct A1 ⊕ A2 (equivalently, a product every  two objects A1 and A2  A1 A2 , or a coproduct A1 A2 ). Our first result on additive categories will be that the abelian group structure on the Hom-sets of an additive category is uniquely determined by C. Let C be an additive category. For every object A of C, consider a biproduct (A ⊕ A, π1,A , π2,A , ε1,A , ε2,A ) of A and A. Let ΔA : A → A ⊕ A be the unique morphism such that π1,A ◦ ΔA = π2,A ◦ ΔA = 1A (it exists and is unique by the universal property of the product (A ⊕ A, π1,A , π2,A )). If f, g : A → B are morphisms in C, then there exists a unique morphism f × g : A ⊕ A → B such that (f × g) ◦ ε1,A = f and (f × g) ◦ ε2,A = g (universal property of the coproduct (A ⊕ A, ε1,A , ε2,A )). In all these constructions, only the category structure on C is involved, hence the composite morphism f × g ◦ ΔA : A → B also depends only on C and not on the abelian group structure on HomC (A, B). Lemma 4.15. Let C be an additive category, A and B objects of C, and f, g ∈ HomC (A, B) morphisms. Then f + g = (f × g) ◦ ΔA . Proof. The proof is by direct calculation: (f × g) ◦ ΔA = (f × g) ◦ 1A⊕A ◦ ΔA = (f × g) ◦ (ε1,A ◦ π1,A + ε2,A ◦ π2,A ) ◦ ΔA = (f ◦ π1,A + g ◦ π2,A ) ◦ ΔA = f ◦ (π1,A ◦ ΔA ) + g ◦ (π2,A ◦ ΔA ) = f + g.



This proves that in an additive category C, the abelian group structure on the Hom-sets is uniquely determined by the category C. Lemma 4.16. Let C and D be additive categories. A covariant functor F : C → D is additive if and only if it preserves biproducts, i.e., for any two objects A1 , A2 of

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C, if (B, π1 , π2 , ε1 , ε2 ) is a biproduct of A1 and A2 in C, then   F (B), F (π1 ), F (π2 ), F (ε1 ), F (ε2 ) is a biproduct of F (A1 ) and F (A2 ) in D. Proof. Suppose first that F is additive. If (B, π1 , π2 , ε1 , ε2 ) is a biproduct, then π1 ◦ ε1 = 1A1 , π2 ◦ ε2 = 1A2 , ε1 ◦ π1 + ε2 ◦ π2 = 1B , so that F (π1 ) ◦ F (ε1 ) = 1F (A1 ) , F (π2 ) ◦ F (ε2 ) = 1F (A2 ) , F (ε1 ) ◦ F (π1 ) + F (ε2 ) ◦ F (π2 ) = 1F (B) . For the converse, let f, g : A → B be morphisms. We must prove that F (f + g) = F (f ) + F (g). Consider a biproduct (A ⊕ A, π1,A , π2,A , ε1,A , ε2,A ) of A and A in C. We know by hypothesis that   F (A ⊕ A), F (π1,A ), F (π2,A ), F (ε1,A ), F (ε2,A ) is a biproduct of F (A) and F (A) in D. If ΔA : A → A ⊕ A is the morphism such that π1,A ◦ ΔA = π2,A ◦ ΔA = 1A , then F (ΔA ) : F (A) → F (A ⊕ A) is a morphism such that F (π1,A ) ◦ F (ΔA ) = F (π2,A ) ◦ F (ΔA ) = 1F (A) . That is, F (ΔA ) = ΔF (A) . Similarly, if f × g : A ⊕ A → B is such that (f × g) ◦ ε1,A = f and (f × g) ◦ ε2,A = g, then F (f × g) : F (A ⊕ A) → F (B) is such that F (f × g) ◦ F (ε1,A ) = F (f ) and F (f × g) ◦ F (ε2,A ) = F (g). That is, F (f × g) = F (f ) × F (g). By Lemma 4.15, we can conclude that F (f + g) = F (f × g) ◦ F (ΔA ) = (F (f ) × F (g)) ◦ ΔF (A) =  F (f ) + F (g). Clearly, if F : C → D is a contravariant functor between two additive categories C and D, then F is additive if and only if it preserves biproducts, in the sense that for any two objects A1 , A2 of C, if (B, π1 , π2 , ε1 , ε2 ) is a biproduct of A1 and A2 in C, then (F (B), F (ε1 ), F (ε2 ), F (π1 ), F (π2 )) is a biproduct of F (A1 ) and F (A2 ) in D. By Lemma 4.16, we can construct a functor V from the category of all additive categories with additive functors as morphisms into the category of all large commutative monoids. It associates to an additive category A the large monoid V (A) and to any additive functor F : A → B the monoid morphism V (F ) : V (A) → V (B) that sends an object A of V (A) to the unique object of V (B) isomorphic to F (A).

4.4 Splitting Idempotents Recall that idempotents split in a category C if for every object C of C and every endomorphism e : C → C in C with e2 = e, there exist an object A and two morphisms f : A → C and g : C → A such that e = f g and gf = 1A . We will see in Proposition 4.17 that if C is an additive category, idempotents split in C if and only if every idempotent has a kernel in C, that is, for every

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object A and every morphism f : A → A in C with f 2 = f , f has a kernel. This explains the link between splitting idempotents and direct-sum decompositions in an additive category. More precisely, if an object A in an additive category C in which idempotents split decomposes as a biproduct B ⊕ C in C, then the composite morphism of the projection of A onto B and the inclusion of B into A is a splitting idempotent. Conversely, if g : A → B and h : B → A are morphisms ∼ B ⊕ker g with gh = 1B , then hg : A → A is an idempotent endomorphism and A = (Proposition 4.17). For example, every full subcategory C of the category Mod-R of all right modules over a ring R is preadditive. A full subcategory C of Mod-R is an additive category if and only if it contains a zero module and for any pair of modules A, B ∈ Ob(C), there is a module in Ob(C) isomorphic to the direct sum A ⊕ B. Idempotents split in C if and only if for every A ∈ Ob(C) and every direct summand B of A, there exists in Ob(C) a module isomorphic to B. Every category can be embedded in a category in which idempotents split constructed in the following way. For any category C, let C be the category whose objects are the pairs (C, e), where C ∈ Ob(C) and e is an idempotent of EndC (C). The morphisms (B, e) → (B  , e ) in C are all the morphisms ϕ : B → B  in C with e ϕe = ϕ. Thus HomC((C, e), (C  , e )) is a subset of HomC (C, C  ). It is easily seen that C is a category in which idempotents split. If C is preadditive, then HomC((C, e), (C  , e )) is a subgroup of HomC (C, C  ), and C is preadditive. Define the embedding functor F : C → C by F (C) := (C, 1C ) for every object C of C. The functor F is full and faithful and has the following universal property. For every preadditive category B in which idempotents split and every additive functor G : C → B, there is a functor H : C → B, unique up to natural isomorphism, such that HF = G. If C is additive, then C is additive, and the functor F has the following universal property. For every additive functor G : C → A of C into an additive category A in which idempotents split, there exists a functor H : C → A, unique up to natural isomorphism, such that HF = G. Proposition 4.17. The following conditions are equivalent for an additive category A: (1) Idempotents have kernels in A; that is, for any object B of A, every morphism e : B → B with e2 = e has a kernel in A. (2) Idempotents split in A. Moreover, if these equivalent conditions hold, then for any two morphisms f : A → B and g : B → A in A with gf = 1A , one has that (a) the endomorphism f g : B → B is idempotent, (b) the kernel k : K → B of f g is also a kernel for g, and (c) (f, k) : A ⊕ K → B is an isomorphism. Proof. (1) ⇒ (2). Let e : B → B be an idempotent in A and let f : A → B be a kernel of the idempotent 1B − e. Then (1B − e)e = 0, so that there exists a unique

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morphism g : B → A with e = f g. Now (1B − e)f = 0 implies that f = ef = f gf . But kernels are monomorphisms, hence 1A = gf . (2) ⇒ (1). Assume that (2) holds. Let e : B → B be an idempotent. Since e splits, there exist f : A → B and g : B → A with f g = 1B − e and gf = 1A . Then f is a kernel of e, because ef = (1B − f g)f = f − f gf = 0; and if t : D → B is such that et = 0, then f (gt) = (1B − e)t = t. Moreover, if t is another morphism with f t = t, then t = gf t = gt. Thus f is a kernel of e. For the second part of the statement, suppose that (1) holds. Let f : A → B and g : B → A be morphisms with gf = 1A . Trivially, (a) holds. If k : K → B is a kernel of f g, then f gk = 0, so that gk = gf gk = 0. In order to prove (b), let k  be a morphism with gk  = 0. Then f gk  = 0, so that there is a unique morphism k  in A with kk  = k  . This shows that (b) holds. For (c), consider the biproduct (A ⊕ K, πA , πK , ιA , ιK ) of A and K and the morphism (f, k) : A⊕K → B. Now g(1B −f g) = 0 and (b) imply that there exists a unique h : B → K with 1B −f g = kh. We will now show that ιA g+ιK h : B → A⊕K is the inverse morphism of (f, k) = f πA + kπK . In order to prove that β := (ιA g + ιK h)(f πA + kπK ) is the identity morphism of A ⊕ K, notice that β = ιA gf πA + ιA gkπK + ιK hf πA + ιK hkπK = ιA πA + ιK hf πA + ιK hkπK . But khf = (1B − f g)f = 0. The kernel k is necessarily a monomorphism, so that hf = 0. Thus β = ιA πA + ιK hkπK . Moreover, khk = (1B − f g)k = k. Since k is a monomorphism, it follows that hk = 1K . Therefore β = 1A⊕K . In order to conclude the proof that ιA g +ιK h is an inverse of (f, k), it remains to notice that (f πA + kπK )(ιA g + ιK h) = f g + kh = 1B .  Additive categories satisfying the equivalent conditions of Proposition 4.17 are also called categories with split idempotents, or, sometimes, idempotent complete categories [Mitchell, p. 11], or amenable (Freyd; cf. [Mitchell, p. 12]). Hence, for any category C, we will call the category C introduced at the beginning of this section the idempotent completion of C. We will now generalize Theorem 2.35 from the category Mod-R to an arbitrary additive category with split idempotents. Let A be an object of a preadditive category C and let add(A) be the subclass of Ob(C) consisting of all objects B ∈ Ob(C) for which there exist an integer n n > 0 and morphisms f1 , . . . , fn : A → B and g1 , . . . , gn : B → A in C with i=1 fi gi = 1B . If C is additive with splitting idempotents, then add(A) is the class of objects of the smallest additive full subcategory of C with splitting idempotents containing A and closed under isomorphism. For instance, when C = Mod-R, then add(RR ) is the class proj-R of all finitely generated projective right R-modules. We denote by add(A) not only the subclass of Ob(C), but also the full subcategory of C whose class of objects is add(A).

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Lemma 4.18. Let A be a nonzero object of a preadditive category C. Set E := EndC (A). Consider the additive functor F := HomC (A, −) : C → Mod-E. Then the following properties hold: (a) The functor F induces a full and faithful functor add(A) → proj-E. (b) If C is an additive category with splitting idempotents, then F induces an equivalence add(A) → proj-E. Proof. Let B be an arbitrary object of add(A), so that there exist f1 , . . . , fn : A → B with

n i=1

and g1 , . . . , gn : B → A

fi gi = 1B . If we apply F to this identity, we get that n 

F (fi )F (gi ) = 1F (B) ,

i=1

where F (fi ) : F (A) → F (B) and F (gi ) : F (B) → F (A) are right E-module mor∼ En , phisms. Thus the module F (B) turns out to be a direct summand of F (A)n = E hence a finitely generated projective right E-module. In order to see that the restriction of F to add(A) is a faithful functor, let f : B → B  be a morphism of add(A) f h = 0 for every h ∈ HomC (A, B). Since with F (f ) = 0, that is, n (f fi )gi = 0. Thus F is faithful. In order 1B = ni=1 fi gi , we have f = f 1B = i=1 to prove that the restriction of F is full, let B, B  be two objects in add(A) and let  ϕ : HomC (A, B) → Hom n C (A, B ) be a right E-module morphism. Define f : B →  B by setting f := i=1 ϕ(fi )gi . We want to show that F (f ) = ϕ, that is, that F (f )(f  ) = ϕ(f  ) for every f  ∈ HomC (A, B). Now ϕ is a right n EndC (A)-module n morphism, so that F (f )(f  ) = f f  = i=1 ϕ(fi )gi f  = ϕ( i=1 fi gi f  ) = ϕ(f  ). This concludes the proof of (a). Now let C be an additive category with splitting idempotents. Let P be a finitely generated projective right E-module. Then there are morphisms αi : P → n given EE and βi : EE → P with 1P = ni=1 βi αi . Thus the endomorphism of EE by left multiplication by the matrix (αi βj ) is an idempotent endomorphism with image P . Since C is additive and the restriction of F to add(A) is full by (a), there is an endomorphism f of An in C such that F (f ) = (αi βj ). Again, the fact that the restriction of F to add(A) is faithful implies that f must be idempotent, so that f splits. Let g : An → B and h : B → An be morphisms in C with hg = f and gh = 1B . Then, for the right E-module morphisms F (g) : F (An ) → F (B) and F (h) : F (B) → F (An ), one gets that F (h)F (g) = F (f ) and F (g)F (h) = 1F (B) . Hence F (g) is onto, so that F (h) and F (f ) have the same image. Now the image of F (f ) = (αi βj ) is the projective module P , and F (g)F (h) = 1F (B) implies that  the image of F (h) is isomorphic to F (B). Thus P ∼ = F (B), as desired. Thus, if A is an additive category and idempotents split in A, the submonoid V (add(A)) of V (A) is a monoid canonically isomorphic to V (proj-E) = V (E). In particular, for A = Mod-R, in order to study the direct-sum decompositions of an

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arbitrary module MR , we can always suppose it finitely generated and projective (but we must change the base ring!). We will now improve this idea in order to apply it not to a single module MR but to an arbitrary skeletally small preadditive category P. The first thing we must do is to associate to each skeletally small preadditive category P a ring over which we will then consider projective modules. Recall that a ring R (possibly without 1) is said to have enough idempotents if there exists a set { ei | i ∈ I } of pairwise orthogonal idempotents of R such that R = ⊕i∈I ei R = ⊕i∈I Rei [Fuller]. Let P be a skeletally small preadditive category and fix an abelian group structure on each Hom-set making the composition Zbilinear. Fix a skeleton S of P, and associate to P the (Gabriel) functor ring F (P) := ⊕X∈Ob(S) ⊕Y ∈Ob(S) HomP (X, Y ) of P [Gabriel], which turns out to be a ring with enough idempotents (the suitable set of idempotents is given by the identities 1X , where X ranges over Ob(S). If f ∈ HomP (X, Y ) and f  ∈ HomP (X  , Y  ), the product f f  in F (P) is 0 if X = Y  and is the composition f ◦ f  if X = Y  , and then multiplication extends by Z-bilinearity. If we choose different skeletons of P, the rings we obtain turn out to be isomorphic, but if we choose different group structures on the Hom-sets we can get nonisomorphic rings (for instance, for the category C with one object of Example 4.1, we can get both R and S as functor rings). Since we are now dealing with rings (possibly) without identity, some care is needed as far as modules are concerned. Recall that a right module MR over a ring R (possibly without identity) is said to be a unitary module if M = M R. This definition coincides with the definition we are used to in the case that R has an identity, because if R is a ring with identity 1R and MR is a module unitary in the sense that M = M R, then for every for suitable x ∈ MR , one has x = i xi ri xi ∈ MR and ri ∈ R, so that x·1R = ( i xi ri )·1R = i xi ·(ri ·1R ) = i xi ri = x. Let Mod-R be the category whose objects are all right unitary R-modules. Two rings R and S with enough idempotents are said to be Morita equivalent if the categories Mod-R and Mod-S are equivalent. It is possible to prove that Morita equivalence is a right/left symmetric condition, that is, the categories R -Mod and S -Mod of left unitary modules are equivalent if and only if Mod-R and Mod-S are equivalent. Similarly to the case of rings with identity, it is also possible to prove that the rings R and S are Morita equivalent if and only if there is a surjective Morita context (R, S, M, N, ϕ, ψ), that is, if and only if there are unital bimodules R MS and S NR and surjective S-bimodule and R-bimodule morphisms ϕ : N ⊗R M → S and ψ : M ⊗S N → R that satisfy the compatibility conditions ϕ(n ⊗ m)n = nψ(m ⊗ n ) and m ϕ(n ⊗ m) = ψ(m ⊗ n)m for every m, m ∈ M , n, n ∈ N . Notice that if R is a ring with enough idempotents, then RR is a projective right R-module, i.e., it is a projective object in Mod-R, but it not finitely generated in general. It is possible to prove that the finitely generated projective unitary right R-modules are those isomorphic to a direct summand of (⊕i∈F ei R)n for some finite subset F of I and some n ≥ 0.

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Another difficulty in associating to every skeletally small preadditive category P with a fixed group structure on the Hom-sets its functor ring F (P), which is a ring with enough idempotents, is that two inequivalent categories can have the same functor ring. For instance, if P is any skeletally small preadditive category with at least two nonisomorphic objects, R is the functor ring of P, and R∗ is the category with one object whose endomorphism ring is R, then P and R∗ are two inequivalent categories with the same functor ring R. All these difficulties disappear if we restrict our attention to skeletally small additive categories in which idempotents split. Consider the correspondence {skeletally small additive categories in which idempotents split} → {rings with enough idempotents}. It turns out to be a one-to-one correspondence between the equivalence classes of skeletally small additive categories in which idempotents split, that is, the skeletally small additive categories in which idempotents split up to category equivalence, and Morita equivalence classes of rings with enough idempotents. The inverse oneto-one correspondence associates to every ring R with enough idempotents the category proj-R of all finitely generated projective unitary right R-modules. Finally, let A be a skeletally small additive category in which idempotents split and let R := F (A) be its functor ring. There is a functor A → proj-R that associates: (1) to each object A of A with isomorphic representative A in S the finitely generated projective unitary right R-module 1A R, and (2) to each morphism g : A → A in A the right R-module morphism 1A R → 1A R defined by left multiplication by the element of F (A) given by the morphism A → A  corresponding to g. The functor A → proj-R turns out to be a category equivalence.

4.5 Product and Coproduct of Categories Product category. Let Cλ , λ ∈ Λ, be categories indexed in a class Λ, with possibly Cλ = Cμ for different indices λ, μ ∈ Λ. 

The product category λ∈Λ Cλ is defined as follows. Let λ∈ΛOb(Cλ ) be the union of the classes Ob(Cλ ). The objects of the product category λ∈Λ Cλ are all mappings f : Λ → λ∈Λ Ob(Cλ ) such that f (λ) ∈ Ob(Cλ ) for every λ ∈ Λ. The morphisms in the product category are defined by taking as Hom-sets   λ∈Λ

  Cλ (Aλ , Bλ ). Cλ ((Aλ )λ∈Λ , (Bλ )λ∈Λ ) = λ∈Λ

The composition of morphisms is defined componentwise. Therefore:  (1) The product category λ∈Λ Cλ always exists in NFUP for any class Λ.  (2) The product category λ∈Λ Cλ exists in NBG and in MK only when Λ is a set.

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For each index μ ∈ Λ, there is a canonical projection functor  Pμ : Cλ → Cμ . λ∈Λ

When the categories Cλ are preadditive, the functors Pλ are additive functors. The product category has the following universal property.  Proposition 4.19. Let Cλ , λ ∈ Λ, be categories indexed in a class Λ, λ∈Λ Cλ the product category, and Pμ : λ∈Λ Cλ → Cμ the canonical projections. If B is an arbitrary category and Fλ : B → Cλ , λ ∈ Λ,  is a class of functors indexed in Λ, then there exists a unique functor F : B → λ∈Λ Cλ with Pλ F = Fλ for all λ ∈ Λ. In other words, the product category is the product in the category Cat of all categories. The category Cat contains the subcategory Preadd, whose objects are all preadditive categories and whose morphisms are all additive functors. The category Preadd contains the full subcategory Add, whose objects are all additive categories.  If all the categories Cλ are preadditive (or additive), the product category λ∈Λ Cλ is also preadditive (or additive, respectively), and the product category is the product in the categories Preadd (or Add, respectively). If Λ is  the empty class, then the product category λ∈Λ Cλ turns out to be the category with one object and one morphism (the identity morphism of the unique object).   Also, V ( λ∈Λ Cλ ) ∼ = indicates a bijection if = λ∈Λ V (Cλ ). Here the symbol ∼ the categories Cλ are arbitrary categories or preadditive categories, and indicates a monoid isomorphism if the categories Cλ are additive. Coproduct in Cat. Let us move on to the notion dual to that of product: coproduct in the category Cat of all categories. The coproduct in the category sense. Let Cλ , λ ∈ Λ, be Cat is the disjoint union of categories, in the following

categories indexed in a class Λ. The disjoint union ˙ λ∈Λ Cλ of the categories Cλ is a category whose class of objects is the disjoint union of the classes Ob(Cλ ) and with class of morphisms of an object Aλ ∈ Ob(Cλ ) into another object Bμ ∈ Ob(Cμ ) the empty set if λ = μ and HomCλ (Aλ , Bμ ) if λ = μ. The composition is the obvious one induced by the compositions in the categories Cλ . Therefore:

(1) The category ˙ Cλ , disjoint union of the categories Cλ , λ ∈ Λ, always λ∈Λ

exists in NFUP and MK, for every class Λ. (2) It exists in NBG when Λ is a set.

(3) There is a canonical bijection between the class V ( ˙ λ∈Λ Cλ ) and the disjoint

union ˙ λ∈Λ V (Cλ ) of the skeletons V (Cλ ), λ ∈ Λ.

Coproduct in Preadd. We have seen that the product in the categories Cat, Preadd, and Add essentially coincide. We will now see what happens as far as the coproduct of categories is concerned. Notice that the coproduct of preadditive categories in the category Cat of all categories we have met above is not a pread-

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ditive category in general (the disjoint union of a class of preadditive categories is not necessarily a preadditive category). a preadditive Let C λ be a preadditive category for every λ in a class Λ. Define

category ˙ λ∈Λ Cλ whose class of objects is the disjoint union ˙ λ∈Λ Ob(Cλ ) of the classes Ob(Cλ ) and whose group of morphisms of Aλ ∈ Ob(Cλ ) into Bμ ∈ Ob(Cμ ) is the trivial group with one element for λ = μ and the group HomCλ (Aλ , Bμ ) for  λ = μ. The preadditive category ˙ λ∈Λ Cλ constructed in this way always exists  in NFUP and MK. It exists in NBG if Λ is a set. Notice that ˙ λ∈Λ Cλ does not depend on the additive structure on the Hom-sets of the categories Cλ . If we fix a group structure on the Hom-sets of the categories Cλ , there  is a canonically induced additive structure on the Hom-sets of the category ˙ λ∈Λ Cλ . A class of additive functors Fλ : Cλ → D to another preadditive category  D corresponds to an additive functor from the preadditive category ˙ λ∈Λ Cλ into D. More  precisely, for each μ ∈ Λ, there is a canonical embedding functor Eμ : Cμ → ˙ λ∈Λ Cλ . For any preadditive category D, there is a bijective correspondence between the product of all the  classes of the additive functors Cλ → D and the class of the additive functors ˙ λ∈Λ Cλ → D. The inverse correspondence  associates to an additive functor F : ˙ λ∈Λ Cλ → D the element (F Eλ )λ∈Λ in  the product. Thus ˙ λ∈Λ Cλ is the coproduct of the categories Cλ in the cate gory Preadd of all preadditive categories. The skeleton of ˙ λ∈Λ Cλ is the disjoint union of the skeletons V (Cλ ) in which the zero objects are identified. More pre cisely, there is a canonical bijection between a skeleton V ( ˙ λ∈Λ Cλ ) and the factor

class ˙ λ∈Λ V (Cλ )/ ∼, where ∼ is the equivalence relation on the class ˙ λ∈Λ V (Cλ ) defined as follows: if λ, μ ∈ Λ, Aλ ∈ V (Cλ ) and Bμ ∈ V (Cμ ), then Aλ ∼ Bμ if and only if either λ = μ and Aλ = Bμ , or Aλ is a zero object in Cλ and Bμ is a zero object in Cμ . In particular, if the categories Cλ do not contain zero objects, then 

V ( ˙ λ∈Λ Cλ ) is the disjoint union ˙ λ∈Λ V (Cλ ).

4.6 Embedding into an Additive Category Let C be a full subcategory of an additive category A. We will denote by C  the additive full subcategory of A generated by C and closed under isomorphism. It is the full subcategory of A whose objects are the objects A ∈ Ob(A) for which there exist an integer n ≥ 0, objects U1 , . . . , Un ∈ Ob(C), and morphisms πi : A → Ui , εi : Ui → A in A for i = 1, . . . , n such that (A, πi , εi ) is a biproduct of U1 , . . . , Un in A. Using the axiom of choice for classes, we can fix a biproduct representation (A, πi , εi ) for every object A of C  , where nA is a nonnegative integer, U1A , . . . , UnAA ∈ Ob(C), and πi : A → UiA , εi : UiA → A, i = 1, . . . , nA , are morphisms in A. We assume that for every A ∈ Ob(C), the fixed biproduct representation of A is (A, 1A , 1A ).

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127

Proposition 4.20. Let A and D be additive categories, C a full subcategory of A, C  the additive full subcategory of A generated by C and closed under isomorphism, and F : C → D be an additive functor. Then there exists an additive functor F  : C  → D that extends F to C  . Such an extension F  is unique up to natural isomorphisms in the following sense. An additive functor F  : C  → D extends F if and only if there exists a natural isomorphism η : F  → F  with ηU = 1U for every U ∈ Ob(C). Proof. Define the functor F  as follows. For every object A in C  , let (A, πi , εi ) be the fixed biproduct representation of A, where U1A , . . . , UnAA ∈ Ob(C), nA ≥ 0, πi : A → UiA and εi : UiA → A. With the axiom of choice for classes, we can fix a biproduct (A , πi , εi ) in D of the objects F (U1A ), . . . , F (UnAA ). In the case nA = 1, we can suppose that the chosen biproduct is (F (A), 1F (A) , 1F (A) ). Thus, for U ∈ Ob(C), we have that (U  , πi , εi ) = (F (U ), 1F (U) , 1F (U) ), that is, U  = F (U ). Define F  : C  → D by setting F  (A) := A for every A ∈ Ob(C  ). If f : A → B is a morphism in C  , then f is a matrix of morphisms fj,i := πj f εi : UiA → UjB in C with i = 1, . . . , nA and j = 1, . . . , nB . Correspondingly, F (fj,i ) : F (UiA ) → F (UjB ) define a unique morphism the morphisms   F (f ) := i,j εj F (fj,i )πi : A = F  (A) → B  = F  (B). Set F  (f ) := f  . Let F  : C  → D be any additive functor that extends F . Since additive functors preserve biproducts (Lemma 4.16) and every object A of C  is a biproduct (A, πi , εi ) of U1A , . . . , UnAA in C  , it follows that (F  (A), F  (πi ), F  (εi )) is a biproduct of the objects F  (UiA ) = F (UiA ) of D. Thus we have two biproducts (F  (A), πi , εi ), (F  (A), F  (πi ), F  (εi )) of the same objects of D. By the uniqueness of biproduct, there is an isomorphism ηA : F  (A) → F  (A) such that F  (πi )ηA = πi and ηA εi = F  (εi ) for all i = 1, . . . , nA . In order to show that the isomorphisms ηA define a natural isomorphism η : F  → F  , fix a morphism f : A → B in C  . We must prove that F  (f )ηA = ηB F  (f ). For this, it suffices to show that F  (πj )F  (f )ηA εi = F  (πj )ηB F  (f )εi for every i = 1, . . . , nA and j = 1, . . . , nB . This is easily seen, because F  (πj )F  (f )ηA εi = F  (πj )F  (f )F  (εi ) = F (πj f εi ) = F (fj,i ) = πj F  (f )εi = F  (πj )ηB F  (f )εi . If U ∈ Ob(C), then F  E(U ) = F (U ) = F  E(U ). The fixed biproduct that defines F  (U ) is (F (U ), 1F (U) , 1F (U) ), and the same holds for F  (U ). Thus ηE(U) = 1F (U) . Conversely, let F  : C  → A be an additive functor and η : F  → F  a natural isomorphism with ηE(U) = 1F (U) for every U ∈ Ob(C). Then F  (E(U )) = F  (E(U )), so that F  E = F  E = F on the objects of C. If ϕ : U → V is a morphism in C, then E(ϕ) : E(U ) → E(V ) is a morphism in C  . By the naturality of η : F  → F  , we get that F  (E(ϕ))ηE(U) = ηE(V ) F  (E(ϕ)), that is, F  (E(ϕ)) = F  (E(ϕ)), so that F  E(ϕ) = F (ϕ). This concludes the proof that  F  E = F . The lack of uniqueness of F  in the statement of Proposition 4.20 depends on the lack of uniqueness of coproduct of objects in the category D. The coproduct of

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two objects in a category is defined only up to isomorphism, and this implies that the functor F  in the previous proposition is unique only up to natural isomorphism of functors. Every preadditive category C can be embedded into an additive category as a full subcategory. In the terminology of [Mac Lane, p. 198, Exercise 6], it is possible to construct the free additive category Mat(C). For any preadditive category C, let Mat(C) be the additive category whose objects are all n-tuples (U1 , . . . , Un ) of objects Ui of C for n ≥ 0, and whose morphisms from such an n-tuple (U1 , . . . , Un ) to an m-tuple (V1 , . . . , Vm ) are the m×n matrices (ϕij ) of morphisms ϕij : Uj → Vi of C. Since the abelian group structure on the Hom-sets is unique for additive categories and not necessarily unique for the preadditive ones, it is clear that the construction of the additive category Mat(C) also depends on the abelian group structure we fix on the Hom-set of the preadditive category C. For instance, let C be the category of Example 4.1. If we fix on EndC (∗) the abelian group structure corresponding to R, we get as Mat(C) a category equivalent to the full subcategory of Mod-R whose objects are all free right R-modules, and if we fix on EndC (∗) the abelian group structure corresponding to S, we get as Mat(C) a category equivalent to the full subcategory of Mod-S whose objects are all free right S-modules. These two categories are not equivalent. More generally, if C is any full subcategory of Mod-R, then Mat(C) is equivalent to the full subcategory of Mod-R whose objects are all right R-modules that are direct sums of finitely many modules in Ob(C). Hence, fix the group structure on the Hom-sets of the preadditive category C. There is a corresponding embedding functor E : C → Mat(C), which is an additive, full, faithful functor. It sends the object U of C to the 1-uple (U ) ∈ Ob(Mat(C)). The following is an immediate corollary of Proposition 4.20. Corollary 4.21. Let C be a preadditive category and E : C → Mat(C) the embedding functor. For every additive category D and every additive functor F : C → D, there exists an additive functor F  : Mat(C) → D with F = F  E. Such an additive functor F  is unique up to natural isomorphism, in the following sense. An additive functor F  : Mat(C) → D has the property that F = F  E if and only if there exists a natural isomorphism η : F  → F  with ηE(U) = 1F (U) for every U ∈ Ob(C). Notice that if the preadditive category C is already an additive category, the embedding functor E : C → Mat(C) defined above is not the identity functor, but only a categorical equivalence, and Mat(C) is a category containing C but with several new objects: a new zero object, obtained for n = 0, and new biproducts, obtained for n ≥ 2. Also notice what happens to the functor V (−) when we apply it to the embedding functor E : C → Mat(C). As we have already remarked on p. 19, if C is any preadditive category, the skeleton V (C) is a class with an operation induced by biproduct of objects, which is only a partially defined operation. Thus V (E) : V (C) → V (Mat(C)) is an injective mapping of this class V (C) with its

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partially defined operation into the commutative monoid V (Mat(C)). For any additive category D, every additive functor F : C → D extends to an additive functor F  : Mat(C) → D, which is not unique. Since it is unique up to natural isomorphism, the monoid morphism V (F  ) : V (Mat(C)) → V (D) is uniquely defined, that is, it does not depend on F  . Proposition 4.22. Let C be a preadditive category, C  an additive category, and E  : C → C  an additive functor. Assume that for every additive category D and every additive functor F : C → D, there exists an additive functor F  : C  → D, unique up to a natural isomorphism, with F = F  E  . Then there exists a category equivalence G : C  → Mat(C) with E  = EG. Proof. Apply the hypothesis to the additive category Mat(C) and the additive functor E : C → Mat(C), getting an additive functor F  : C  → Mat(C) with E = F  E  . By Corollary 4.21, applied to the additive category C  and the additive functor E  : C → C  , there exists an additive functor G : Mat(C) → C  with E  = GE. Thus E = F  GE. By the uniqueness up to isomorphism of Corollary 4.21,  there exists a natural isomorphism η : 1Mat(C) → F  G. Similarly for GF  . In this section, we have embedded each preadditive category C into an additive category, as a full subcategory. The additive category has a unique additive group structure on the Hom-sets, which induce an additive group structure on the Hom-sets of C. Thus the various additive group structures on the Hom-sets of C correspond to the embeddings of C into additive categories. Notice that for each additive category A containing a preadditive category C, the additive full subcategory of A generated by C and closed under isomorphism is equivalent to Mat(C).

4.7 Weak Coproduct of Additive Categories Let Cλ bean additive category for every λ in a class Λ. Construct the preadditive category ˙ λ∈Λ Cλ , coproduct in Preadd, as in Section 4.5. Now construct the cate  ˙ ˙ gory λ∈Λ Cλ ), like in Section 4.6. We will denote this category Mat( λ∈Λ Cλ ) Mat( w by λ∈Λ Cλ , and it the weak coproduct. There is a canonical embedding funccall w tor Fλ : Cμ → λ∈Λ Cλ for each μ ∈ Λ. It is the composition of the canonical    functor Eμ : Cμ → ˙ λ∈Λ Cλ and the canonical functor ˙ λ∈Λ Cλ → Mat( ˙ λ∈Λ Cλ ). If we have additive functors Gλ : Cλ → D into an additive  category D, then the functors Gλ canonically induce a unique additive functor ˙ λ∈Λ Cλ → D by the universal the coproduct in Preadd. This functor extends to a functor w property of  G of λ∈Λ Cλ = Mat( ˙ λ∈Λ Cλ ) into D, which is unique up to a natural isomorphism that is the identity on the objects Fλ (U ) for every λ ∈ Λ and U ∈ Ob(Cλ ). w The functor G : λ∈Λ Cλ → D is such that GFλ = Gλ for every λ ∈ Λ. category for every λ in a Thus we have proved that if Cλ is an additive w class Λ, then there exist an additive category λ∈Λ Cλ and embedding functors

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w Fμ : Cμ → λ∈Λ Cλ for all μ ∈ Λ with the property that if Gλ : Cλ → D is an additive functor into a fixed w additive category D for every λ ∈ Λ, then there exists an additive functor G : λ∈Λ Cλ → D such that GFλ = Gλ for every λ ∈ Λ. Notice that the functor G is not unique, but unique only up to a natural isomorphism that w is the identity on the objects Fλ (U ) for every λ ∈ Λ and U ∈ Ob(Cλ ). Thus λ∈Λ Cλ is “almost” a coproduct of the categories Cλ in the category Add of all additive categories. w Cλ is the additive large monoid that is the direct sum The skeleton of λ∈Λ of the skeletons V (Cλ ). Here some care is needed, because we are dealing with the direct sum of a class of large monoids. Thus let Mλ be a (possibly large) monoid for every index λ ranging over a class Λ. It is possible to define the direct sum of the monoids Mλ as follows. Let ⊕λ∈Λ Mλ be the class having as elements all finite sets { (λ1 , a1 ), (λ2 , a2 ), . . . , (λn , an ) }, where λ1 , . . . , λn are distinct elements of Λ, ai is a nonzero element of Mλi for every i = 1, . . . , n, and n ≥ 0 is an integer. If λ1 , . . . , λn , μ1 , . . . , μm , ν1 , . . . , νp are distinct elements of Λ, define {(λ1 , a1 ), . . . , (λn , an ), (μ1 , b1 ), . . . , (μm , bm )} + {(λ1 , c1 ), . . . , (λn , cn ), (ν1 , d1 ), . . . , (νp , dp )} = {(λ1 , a1 + c1 ), . . . , (λn , an + cn ), (μ1 , b1 ), . . . , (μm , bm ), (ν1 , d1 ), . . . , (νp , dp )}. Then ⊕λ∈Λ Mλ with this operation becomes a large commutative monoid, called the direct sum of the monoids Mλ . If all the monoids Mλ are equal to the same monoid M , we denote the direct sum ⊕λ∈Λ M by M (Λ) . w Remark 4.23. The additive category λ∈Λ Cλ has a weak form of universal propw erty: there does not exist a unique functor λ∈Λ Cλ → D, it is unique only up to a natural isomorphism. Let us prove that except for trivial cases, a coproduct in Add in the “classical form” of the universal property, that is, with uniqueness, does not exist. We will show that if C1 , C2 are additive categories, each with at least a nonzero object, then there does not exist an additive category C with two additive functors F1 : C1 → C, F2 : C2 → C, with the property that for every additive category D and additive functors G1 : C1 → D and G2 : C2 → D, there exists a unique additive functor G : C → D with GF1 = G1 and GF2 = G2 . Assume the contrary, and let C1 , C2 , C be additive categories, Ai ∈ Ob(Ci ) nonzero objects for i = 1, 2, and F1 : C1 → C, F2 : C2 → the univerC functors with  sal property of coproduct. As usual, we will write C1 w C2 instead of w i∈{1,2} Ci . w C2 be the canonical embedding functors, i = 1, 2. By the uniLet Fi : Ci → C1 w versal property of C, there is a unique additive functor G : C → C1 C2 such that F are full and faithful, the objects GF1 = F1 and GF2 = F2 . Since the functors i w Fi (Ai ), i = 1, 2, are nonzero objects of C1 C2 , and there is no nonzero morphism (Ai ), i = 1, 2, are nonisomorphic them. Thus in C1 w C2 between w the objects Fi w w objects of C1 C2 → Mat(C1 C2 . Let E : C1 C2 ) be the embedding functor. ) Let C be a coproduct of F1 (A1 ) and F2 (A 2 w in C. The two nonisomorphic objects EFi (Ai ) = (Fi (Ai )), i = 1, 2, of Mat(C1 C2 ) have at least two different

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w biproducts in Mat(C1 C2 ). One is the pair (F1 (A1 ), F2 (A2 )); the other is the objects (Fi (Ai )), i = 1, 2, of (G(C)). Notice that the two nonisomorphic 1-tuple  w w Mat(C1 C2 ) do not have a biproduct in Mat(C1 C2 ) of the form (F1 (X1 )) w for some X1 ∈ Ob(C1 ), for otherwise, F1 (X1 ) would be a biproduct in C1 C2 of F1 (A1 ) and F2 (A2 ), which is not possible because X1 cannot be a biproduct of  A1 and A2in C1 ˙ C2 . Similarly, the two nonisomorphic wobjects (Fi (Ai )), i = 1, 2, w of Mat(C1 C2 ) of the form (F2 (X2 )) C2 ) do not have a biproduct in Mat(C1 for some X2 ∈ Ob(C2 ). By Proposition 4.20, there is an additive functor  w  w     H : Mat C1 C2 C2 → Mat C1 w that is the identity on the full subcategory of Mat(C1 C2 ), whose objects are all the objects (F1 (X1 )), X ∈ Ob(C1 ), and all the objects (F2 (X2 )), X2 ∈ Ob(C2 ), and that is a fixed-point-free permutation on the biproducts of (F1 (A1 )) and (F2 (A2 )). w Set D := Mat(C1 C2 ). Then EG and HEG are two additive functors of C into D, which are different because EG(C) = HEG(C). But EGF1 = HEGF1 and EGF2 = HEGF2 . This contradicts the universal property of C, which concludes our proof. w Thus the skeleton of λ∈Λ Cλ is the additive large monoid that is the direct sum of the large monoids V (C λ ). Notice that the weak form of the universal propw erty of the weak coproduct λ∈Λ Cλ is sufficient to ensure that if Gλ : Cλ → D is an additive functor into a fixed w additive category D for every λ ∈ Λ, then the V monoid morphism V ( (G) : λ∈Λ Cλ ) → V (D) defined by the additive functor w G : λ∈Λ Cλ → D, such that GFλ = Gλ for every λ ∈ Λ, is uniquely defined, that is, does not depend on the functor G. It is the monoid morphism of the coproduct of the monoids V (Cλ ) defined by the monoid morphisms V (Gλ ) : V (Cλ ) → V (D). ∼ between two functors means “naturally In the next proposition, the symbol = isomorphic.” Proposition 4.24. Let Λ be a class, Cλ an additive category for every λ ∈ Λ, C an additive category, and Fλ : Cλ → C additive functors. The following conditions are equivalent: (a) For every additive category D and additive functors Gλ : Cλ → D, there exists ∼ Gλ , and for every additive an additive functor G : C → D such that GFλ =   ∼  functor G : C → D with G Fλ = Gλ for every λ ∈ Λ, there exists a natural isomorphism η : G → G . w (b) There exists a category equivalence E : λ∈Λ Cλ → C such that Fλ = EFλ for every λ ∈ Λ. Proof. (a) ⇒ (b) Assume that (a) holds. Apply the “universal w property” of  w  : F to the functors C , getting an additive functor E λ λ λ∈Λ λ∈Λ Cλ → D such

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that EFλ = Fλ for every λ ∈ Λ. Apply the “universal property” of C of hypothesis (a) to the functors Fλ , getting an additive functor G : C → D such that GFλ ∼ = Fλ . To conclude the proof of (b), it remains to show that EG and GE are naturally isomorphic to the identities. ∼ Fλ , there is a natural isomorphism θ(λ) : GF  → Fλ . Thus Since GFλ = λ (λ)

(λ)

the position θA := E(θA ) for every A ∈ Ob(Cλ ) defines a natural isomorphism (λ)

θ : EGFλ → EFλ = Fλ . By the “uniqueness” part in (a), it follows that there is a natural isomorphism EG ∼ = 1. We now show that GE is naturally isomorphic to the identity. From EFλ = Fλ and GFλ ∼ = Fλ , it follows that GEFλ ∼ = Fλ . Thus, for every λ, there is a natural isomorphism η (λ) : GEFλ → Fλ . Define w a natural isomorphism η : GE → 1, where 1 denotes the identity functor of λ∈Λ Cλ , as follows. Every object X of w

˙ λ∈Λ Ob(Cλ ). Thus for every λ∈Λ Cλ is an n-tuple (A1 , . . . , An ) of objects Ai of i = 1, . . . , n, there is an index λi ∈ Λ with Ai ∈ Ob(Cλi ). Since GE wis an additive functor and X is the biproduct of the objects (Ai ) = Fλi (Ai ) of λ∈Λ Cλ via the canonical morphisms πi : X → (Ai ) and εi : (Ai ) → X, itfollows that GE(X) w is the biproduct of the objects GE(Ai ) = GEFΛi (Ai ) of λ∈Λ Cλ via the morphisms GE(πi ) : GE(X) → GE((Ai )) = GEFΛi (Ai ) and GE(εi ) : GEFΛi (Ai ) = GE((Ai )) → GE(X). The natural isomorphism η : GE → 1 is defined by setting n (λ ) ηX : GE(X) → X, ηX = i=1 εi ηAii GE(πi ). We leave to the reader to check the naturality of η. w (b) ⇒ (a) Let E : λ∈Λ Cλ → C be a category equivalence. Assume Fλ = EFλ for every λ ∈ Λ. Let Gλ : Cλ → D be additive wfunctors into an additive category D.  Then there exists an additive functor G : λ∈Λ Cλ → D such that G Fλ = Gλ for w every λ ∈ Λ. Let E  : C → λ∈Λ Cλ be a quasi-inverse of E. Set G := G E  : C → D. In order to prove that GFλ = G E  EFλ is naturally isomorphic to Gλ = G Fλ for  any λ ∈ Λ, let η  : E  E → 1 be a natural ηX : E  E(X) → X is w isomorphism, so that   an isomorphism for every X ∈ Ob( λ∈Λ Cλ ). Then η : G E EFλ → G Fλ , defined by ηU := G(ηF λ (U) ) for every U ∈ Ob(Cλ ) is the required natural isomorphism. As far as the “uniqueness” in (a) is concerned, let G : C → D be an additive functor with G Fλ ∼ = Gλ for every λ ∈ Λ, so that G Fλ ∼ = G Fλ , that is, G EFλ ∼ =  G EFλ for every λ ∈ Λ. As in the last paragraph of the proof of (a) ⇒ (b), it is possible to glue together the natural isomorphisms η (λ) : G EFλ → G EFλ , getting a natural isomorphism η : G E → G E. Then η  : G EE  → G EE  , defined  by ηC := ηE(C) for every C ∈ Ob(C), is a natural isomorphism. Now EE  ∼ = 1, so  that G EE  ∼ = G and G EE  ∼ = G . It follows that G ∼ = G . w We will call any category equivalent to the category λ∈Λ Cλ the weak coproduct of the categories Cλ . As we have seen in Remark 4.23, a coproduct in Add strictly speaking usually does not exist.

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4.8 Constructions Equivalent to the Construction of Weak Coproduct In this section, we will present some constructions wthat yield, for additive categories Cλ , categories equivalent to the category λ∈Λ Cλ . In general, the same constructions can be carried out for arbitrary preadditive categories Cλ . The weak direct sum ⊕λ∈Λ Cλ . Let Cλ be a preadditive category for every λ in a class Λ. We will call the following preadditive category the weak direct sum ⊕λ∈Λ Cλ of the preadditive categories Cλ . The objects of ⊕λ∈Λ Cλ are the finite sets {(λ1 , A1 ), (λ2 , A2 ), . . . , (λn , An )} of pairs (λi , Ai ), i = 1, . . . , n, where λ1 , . . . , λn are distinct elements of Λ, n ≥ 0 is an integer, and Ai is a nonzero object of Cλi for every i = 1, 2, . . . , n. The group of all morphisms of an object { (λ1 , A1 ), (λ2 , A2 ), . . . , (λn , An ) } into an object { (μ1 , B1 ), (μ2 , B2 ), . . . , (μm , Bm ) } of ⊕λ∈Λ Cλ is the additive group  HomCλi (Ai , Bj ). i=1,...,n j=1,...,m λi =μj

As for the category Mat(C) defined in Section 4.6, it is convenient to think of morphisms in ⊕λ∈Λ Cλ as matrices. The composition is given by matrix multiplication. This notation and the name weak direct sum were introduced in [Facchini and Perone 2011]. The weak direct sum ⊕λ∈Λ Cλ of the preadditive categories Cλ is a preadditive category with a unique zero object w (the empty set). When the categories w Cλ are additive, so that the category λ∈Λ Cλ is defined and is additive, then λ∈Λ Cλ and w ⊕λ∈Λ Cλ are equivalent categories. A category equivalence E : λ∈Λ Cλ → ⊕λ∈Λ Cλ w  is defined as follows. If (U1 , . . . , Un ) is an object of λ∈Λ Cλ = Mat( ˙ λ∈Λ Cλ ), the equivalence E sends (U1 , . . . , Un ) to ∅ ∈ Ob(⊕λ∈Λ Cλ ) if n = 0. Otherwise, that is, when n > 0, for every i = 1, . . . , n there exists a unique index λ ∈ Λ such that Ui ∈ Ob(Cλ ). Then the equivalence E sends (U1 , . . . , Un ) to { (λ, ⊕λ = λi Ui ) | λ ∈ Λ and λ = λi for some i = 1, . . . , n }, where ⊕λ = λi Ui denotes any biproduct in Cλ of the objects Ui , where i ranges over the finite set of the indices i = 1, . . . , n with λi = λ. f  The subcategory λ∈Λ Cλ of λ∈Λ Cλ . Let Λ be a class. For every index λ ∈ Λ, f let Cλ be a preadditive category with at least  a zero object. Let λ∈Λ Cλ be the full subcategory of the product category λ∈Λ Cλ whose objects are all Λ-tuples (Aλ )λ∈Λ for which Aλ ∈ Ob(Cλ ) is a zero object in Cλ for every λ ∈ Λ except for finitely many indices λ ∈ Λ. Proposition 4.25. Let fCλ be an additive w category for every index λ in a class Λ. Then the categories λ∈Λ Cλ and λ∈Λ Cλ are equivalent.

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w f An equivalence F of the category λ∈Λ Cλ into the category λ∈Λ Cλ can f Cλ and {λ1 , . . . , λn } is the be defined as follows. If (Aλ )λ∈Λ is an object of λ∈Λ finite subclass of Λ consisting of all λ ∈ Λ with Aλ a nonzero f object in Cλ , the equivalence F can be defined to send the object (Aλ )λ∈Λ of λ∈Λ Cλ to the object w  (A1 , A2 , . . . , An ) of λ∈Λ Cλ = Mat( ˙ λ∈Λ Cλ ). a finite set, Proposition 4.25, it follows that when Λ is  Notice that from  w the product category λ∈Λ Cλ and the weak coproduct category λ∈Λ Cλ are equivalent, wso that in particular, when Λ is the empty set, the weak coproduct category λ∈Λ Cλ is just any category in which every object is a zero object.

4.9 Ideals in a Preadditive Category An ideal of a preadditive category C assigns to every pair A, B of objects of C a subgroup I(A, B) of the abelian group HomC (A, B) with the property that for every ϕ : C → A, ψ : A → B, and ω : B → D with ψ ∈ I(A, B), one has that ωψϕ ∈ I(C, D). If I is an ideal of a preadditive category C, the factor category C/I has the same objects of C, for A, B ∈ Ob(C) = Ob(C/I), the group of morphisms A → B in C/I is HomC/I (A, B) := HomC (A, B)/I(A, B), and the composition is that induced by the composition of C. Example 4.26 (Ideal generated by a class of morphisms). Let M = { ψλ : Aλ → Bλ | λ ∈ Λ } be a nonempty class of morphisms in a preadditive category C. The ideal generated by M is the ideal I of C defined as follows. For every pair A, B of objects of C, the subgroup I(A, B) nof HomC (A, B) consists of all the morphisms that can be ωi ψλi ϕi , where n ≥ 1 is an integer, λ1 , . . . , λn are not written in the form i=1 necessarily distinct indices in Λ, and ϕi : A → Aλi , ωi : Bλi → B are morphisms in C. The ideal generated by M is the smallest ideal I of C such that ψλ ∈ I(Aλ , Bλ ) for every λ ∈ Λ. Example 4.27 (Trace ideals). Let C be a preadditive category and U a subclass of Ob(C). The trace Tr(U) of U in C is the ideal of C defined as follows. For any pair A, B of objects of C, Tr(U)(A, B) := C∈U Hom(C, B) Hom(A, C). For instance, the improper ideal HomC of C coincides with Tr(Ob(C)). The zero ideal of C coincides with Tr(∅). We leave to the reader to show that for any subclass U of Ob(C), where C is a preadditive category, the trace Tr(U) is the smallest ideal I of C with I(B, B) = End(B) for every object B ∈ U. In other words, the trace Tr(U) is the ideal of C generated by the class of morphisms { 1A | A ∈ U }. Example 4.28. In this chapter, we have seen that from a preadditive category C, we can construct the free additive category Mat(C) and the idempotent completion

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C of C (Sections 4.4 and 4.6). Since morphisms between finite direct sums are matrices of morphisms, and a matrix is in an ideal I if and only if all entries of the matrix are in the ideal I, it is clear that both constructions do not change the ideals of the category. This is statement (a) in the following lemma: Lemma 4.29. Let C be a preadditive category.  (a) Every ideal I of C extends to an ideal I + of Mat(C) and to an ideal I of C. (b) There is a natural order-preserving one-to-one correspondence between ideals  of C, ideals of Mat(C), and ideals of C. To prove statement (b), notice that the restriction of an ideal of Mat(A) to A yields an inverse to the extension of an ideal I of A to the ideal I + of Mat(A).  A similar behavior occurs for the idempotent completion A. Example 4.30 (The ideal of the category associated to an ideal of the endomorphism ring of an object). Let C be a preadditive category, A an object of C, and I a two-sided ideal of the ring EndC (A). Let AI be the ideal of the category C defined as follows. A morphism f : X → Y in C belongs to AI (X, Y ) if and only if βf α ∈ I for every pair of morphisms α : A → X and β : Y → A in the category C. The ideal AI is called the ideal of C associated to I [Facchini and Pˇr´ıhoda 2009, Facchini and Pˇr´ıhoda 2011a]. Remark 4.31. The ideal AI of Example 4.30 is the greatest of the ideals I  of C with I  (A, A) ⊆ I. Thus the ideal AI associated to I must not be confused with the ideal of C generated by I, which is the smallest of the ideals I  of C with I ⊆ I  (A, A). It is easily seen that AI (A, A) = I. If A is an object of C, the ideals associated to two distinct ideals of EndC (A) are obviously two distinct ideals of the category C. Example 4.32 (Jacobson radical). We need the following lemma. Lemma 4.33. Let A be a preadditive category, A, B objects of A, and f : A → B a morphism. The following conditions are equivalent: (a) 1A − gf has a left inverse for every morphism g : B → A; (b) 1B − f g has a left inverse for every morphism g : B → A; (c) 1A − gf has a two-sided inverse for every morphism g : B → A. Proof. (a) ⇒ (b). If h(1A − gf ) = 1A , then (1B + f hg)(1B − f g) = 1B . (b) ⇒ (a). If h (1B − f g) = 1B , then (1A + gh f )(1A − gf ) = 1A . (a) ⇒ (c). Fix g : B → A. By (a), there exists h ∈ EndC (A) such that h(1A − gf ) = 1A . Thus h = 1A − (−hg)f . By (a) again, there exists h ∈ EndC (A) such that h h = 1A . In particular, h is a left inverse of h in the ring EndC (A) and 1A − gf is a right inverse of h. It follows that h = 1A − gf , hence (1A − gf )h = 1A . Thus h is a two-sided inverse of 1A − gf . (c) ⇒ (a) is trivial. 

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Let J (A, B) be the set of all morphisms f ∈ HomA (A, B) satisfying the equivalent conditions of the lemma. Then J turns out to be an ideal of the category A, called the Jacobson radical of A [Mitchell, p. 21]. Notice that J (A, A) = J(EndA (A)) for every object A of A. The quotient category A/J has zero Jacobson radical.

4.10 The Krull–Schmidt Property The next result shows that additive categories in which idempotents split are the proper setting in which to study finite direct-sum decompositions. Theorem 4.34 (Krull–Schmidt–Azumaya [Bass 1968, p. 20]). Let A be an additive category in which idempotents split. Let A1 , . . . , An be a finite family of nonzero objects of A with local endomorphism rings. Then: (1) Every direct summand of A1 ⊕· · ·⊕An is a finite direct sum of indecomposable objects. (2) If A1 ⊕ · · · ⊕ An ∼ = B1 ⊕ · · · ⊕ Bm with the objects Bj indecomposable, then ∼ Bσ(i) for n = m and there is a permutation σ of {1, 2, . . . , n} such that Ai = every i = 1, 2, . . . , n. Proof. Let A be the object A1 ⊕ · · · ⊕ An of A. We can suppose without loss of generality that A is the category proj-E, where E is the endomorphism ring EndA (A) (Theorem 2.35). That is, we can suppose A, A1 , . . . , An finitely generated projective right modules over a semiperfect ring E (cf. p. 100). Over a semiperfect ring, a finitely generated projective module is indecomposable if and only if it is the projective cover of a simple module. Now E semiperfect implies E semilocal, so that E/J(E) is semisimple Artinian and has finitely many simple right modules S1 , . . . , Sp up to isomorphism. Thus we have that the monoids with order-unit (V (add(A)), A),

(V (proj-E, EE ),

(V (proj-E/J(E), E/J(E)),

and (Np0 , (n1 , . . . , np )) are isomorphic. Here ni is the number of composition factors isomorphic to Si in a composition series of E/J(E). From this the theorem follows immediately. Let us explain this in more detail. For any finitely generated projective module PE over the semiperfect ring E, the module PE /PE J(E) is the direct sum of finitely many copies t1 , . . . , tp of S1 , . . . , Sp , respectively. If P1 , . . . , Pp are the projective covers of S1 , . . . , Sp , then PE is the sum of finitely many copies t1 , . . . , tp of P1 , . . . , Pp by Theorem 2.20. In particular, every direct summand of EE is a finite direct sum of indecomposable objects isomorphic to P1 , . . . , Pp , which implies (1), and every indecomposable direct summand of EE is isomorphic to some Pj . Thus ∼ A1 ⊕ · · · ⊕ An ∼ the isomorphisms A = = B1 ⊕ · · · ⊕ Bm in A with the objects Bj in∼ Q ⊕ decomposable in A yield corresponding isomorphisms EE ∼ = Q1 ⊕ · · · ⊕ Qn = 1

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· · ·⊕Qm in proj-E with the modules Qj indecomposable in proj-E. These, in turn, n   yield isomorphisms E/J(E) ∼ /Qm J(E) = S1n1 ⊕ · · · ⊕ Sp p ∼ = Q1 /Q1 J(E) ⊕ · · · ⊕ Qm between direct-sum decompositions into simple modules in Mod-E/J(E) with the objects Qj /Qj J(E) simple right E/J(E)-modules. Thus (2) follows from the cor responding statement for simple modules (Theorem 2.20). We say that the Krull–Schmidt property holds in an additive category A if every object of A is a biproduct of finitely many indecomposable objects, and A1 ⊕ · · · ⊕ An ∼ = B1 ⊕ · · · ⊕ Bm , where A1 , . . . , An , B1 , . . . , Bm are indecomposable objects of A, implies that n = m and there is a permutation σ of {1, 2, . . . , n} ∼ Bσ(i) for every i. with Ai = We have already considered on p. 130 the direct sum of monoids Mλ , when λ ranges over a class Λ, that is, the direct sum of a class of monoids. If all the monoids Mλ are equal to the same monoid M , we denote the direct sum ⊕λ∈Λ M by M (Λ) . We will say that a (possibly large) monoid is free if it is isomorphic to (Λ) the monoid N0 for some class Λ. It is easy to see that the following holds: Proposition 4.35. The Krull–Schmidt property holds in an additive category A if and only if the skeleton V (A) of A is a free monoid. The aim of this section is to present a characterization of additive categories with splitting idempotents in which the Krull–Schmidt property holds. Notice that if A is an additive category in which the Krull–Schmidt property holds, so that ∼ N(Λ) for some class Λ, then Λ is necessarily in one-to-one correspondence V (A) = 0 with a class of representatives up to isomorphism of the indecomposable objects of A. Example 4.36. For a ring R, there can be a proper class of pairwise nonisomorphic modules with local endomorphism ring. For instance, let R := Z be the ring of integers. We will show that there is a proper class, that is, a class that is not a set, of pairwise nonisomorphic abelian groups whose endomorphism ring is local. For this, we need a notion that will not be explained in detail in this book, the notion of cotorsion-free abelian group. Let Z(p) be the localization of Z at its maximal ideals (p). Then Z(p) is a cotorsion-free abelian group, so that every free Z(p) -module F is also a cotorsion-free abelian group. It follows that the trivial extension Z(p) ∝ F of Z(p) by F is a local commutative ring whose additive group is cotorsion-free. Now F can be any free Z(p) -module of arbitrary rank. By [G¨obel and Trlifaj, Theorem 1.7], every ring whose additive group is cotorsionfree is the endomorphism ring of a suitable abelian group. Thus we get a proper class of pairwise nonisomorphic abelian groups GF whose endomorphism rings ∼ Z(p) ∝ F are local. This shows that there is a proper class of pairwise End(GF ) = nonisomorphic abelian groups whose endomorphism rings are local. Notice that the fact that the Krull–Schmidt property holds in an additive ∼ N(Λ) , does not imply that there exists some decategory A, that is, V (A) = 0 composition of A indexed in Λ. For instance, let R be a ring and let A be the full

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subcategory of Mod-R whose objects are all right R-modules of finite composition length, so that the Krull–Schmidt property holds in A. Let { Aλ | λ ∈ Λ } be a set of representatives of the isomorphism classes of the right R-modules of finite (Λ) composition length. Then V (A) ∼ = N0 , but this decomposition of V (A) does not correspond to a decomposition of A as a weak coproduct of categories Aλ , essentially because there can be nonzero homomorphisms Aλ → Aμ for λ = μ. Recall (Section 2.3) that a ring R is said to have the invariant basis property n ∼ m (or the invariant basis number, IBN) if for every integer n, m ≥ 0, RR = RR implies n = m. Proposition 4.37. Let A be an additive category. The following conditions are equivalent: (a) Every object of A is a biproduct of finitely many indecomposable objects, the category A has exactly one indecomposable object up to isomorphism, and An ∼ = Am implies n = m for every (some) indecomposable object A of A. (b) The monoid V (A) is isomorphic to the additive monoid N0 . (c) There exists a ring R such that R has IBN and A is equivalent to the full subcategory FR of Mod-R whose objects are all finitely generated free right R-modules. Proof. The implications (a) ⇔ (b) and (c) ⇒ (a) are trivial. (a) ⇒ (c) Assume that A is a category and that condition (a) holds. In ∼ Am imparticular, let A be an indecomposable object of A for which An = plies n = m. For every object B of A, HomA (A, B) is a finitely generated free right module over the endomorphism ring R = HomA (A, A) of A. We will now show that the functor HomA (A, −) : A → Mod-R is full. If B, C ∈ Ob(A) and f : HomA (A, B) → HomA (A, C) is a right R-module morphism, then B ∼ = An and n m ∼ ∼ Rm ∼ C = A for suitable n and m, so that HomA (A, B) = RR and HomA (A, C) = R are free R-modules. Thus f corresponds to a matrix with entries in R = EndA (A), ∼ An → C ∼ so that f = HomA (A, f  ) for a morphism f  : B = = Am corresponding to the same matrix. It follows that the functor HomA (A, −) : A → FR is a cat∼ Am implies n = m in egory equivalence. In particular, from the condition An =  (a), we get that the ring R has IBN. We shall call any additive category A satisfying the equivalent conditions of Proposition 4.37 an IBN category. Every IBN category is necessarily skeletally small. We will say that an additive functor F : A → B between two additive categories A and B is: ∼ F (A ) (a) isomorphism reflecting if, for every pair A, A of objects of A, F (A) =  ∼ implies A = A ; (b) essentially surjective if every object of B is isomorphic to F (A) for some object A of A;

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(c) a weak equivalence if it is isomorphism reflecting and essentially surjective. In particular, every equivalence F : A → B is a weak equivalence. Theorem 4.38. The following conditions are equivalent for an additive category A in which idempotents split: (a) The Krull–Schmidt property holds in A. (b) There exist a class Λ,  an IBN category Aλ for every λ ∈ Λ, and a weak equivalence F : A → w λ∈Λ Aλ . Proof. (a) ⇒ (b) Let A be an additive category in which idempotents split and with the Krull–Schmidt property. Let { Aλ | λ ∈ Λ } be a class of representatives up to isomorphism of the indecomposable objects of A. For every index λ ∈ Λ, let Uλ be the subclass of Ob(A) consisting of all objects C ∈ Ob(A) such that no direct summand isomorphic to Aλ appears in a decomposition of C as a biproduct of indecomposable objects. Let Tr(Uλ ) be the trace of Uλ (Example 4.27), and consider the factor category Aλ := A/Tr(Uλ). Let Fλ : A → Aλ denote the canonical functor and let F = λ∈Λ Fλ : A → λ∈Λ Aλ denote the product functor. For every μ ∈ Λ and μ = λ, one has that Aμ ∈ Uλ , so that the identity of Aμ is in Tr(Uλ )(Aμ , Aμ ). It follows that Aμ becomes the zero object in the factor category Aλ . Hence, for every object A of A, the image F (A) is an object of the full subcatf  egory λ∈Λ Aλ of λ∈Λ Aλ . Thus F can be viewed as a functor of the category f Aλ . To conclude the proof of (a) ⇒ (c), it remains to A into the category λ∈Λ show thatall the categories Aλ are IBN categories and that the product functor f F : A → λ∈Λ Aλ is a weak equivalence. In order to prove that the categories Aλ are IBN, we will show that they satisfy condition (a) in Proposition 4.37. We have already remarked that the image of Aμ in Aλ is a zero object of Aλ for every μ = λ. It follows that every object of Aλ is a direct sum of finitely many copies of Aλ . We must show that Aλ is ∼ Am in Aλ implies n = m. If Aλ is not indecomposable in Aλ , and that Anλ = λ indecomposable in Aλ , then Aλ = B ⊕ C for nonzero objects B, C ∈ Ob(Aλ ), so p ∼ Aq in A for suitable positive that, without loss of generality, B ∼ = Aλ and C = λ integers p, q. Therefore it suffices to show that Anλ ∼ = Am λ in Aλ implies n = m. m Assume Aλn ∼ = Aλm in Aλ . Then there exist morphisms f : Aλn → Am λ and g : Aλ → n n n Aλ in the category A such that gf −1 ∈ Tr(Uλ )(Aλ , Aλ ). Thus there exist an object C in Uλ and morphisms h ∈ A(Aλn , C) and ∈ A(C, Anλ ) such that gf − 1 = h. Hence 1 = gf − h, that is, the composite morphism f ) (−h (g ) n Anλ −→Am λ ⊕ C −→Aλ

is the identity of Aλn . By Proposition 4.17, the idempotent endomorphism   of Aλm ⊕ C has a kernel kk : K → Am λ ⊕ C and   f k : Aλn ⊕ K → Am λ ⊕C −h k 



 (g )

f −h

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140

is an isomorphism. Since C ∈ Uλ , counting the number of direct summands isomorphic to Aλ in the isomorphic objects Anλ ⊕ K ∼ = Am λ ⊕ C, we find that n ≤ m. n m ∼ An ⊕ C  for Inverting the roles of Aλ and Aλ , we similarly find that Aλm ⊕ K  = λ   suitable K ∈ Ob(A) and C ∈ Uλ , so that m ≤ n. This shows that n = m and that the factor categories Aj are IBN. In order to conclude the proof, we must show that F =

 λ∈Λ

Fλ : A →

w λ∈Λ

Aλ

is a weak equivalence. wSince Fλ (Aμ ) is a zero object for every λ = μ, it is clear that every object of λ∈Λ Aλ is isomorphic to F (A) for some A ∈ Ob(A). Assume ∼ Fλ (B) in Aλ for A, B ∈ Ob(A) are such that F (A) ∼ = F (B). Then Fλ (A) = ∼ all λ ∈ Λ. Since the Krull–Schmidt property holds in A, we know that A = nt mt n1 m1 ∼ Aλ1 ⊕ · · · ⊕ Aλt and B = Aλ1 ⊕ · · · ⊕ Aλt for suitable distinct indices λi ∈ Λ and suitable nonnegative integers ni and mi . Thus Fλ (A) ∼ = Fλ (B) for every λ implies that Fλi (A) ∼ = Fλi (B) for every i, that is, Aλnii ∼ = Aλmii in Aλi , which is an IBN category. Hence ni = mi . Since this is true for every i = 1, . . . , t, we get that A∼ = B, and F is isomorphism reflecting. This proves that F is a weak equivalence. (b) ⇒ (a) Every additive functor F : A → B between two additive categories A and B induces a monoid homomorphism V (F ) : V (A) → V (B). The functor F is isomorphism reflecting if and only if the mapping V (F ) is injective, and F is a weak equivalence if and only if V (F ) is bijective. Therefore, if there is a weak w  ∼ ∼ (Λ) ∼ equivalence A → w λ∈Λ Aλ , then V (A) = V ( λ∈Λ Aλ ) = ⊕λ∈Λ V (Aλ ) = N0 because all the categories Aλ are IBN categories. Proposition 4.35 allows us to  conclude. The previous theorem shows that the notion of IBN ring seems to be related to categories with the Krull–Schmidt property in a more natural way than the notion of local ring. Example 4.39. Let A be an additive category in which idempotents split and in which the endomorphism ring of every nonzero object of A is semiperfect. By Theorem  4.34, the Krull–Schmidt property holds in A. A weak equivalence F: A → w λ∈Λ Aλ like that in condition (b) of Theorem 4.38 is given by the canonical projection F : A → A/J , where J denotes the Jacobson radical of A. It is easy to see that if { Aλ | λ ∈ Λ } is a class of representatives up to isomorphism of the indecomposable objects of A, then J (Aλ , Aμ ) = HomA (Aλ , Aμ ) for λ = μ, and J (Aλ , Aλ ) is the maximal ideal of the local ring EndA (Aλ ). It follows that w the factor category A/J is equivalent to the weak coproduct λ∈Λ Aλ , where each Aλ is the IBN category of all finite-dimensional right vector spaces over the division ring EndA (Aλ )/J (Aλ , Aλ ).

4.11. Annihilating a Class of Objects

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4.11 Annihilating a Class of Objects Let A be an additive category. Fix a subclass U of Ob(A) closed under finite direct sums. Let IU denote the trace Tr(U) of U, that is, the ideal of A generated by the class of morphisms { 1A | A ∈ U } (Example 4.27). It is easy to check that for every A, B ∈ Ob(A), one has that a morphism f ∈ HomA (A, B) is in IU (A, B) if and only if it factors through some object of U, that is, if there exist C ∈ U, g ∈ HomA (A, C), and h ∈ HomA (C, B) with f = hg. More generally, if the category A is only a preadditive category and U is a subclass of Ob(A), the trace IU of U consists, for every A, B ∈ Ob(A), of all morphisms f ∈ HomA (A, B) such that there exist n > 0, C1 , . . . , Cn ∈ U, gi ∈ HomA (A, Ci ), and hi ∈ HomA (Ci , B) for i = 1, . . . , n with f = h1 g1 + · · · + hn gn . It is well known that if AR , BR and MR = AR ⊕ BR are modules, and NR is a submodule of MR containing AR , then NR = AR ⊕ (BR ∩ NR ). In the next lemma, we give a categorical presentation of this elementary fact. It holds for any category, not only for the preadditive ones. Lemma 4.40. Let A be an object of a category C. Let e be an idempotent endomorphism of A and suppose e = gf and f g = 1C for an object C of C and suitable morphisms f : A → C and g : C → A. Assume that e = βα for suitable X ∈ Ob(C), α : A → X, and β : X → A. Then γ := f β : X → C and δ := αg : C → X are morphisms in C with e := δγ an idempotent endomorphism of X and γδ = 1C . Proof. γδ = f βαg = f eg = f gf g = 1C 1C = 1C . Therefore e := δγ is idempotent.  Proposition 4.41. Let A be an additive category in which idempotents split and let U be a subclass of Ob(A) closed under finite direct sums and direct summands. For every A, B ∈ Ob(A), A and B are isomorphic in A/IU if and only if there ∼ B ⊕ D. exist C, D ∈ U with A ⊕ C = Proof. One implication is trivial, because if A ⊕ C and B ⊕ D are isomorphic in A, then they are isomorphic in the factor category A/IU , and in this category C and D are zero. For the other implication, assume A and B are isomorphic in A/IU . Let f : A → B and g : B → A be morphisms in A such that 1A − gf ∈ IU (A, A) and 1B − f g ∈ IU (B, B). Let C ∈ U and u : A → C, p : C → A be such that 1A − gf = pu. Similarly, let D ∈ U and v : B → D, q : D → B be such that 1B − f g = qv. From Proposition 4.17, we get that 1A = (g, p) uf implies that (a)   e := uf (g, p) is an idempotent endomorphism of B ⊕ C; (b) the kernel k : K → B ⊕ C of e is also a kernel for (g, p); and (c) A ⊕ K ∼ = B ⊕ C. Thus it suffices to prove that K belongs to U. We want to apply Lemma 4.40 to the idempotent endomorphism 1B⊕C − e of B ⊕ C. For this, notice that 1B⊕C − e factors through k, because e(1B⊕C − e) = 0 and k is a kernel of e, so that 1B⊕C − e = k for a unique : B ⊕ C → K. Thus

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1B⊕C k − ek = k k, that is, k = k k. Now the kernel k is a monomorphism; hence 1K = k. Moreover,       f 1B − f g −f p qv −f p 1B⊕C − (g, p) = = −ug 1C − up −ug 1C − up u ⎛ ⎞ 0 v   ⎜ ⎟ 0 q −f p 0 1C ⎟ ⎜ 0 = , 1C 1C − up ⎝ −ug 0 ⎠ 0 0 0 1C where

and



q 0

−f p 0 0 1C ⎛

v ⎜ 0 ⎜ ⎝ −ug 0

0 1C − up

 : D⊕C ⊕C⊕C →B⊕C

⎞ 0 ⎟ 1C ⎟ : B ⊕ C → D ⊕ C ⊕ C ⊕ C. 0 ⎠ 1C

By Lemma 4.40, there is an endomorphism e of D ⊕ C ⊕ C ⊕ C that factors through K, that is, e = δγ and γδ = 1K for suitable γ : D ⊕ C ⊕ C ⊕ C → K and δ : K → D ⊕ C ⊕ C ⊕ C. By Proposition 4.17, K is a direct summand of  D ⊕ C ⊕ C ⊕ C, so that K ∈ U, as we wanted to show. Let R be a ring, Mod-R the category of right R-modules, and U := Proj-R the class of all projective right R-modules. The factor category (Mod-R)/IU is called the stable category, and is denoted by Mod-R. Two modules M and N are said to be stably isomorphic if they are isomorphic objects in the stable category Mod-R. If mod-R is the full subcategory of Mod-R whose objects are all finitely presented right R-modules, one defines the stable category mod-R as the factor category of the full subcategory mod-R modulo the ideal of all morphisms that can be factored through a projective module, that is, the factor category (mod-R)/IU . Similarly for finitely presented left R-modules, where one gets the category R-mod. If A, B are additive categories and F : A → B is an additive functor, we say that F is: (1) direct-summand reflecting if for every pair A, B of objects of A with F (A) isomorphic to a direct summand of F (B), A is isomorphic to a direct summand of B. (Here, if A and B are objects of an additive category C, we say that A is isomorphic to a direct summand of B if there exists an object C of C such that B is a biproduct of A and C.) (2) weakly direct-summand reflecting if for every pair A, B of objects of A with F (A) isomorphic to a direct summand of F (B), there exists an object C of A with F (C) = 0 and A isomorphic to a direct summand of B ⊕ C. Obviously, direct-summand reflecting implies weakly direct-summand reflecting.

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Notice that every additive functor F : A → B induces a monoid homomorphism V (F ) : V (A) → V (B) between the (possibly large) additive monoids V (A) and V (B). The functor F is isomorphism reflecting if and only if V (F ) is an injective mapping, essentially surjective if and only if V (F ) is a surjective mapping, and it is direct-summand reflecting if and only if the monoid morphism V (F ) is a divisor homomorphism, weakly direct-summand reflecting if and only if the monoid morphism V (F ) is an essential morphism, a weak equivalence if and only if V (F ) is a monoid isomorphism. If A is an additive category, N is a (possibly large) monoid, ϕ : V (A) → N is a monoid morphism, and Uϕ := { A ∈ Ob A | ϕ(A) = 0 }, Iϕ will denote the two-sided ideal IUϕ of A. Theorem 4.42. Let A be an additive category in which idempotents split, N a directly finite, reduced (possibly large) additive monoid, and ϕ : V (A) → N an essential monoid morphism. Then the position A → ϕ(A) for all A ∈ Ob(A/Iϕ ) defines a monoid isomorphism V (A/Iϕ ) → ϕ(V (A)). Proof. To prove the theorem, we must show that for every A, B ∈ Ob(A), A is isomorphic to B in A/Iϕ if and only if ϕ(A) = ϕ(B). We claim that if A ≤ B in V (A/Iϕ ), then ϕ(A) ≤ ϕ(B). To prove the claim, suppose A ≤ B in V (A/Iϕ ). That is, there exist morphisms f : A → B and f  : B → A in A such that f  f − 1A ∈ Iϕ (A, A). Equivalently, there exist an object C ∈ Uϕ and morphisms g ∈ A(A, C), h ∈ A(C, A) with f  f − 1A = hg. Thus 1A is the composite mapping of the morphisms A

f ) (−g

/ B ⊕ C (f



, h)

/ A.

By Proposition 4.17, the morphism (f  , h) : B ⊕ C → A has a kernel K in A and K ⊕ A and B ⊕ C are isomorphic objects in A. Hence, applying ϕ(−), we find that ϕ(A) ≤ ϕ(K ⊕ A) = ϕ(B ⊕ C) = ϕ(B) + ϕ(C) = ϕ(B). This concludes the proof of the claim. We are ready to prove that for every A, B ∈ Ob(A), A ∼ = B in A/Iϕ if and only if ϕ(A) = ϕ(B). If A ∼ = B in A/Iϕ , then A ≤ B and B ≤ A, so that from the claim, we know that ϕ(A) ≤ ϕ(B) and ϕ(B) ≤ ϕ(A). Since N is directly finite, the algebraic preorder ≤ is a partial order on N , so that ϕ(A) = ϕ(B), as desired. As a consequence of the implication we have proved in the previous paragraph, we see that the category A/Iϕ is directly finite, that is, X ∼ = X ⊕ Y implies Y = 0 for every X, Y ∈ Ob(A/Iϕ ), because if X, Y ∈ Ob(A) and X ∼ = X ⊕ Y in A/Iϕ , then ϕ(X) = ϕ(X) + ϕ(Y ), so that ϕ(Y ) = 0. Thus Y ∈ Uϕ , and Y = 0 in A/Iϕ .

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Finally, to prove the remaining implication, assume that A and B are objects of A such that ϕ(A) = ϕ(B). Now ϕ is an essential homomorphism, so that ϕ(A) ≤ ϕ(B) implies that there exist C, D ∈ Ob(A) with ϕ(C) = 0 and ∼ B in A/Iϕ , that is, A is isomorphic ∼ B ⊕ C in A. Then A ⊕ D = A⊕D = to a direct summand of B. Interchanging the roles of A and B, we get that B is isomorphic to a direct summand of A in A/Iϕ . But as we have seen in the previous paragraph, the category A/Iϕ is directly finite, so that A and B are isomorphic in the category A/Iϕ .  We now combine the ideas developed in this section with the essential morphisms of Section 1.4. Theorem 4.43 (Isomorphism theorem). Let A be an additive category in which idempotents split, B a directly finite additive category, and F : A → B a weakly direct-summand reflecting additive functor. Then: (a) The class U := { A ∈ Ob(A) | F (A) = 0 } is closed under isomorphism, finite direct sums and direct summands, and for every A, B ∈ Ob A and every C ∈ U, A ⊕ B ∼ = A ⊕ C implies B ∈ U. (b) Factor category A/IU is a directly finite category, and the functor F induces an isomorphism reflecting, direct-summand reflecting functor A/IU → B. Conversely, let A be an additive category in which idempotents split. Suppose that U is a subclass of Ob(A) closed under isomorphism, finite direct sums, and direct summands, and with the property that for every A, B ∈ Ob A and every C ∈ U, A ⊕ B ∼ = A ⊕ C implies B ∈ U. Then: (a) The canonical functor P : A → A/IU is weakly direct-summand reflecting. (b) The factor category A/IU is directly finite. (c) U = { A ∈ Ob A | P (A) = 0 }. The proof is direct and easy, and we leave it to the reader.

4.12 Local Functors, Almost Local Functors and Amenable Semisimple Categories The notion of local morphism between two rings is easily generalizable to that of local functor between two preadditive categories. If A, B are preadditive categories and F : A → B is an additive functor, we say that the functor F is local if for every pair A, A of objects of A and every morphism f : A → A in A, F (f ) an isomorphism in B implies f an isomorphism in A. We say that an additive functor F : A → B between preadditive categories is almost local if for every object A of A and every endomorphism f of A, F (f ) an automorphism implies f an automorphism. Thus every local functor is almost local.

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145

Any additive functor F of a preadditive category A into a preadditive category B induces a ring morphism FA : EndA (A) → EndB (F (A)) for every object A of A. An additive functor F : A → B is almost local if and only if the ring morphisms FA are local morphisms for all A ∈ Ob(A). Example 4.44. The composite functor G ◦ F of two local functors F : A → B and G : B → C is a local functor. Conversely, if the composite functor G ◦ F of two additive functors F : A → B and G : B → C is a local functor, then F is a local functor. Example 4.45. Let us show that if A is a preadditive category, J is its Jacobson radical, and F : A → A/J is the canonical functor, then F is a local functor. Let f : A → A be a morphism in A with F (f ) an isomorphism in A/J . Let g : A → A be a morphism in A with F (g) = (F (f ))−1 . Then 1A − gf ∈ J (A, A), so that gf is invertible in A. Similarly, f g is invertible in A. Thus f is an isomorphism in A. More generally, it is easily seen that if A is a preadditive category and I is any ideal of A contained in the Jacobson radical, the canonical functor A → A/I is a local functor. Example 4.46. The kernel of any local functor F : A → B is contained in the Jacobson radical J of A. This generalizes Lemma 3.24(a). Let f : A → A be a morphism in A with f ∈ ker F (A, A ). Let g : A → A be any morphism in A. Then F (1A − gf ) = 1F (A) . Since F is local, it follows that 1A − gf is invertible in A. Thus f ∈ J (A, A ). In particular, if A is a preadditive category and I is any ideal of A, the canonical functor A → A/I is a local functor if and only if I is contained in the Jacobson radical of A. Example 4.47. A full functor F : A → B is a local functor if and only if its kernel is contained in the Jacobson radical J of A. To see this, it suffices to show, by Example 4.46, that if F is a full functor and ker F ⊆ J , then F is local. Now if f : A → A is a morphism in A and F (f ) is an isomorphism in B, then there exists a morphism g : A → A for which F (g) is the inverse of F (f ). Thus both 1A − gf and 1A − f g are in ker F , hence in J . Now it is possible to conclude, as in Example 4.45, that the morphism f is an isomorphism in A. Considering the ring embedding Z → Q as a functor between two preadditive categories with one object, it is easily seen that there exist nonfull nonlocal functors F for which ker F ⊆ J . Example 4.48. Here is an example of an almost local functor that is not a local functor. Let A be the full subcategory of the category Mod-R whose objects are all Artinian modules. Let soc: A → A be the functor that associates to every Artinian module its socle, which is necessarily an essential submodule of the Artinian module. It is easy to construct a proper submodule A of an Artinian module A with soc(A) = soc(A ) (consider, for instance, pZ/p2 Z ⊂ Z/p2 Z). The embedding A → A shows that the functor soc is not local. But if f is an endomorphism of an

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Artinian module A and the restriction of f to the socle is an isomorphism, then f is a monomorphism because the socle is essential, so that f is an automorphism of A by Lemma 3.18. This shows that the functor soc : A → A is almost local. Let F : A → B be an additive full functor of an additive category A into a preadditive category B. Then F is local if and only if it is almost local. To prove this, assume F almost local. Then, for every pair of objects A, A of A, F induces a local morphism EndA (A ⊕ A ) → EndB (F (A ⊕ A )). By Lemma 3.24, the kernel of this morphism is contained in the Jacobson radical of EndA (A ⊕ A ). It follows that any morphism f : A → A in A that is mapped to zero by F is necessarily contained in J (A, A ). That is, the kernel of F is contained in the Jacobson radical J of A. It is now possible to conclude by Exercise 4.47. Example 4.49. A local full functor is isomorphism reflecting, as is easily seen. There is no relation, that is, no trivial implication, between being a directsummand reflecting functor, an isomorphism reflecting functor, or a local functor in general, as we show in the following examples. Example 4.50. Let B be the additive category of all finite-dimensional vector spaces over a field k, and let A be the full subcategory of B whose objects are the nonzero vector spaces. The embedding functor A → B is both isomorphism reflecting and local, but not a weakly direct-summand reflecting functor. Example 4.51. Let A be the category of all finitely generated free abelian groups and let B be the category of all finite-dimensional vector spaces over the field Q of rational numbers. The functor − ⊗ Q : A → B is both isomorphism reflecting and direct-summand reflecting, but is not a local functor. Example 4.52. In this example, we consider a particular case of a construction we have already seen at the end of Section 2.3. Let k be a field and n a positive integer. Let xi , yi (i = 1, 2, . . . , n) be 2n noncommutative indeterminates, and let kxi , yi | i = 1, 2, . . . , n be the free k-algebra with these 2n free generators. Let X be the 1 × n matrix with (1, i) entry the indeterminate xi , and let Y be the n × 1 matrix with (i, 1) entry the indeterminate yi . The 1×1 matrix XY −1 and the n×n matrix Y X − 1n have their entries in the free k-algebra kxi , yi | i = 1, 2, . . . , n. Let I be the two-sided ideal of the algebra kxi , yi | i = 1, 2, . . . , n generated by the 1 + n2 entries of these two matrices XY − 1 and Y X − 1n . Set Rn := kxi , yi | i = 1, 2, . . . , n/I. It is easily seen that the free right Rn -modules Rnn and Rn are isomorphic. (In fact, it would be possible to prove that all projective modules over the k-algebra Rn are free, see [Bergman 1974b, Theorem 6.1], and that a complete set of all nonisomorphic finitely generated free Rn -modules is given by the n modules {0, Rn , Rn2 , . . . , Rnn−1 }.) The position

x1 x1 → x2 →

x2 x1

y1 →

y1 y2 → y1 y2

x3 → x22

y3 → y22

4.12. Local Functors, Almost Local Functors and . . .

147

defines a ring homomorphism R3 → R2 . The functor − ⊗R3 R2 : {0, R3 , R32 } → {0, R2 } turns out to be direct-summand reflecting, but not isomorphism reflecting. Example 4.53. In Chapter 5, we will define, for every Grothendieck category A, its spectral category SpecA, the dual construction A , and two canonical functors P : A → SpecA and F : A → A . The functor P × F : A → SpecA × A is a local functor, as we will see in Proposition 5.42, and it is possible to prove that P × F is neither weakly direct-summand reflecting nor isomorphism reflecting [Facchini 2007, Example 3.4]. Now we will determine, for a full subcategory A of a preadditive category B and an ideal I of B, when the canonical projection functor C : A → A/I is local. Since the functor C is full, this is equivalent to requiring that the kernel I of C be contained in the Jacobson radical J of A. In particular, for every object A of A, it follows that I(A, A) ⊆ J(EndB (A)). In the next theorem, we show that the largest full subcategory of B for which the functor C : C → C/I is local is the full subcategory C of B whose objects are all the objects A of B with I(A, A) ⊆ J(EndB (A)). Theorem 4.54. Let B be a preadditive category and I an ideal of B. Let C be the full subcategory of B whose objects are all the objects A of B with I(A, A) ⊆ J(EndB (A)). Then the ideal I, restricted to the category C, is contained in the Jacobson radical J of C, so that the canonical functor C : C → C/I is local. Moreover, the category C is the largest full subcategory of B with C : C → C/I a local functor. Finally, if B is an additive category, then C is also an additive category, and if B is additive and idempotents split in B, then idempotents also split in C. Proof. In order to show that the ideal I, restricted to the category C, is contained in the Jacobson radical J of C, we must prove that I(A, B) ⊆ J (A, B) for every pair of objects A, B of C. Let f ∈ I(A, B) and let g : B → A be any morphism. Then gf ∈ I(A, A) ⊆ J(End(A)), so that 1A − gf is an automorphism of A. It follows that f ∈ J (A, B), so I ⊆ J on C. It is now obvious that C is the largest full subcategory of B with this property. Now assume B additive. To prove that the full subcategory C of B is additive, we must show the class of all objects A of B with I(A, A) ⊆ J(EndB (A)) is closed under finite coproducts. Notice that   EndB (A) HomB (B, A) EndB (A ⊕ B) = EndB (B) HomB (A, B) 

and K(A ⊕ B, A ⊕ B) =

K(A, A) K(A, B)

K(B, A) K(B, B)



for any ideal K of C. Thus if A and B are objects such that I(A, A) ⊆ J(EndB (A)) and I(B, B) ⊆ J(EndB (B)), then I(A ⊕ B, A ⊕ B) ⊆ J(EndB (A ⊕ B)) by what we have seen in the previous paragraph.

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Finally, suppose that B is additive and idempotents split in B. Let f : C → C be an idempotent endomorphism in C. Then C ∈ Ob(C) and there are an object B ∈ Ob(B) and morphisms g : C → B and h : B → C such that f = hg. Then C = B⊕K for a suitable object K of B. Reasoning as in the previous paragraph, we find that I(B ⊕K, B ⊕K) ⊆ J(EndB (B ⊕K)) implies I(B, B) ⊆ J(EndB (B)).  In the statement of the next theorem, the category of all finite-dimensional right vector spaces over a division ring k will be denoted by vect-k. Theorem 4.55. Let A be an additive category. The following conditions are equivalent: (a) Idempotents split in A, and the endomorphism ring EndA (A) of every nonzero object A of A is semisimple Artinian. (b) There exist a class Λ and, for each wλ ∈ Λ, a division ring kλ such that A is vect-kλ . equivalent to the weak coproduct λ∈Λ Proof. (a) ⇒ (b). Let A be an additive category in which idempotents split and the endomorphism ring of every object is semisimple Artinian. With the axiom of choice for classes, we can find a class { Aλ | λ ∈ Λ } of representatives of the indecomposable objects of A up to isomorphism. Since the endomorphism rings in A are all semisimple Artinian and idempotents split in A, an object A of A is indecomposable if and only if its endomorphism ring EndA (A) is a division ring. For every λ ∈ Λ, let kλ be the endomorphism ring of Aλ . Since the endomorphism rings in A are all semisimple Artinian, every object A of A decomposes as a direct sum of finitely many objects whose endomorphism rings are division rings, hence as a direct sum of objects indecomposable in A. Let λ, μ ∈ Λ ⊕ Aμ ) is isomorphic to the be indices with Hom  A (Aλ , Aμ ) = 0. Then EndA (Aλ  EndA (Aλ ) HomA (Aμ , Aλ ) matrix ring E = . This is a semisimple ArEndA (Aμ ) HomA (Aλ , Aμ ) tinian ring  by (a). Let f : Aλ → Aμ be a nonzero morphism in A. The element  0 0 ∈ E is nonzero and induces by left multiplication a nonzero morphism f 0   1 Aλ 0 E of E into the of right E-modules of the indecomposable right ideal 0 0   0 0 indecomposable right ideal E of E. Now indecomposable right ide0 1 Aμ als are simple right E-modules because the ring E is semisimple Artinian. Thus the nonzero morphism induced by left multiplication is a right E-module isomorphism, and therefore it hasan inverse isomorphism. This inverse isomorphism    1 Aλ 0 0 0 E → E is given by left multiplication by an element 0 0 0 1 Aμ   0 g α ∈ E of the form α = . Then g : Aμ → Aλ is a morphism in A with the 0 0 property that gf = 1Aλ and f g = 1Aμ . Thus Aλ ∼ = Aμ , which implies that λ = μ.

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149

We have thus proved that λ = μ implies HomA (Aλ , Aμ ) = 0. Moreover, the endomorphism rings of the objects Aλ are division rings kλ , and every object A of A is a biproduct of finitely many objects with local endomorphism ring kλ . By Theorem 4.34, for every object A of A there are only finitely many indices λ with A(Aj , A) = 0. The product functor F =



HomA (Aλ , −) : A →

λ∈Λ



vect-kλ

λ∈Λ

f restricts therefore to an equivalence of A into the full subcategory λ∈Λ vect-kλ  f of λ∈Λ vect-kλ (seep. 133). Finally, the category λ∈Λ vect-kλ is equivalent to w the weak coproduct λ∈Λ vect-kλ . The implication (b) ⇒ (a) is trivial.



The additive categories satisfying the equivalent conditions of Theorem 4.55 are called amenable semisimple [Mitchell, p. 20]. Notice that every amenable semisimple category is necessarily abelian.

4.13 An Application: Endomorphism Rings of Finitely Generated Modules over Commutative Semilocal Rings As a first application of almost local functors and amenable semisimple categories, we will prove that every finitely generated module over a commutative semilocal ring has a semilocal endomorphism ring. We need the following elementary fact about finitely generated modules over a commutative ring. Proposition 4.56. Let M be a finitely generated module over a commutative ring R. Let I be an ideal of R such that M I = M . Then there exists a ∈ I such that M (1 + a) = 0. of M . For every j = 1, . . . , n, Proof. Let {x1 , . . . , xn } be a finite set of generators n there exist a1j , . . . , anj ∈ I such that xj = i=1 xi aij . Thus we get n equalities n 

xi (δij − aij ) = 0,

(4.1)

i=1

denote the cofactor of the (i, j) where δij denotes the Kronecker delta. Let Dij n entry of the n×n matrix D = (δij −aij )ij , so that j=1 (δij −aij )Dkj = δik ·det(D).

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Multiply the jth equality in (4.1) by Dkj and sum the n equalities obtained in this n x (δ − aij )) Dkj = 0 for every k = 1, . . . , n. Then way, getting that nj=1 ( i=1   i ij n n i=1 xi j=1 (δij − aij )Dkj = 0, that is, xk det(D) = 0 for every k = 1, . . . , n. Thus M det(D) = 0, and clearly det(D) is of the form 1 + a for some a ∈ I.



Proposition 4.57 ([Vasconcelos 1969, Proposition 1.2]). Every surjective endomorphism of a finitely generated module M over a commutative ring R is an automorphism. Proof. Let f : M → M be a surjective endomorphism of M . Let R[t] be the ring of polynomials in the indeterminate t. The module M can be given an R[t]module structure by defining xt = f (x) for every x ∈ M . Apply the previous proposition to the finitely generated R[t]-module M and to the principal ideal I of R[t] generated by t. Thus there exists a ∈ I such that M (1 + a) = 0, that is, there exist r1 , . . . , rn ∈ R such that M (1+r1 t+· · ·+rn tn ) = 0. Then 1+r1 f +· · ·+rn f n is the zero endomorphism of M . Thus f (x) = 0 implies x = 0 for every x ∈ M ,  that is, f is a monomorphism. Notice that similarly, every surjective endomorphism of a finitely generated right module M over a right Noetherian ring R is an automorphism [Goodearl and Warfield, Corollary 4.25]. Theorem 4.58. The endomorphism ring of any finitely generated module over a commutative semilocal ring is semilocal. Proof. For any ring S, let mod-S denote the full subcategory of Mod-S whose objects are all finitely generated right S-modules. We will show that if R is a commutative semilocal ring, the functor − ⊗R R/J(R) : mod-R → mod-R/J(R) is an almost local functor. In order to prove this, let f : AR → AR be an R-module endomorphism with f ⊗R R/J(R) : AR /AR J(R) → AR /AR J(R) an isomorphism. By Nakayama’s lemma, f must be an epimorphism. By Proposition 4.57, f is an automorphism. This shows that − ⊗R R/J(R) is an almost local functor. Thus, for every finitely generated R-module MR , there is a local morphism End(MR ) → End(MR ⊗R R/J(R)) ∼ = End(MR /MR J(R)). Since R is commutative and semilocal, the ring R/J(R) is a direct product of finitely many fields, so that the endomorphism ring End(MR /MR J(R)) of its finitely generated module MR /MR J(R) is a semisimple Artinian ring. Thus we conclude by Corol lary 3.28. For instance, if R is a commutative semilocal ring with n maximal ideals and AR is an R-module that can be generated with m elements, then codim(End(AR )) ≤ nm,

4.14. Semilocal Categories

151

because every surjective endomorphism of AR is bijective (Proposition 4.57) and m ) (Proposition 3.8(c)). codim(AR ) ≤ codim(RR Remark 4.59. In the proof of Theorem 4.58, we saw that for a commutative semilocal ring R, the functor − ⊗R R/J(R) : mod-R → mod-R/J(R) is almost local. It is not a local functor, as can be seen with the ring R = Z/4Z and the canonical projection Z/4Z → Z/2Z. From Propositions 3.35 and 4.58, we get the next result, which extends [Warfield 1980, Lemma 2.3]. In that lemma, Warfield had considered the case in which S was a (not necessarily commutative) algebra over a semilocal commutative ring R, S was finitely generated as an R-module, and MS was a finitely generated S-module. Proposition 4.60. If S is a (not necessarily commutative) algebra over a semilocal commutative ring R and MS is an S-module that is finitely generated as an Rmodule, then the endomorphism ring of MS is a semilocal ring. From Theorem 4.58, we get that if we consider a finitely generated module MR over a commutative semilocal ring R, then V (MR ) turns out to be a finitely generated reduced Krull monoid. Conversely, every finitely generated reduced Krull monoid arises in this way from a finitely generated module MR over a Noetherian commutative semilocal ring [Wiegand 2001].

4.14 Semilocal Categories Definition 4.61. A semilocal category is a preadditive category with a nonzero object such that the endomorphism ring of every nonzero object is a semilocal ring. The following are examples of semilocal full subcategories of the category Mod-R: (a) The full subcategory of Mod-R whose objects are all Artinian R-modules (Theorem 3.19). (b) If R is a semilocal ring, the full subcategory of Mod-R whose objects are all finitely presented R-modules (Theorem 5.48). (c) If R is a semilocal commutative ring, the full subcategory of Mod-R whose objects are all finitely generated R-modules (Theorem 4.58). Several other examples can be obtained by taking as class of objects one of the several classes of modules with semilocal endomorphism rings that we will see in Chapter 5.

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Let us move on to describe the structure of semilocal categories. For any  This is always preadditive category A, add(A) will denote the category Mat(A). an additive category with splitting idempotents, for any preadditive category A. Remark 4.62. Every additive functor F : A → B, where A, B are preadditive categories, can be extended to two additive functors Mat(F ) : Mat(A) → Mat(B) and add(F ) : add(A) → add(B), but it is possible that F is local, but Mat(F ) and add(F ) are not local functors. For instance, there exist two rings R, S and a local morphism ϕ : R → S that induces a nonlocal morphism M(F ) : Mn (R) → Mn (S) between the corresponding rings of matrices [Facchini and Herbera 2000b, p. 189]. Now let A and B denote the preadditive categories with only one object ∗ and in which the endomorphism rings of ∗ are R and S, respectively. Let F : A → B be the functor with F (∗) = ∗ and F (r) = ϕ(r) for every r ∈ R = EndA (∗). Then the extensions Mat(F ) : Mat(A) → Mat(B) and add(F ) : add(A) → add(B) are not local functors. Proposition 4.63. Let A be a preadditive category with a nonzero object, and let J be its Jacobson radical. Then the category A is semilocal if and only if the factor category A/J is equivalent to a full subcategory of an amenable semisimple category. Proof. Let A be a semilocal category, so that the endomorphism ring of every nonzero object of A is semilocal. Since J (A, A) = J(EndA (A)) for every object A of A, the category A/J has a nonzero object because the image of every nonzero object of A is a nonzero object of A/J , and the endomorphism ring of every nonzero object of A/J is semisimple Artinian. Thus the category A/J is equivalent to a full subcategory of the category add(A/J ). Clearly, all the nonzero objects of add(A/J ) have a semisimple Artinian endomorphism ring. Hence the category add(A/J ) is an amenable semisimple category. For the converse, let A/J be a full subcategory of an amenable semisimple category. The endomorphism ring of every nonzero object in the category A/J is semisimple Artinian. But J (A, A) = J(EndA (A)), so that the endomorphism ring  of every nonzero object of A is semilocal. Thus A is a semilocal category. The next theorem is the categorical version of Corollary 3.28. Theorem 4.64. The following conditions are equivalent for a preadditive category A with a nonzero object. (a) The category A is semilocal. (b) There exists a local functor F : A → B for some semilocal category B. (c) There exists a local functor F : A → B for some amenable semisimple category B. (d) There exists an almost local functor F : A → B for some semilocal category B. (e) There exists an almost local functor F : A → B for some amenable semisimple category B.

4.15. Some Realization Theorems

153

Proof. (a) ⇒ (b) and (a) ⇒ (d). Take as F the identity functor A → A. (b) ⇒ (c). The functor required in (c) is the composition of the following three functors: (1) the local functor F : A → B with B semilocal, which exists by (b); (2) the canonical functor B → B/J , where J is the Jacobson radical of B, which is a local functor by Example 4.45; a full and faithful functor B/J → B  of the factor category B/J into an amenable semisimple category B  , which exists by Proposition 4.63. Full and faithful functors are also local functors. By Example 4.44, the composite functor of these three local functors is a local functor A → B. (c) ⇒ (e) is trivial. (e) ⇒ (a) Assume that (e) holds and fix a nonzero object A of A. We must prove that its endomorphism ring is semilocal. Now the image F (A) of A in the amenable semisimple category B is nonzero. Otherwise the zero morphism A → A would be mapped by F into the zero morphism F (A) = 0 → F (A) = 0, which is an automorphism. Since F is almost local, the zero morphism A → A would be an automorphism of A, which implies that A is a zero object, contradiction. This proves that F (A) is nonzero. Now the local functor F : A → B induces a local morphism FA : End(A) → End(F (A)) of End(A) into the semisimple Artinian ring End(F (A)). It follows that End(A) is a semilocal ring by Corollary 3.28. (d) ⇒ (e). The functor required in (e) is the composition of the functor F : A → B that exists by (d); the canonical functor B → B/J , where J is the Jacobson radical of B, which is a local functor by Example 4.45; and a full and faithful functor B/J → B  of the factor category B/J into an amenable semisimple category B  , which exists by Proposition 4.63. Clearly, the composite functor of  these three almost local functors is an almost local functor A → B  .

4.15 Some Realization Theorems In this section, we will discuss some realization theorems without presenting the complete proofs. Let us give an example to explain what we mean by realization theorem. If R is a semilocal ring and π : R → R/J(R) is the canonical projection of R onto R modulo its Jacobson radical, then π : R → R/J(R) is a surjective local morphism, so that V (π) : V (R) → V (R/J(R)) is an injective divisor homomorphism (Proposition 3.29). Moreover, the injective divisor homomorphism V (π) is a morphism of monoids with order-units of (V (R), RR ) into (V (R/J(R)), R/J(R)R/J(R ). Since R/J(R) is semisimple Artinian, V (R/J(R)) is a finitely generated free ∼ (Nn , (d1 , . . . , dn )) commutative monoid, so that (V (R/J(R)), R/J(R)R/J(R ) = 0 for some n ≥ 1 and positive integers d1 , . . . , dn (the order-units in N0n are exactly the n-tuples of positive integers). Our first realization theorem says that this situation can be “realized in full generality.”

Chapter 4. Additive Categories

154

Theorem 4.65 ([Facchini and Herbera 2000a, Theorem 6.1]). Let k be a field, let (M, u) be a commutative monoid with order-unit, let (d1 , . . . , dn ) be an orderunit in the finitely generated free commutative monoid Nn0 , and let f : (M, u) → (N0n , (d1 , . . . , dn )) be a divisor homomorphism that is injective and a morphism of monoids with order-units. Then there exist a semilocal hereditary k-algebra R and two isomorphisms of monoids with order-units g : (M, u) → (V (R), RR ) and h : (N0n , (d1 , . . . , dn )) → (V (R/J(R)), R/J(R)R/J(R ) such that if π : R → R/J(R) denotes the canonical projection, the diagram of monoids with order-units f

(M, u) g

 (V (R), RR )

/ (Nn , (d1 , . . . , dn )) 0

(4.2)

h

V (π)

 / (V (R/J(R)), R/J(R)R/J(R )

is commutative. The previous theorem also has the following version concerning partially ordered abelian groups with order-units instead of commutative monoids with orderunit (Theorem 4.66). If R is a semilocal ring, then K0 (π) : K0 (R) → K0 (R/J(R)) is an embedding of partially ordered abelian groups with order-units (i.e., it is an injective mapping and induces an order isomorphism of K0 (R) onto the image of K0 (π) endowed with the order induced by the order of K0 (R/J(R))). Since R/J(R) is semisimple Artinian, K0 (R/J(R)) is a finitely generated free abelian group. More precisely, (K0 (R/J(R)), ≤, [R/J(R)R/J(R ]) ∼ = (Zn , ≤, (d1 , . . . , dn )) for some n ≥ and positive integers d1 , . . . , dn . The order on Zn is the componentwise order. Theorem 4.66 ([Facchini and Herbera 2000a, Theorem 6.3]). Let k be a field, let n be a positive integer, let u = (d1 , . . . , dn ) be an order-unit in Zn with the componentwise order, let G be a partially ordered subgroup of Zn with u ∈ G, and let f : (G, ≤, u) → (Zn , ≤, u) be the embedding. Then there exist a semilocal hereditary k-algebra R and two isomorphisms of partially ordered groups with order-units g : G → K0 (R) and h : Zn → K0 (R/J(R)) such that if π : R → R/J(R) denotes the canonical projection, then there is a commutative diagram of partially ordered abelian groups with order-units (G, ≤, u)

f

g

h



(K0 (R), ≤, [RR ])

/ (Zn , ≤, u)

K0 (π)

 / (K0 (R/J(R)), ≤, [R/J(R)R/J(R ]).

(4.3)

4.15. Some Realization Theorems

155

Theorem 4.65 has been generalized in various directions. For instance, Theorem 4.65 will be extended in Theorem 4.68 from the case of finitely generated reduced Krull monoids to that of arbitrary reduced Krull monoids. First we need a lemma. Lemma 4.67. Let R be a ring. A finitely generated projective R-module PR has a semilocal endomorphism ring End(PR ) if and only if it is the projective cover of a finitely generated semisimple right R-module. Proof. If PR is a finitely generated projective module and End(PR ) is semilocal, then PR /PR J(R) is a finitely generated projective R/J(R)-module and its endomorphism ring E := EndR/J(R) (PR /PR J(R)) is isomorphic to End(PR )/J(End(PR )) by Corollary 2.25. Thus E is semisimple Artinian, so that there exists a complete set of orthogonal idempotents e1 , . . . , en ∈ E, each ei E is a simple E-module, PR /PR J(R) is the direct sum of its n R/J(R)-submodules ei (PR /PR J(R)), and EndR/J(R) (ei (PR /PR J(R))) ∼ = ei Eei is a division ring for every i. Thus PR /PR J(R) is a direct sum of n finitely generated projective indecomposable R/J(R)-modules each of which has an endomorphism ring that is a division ring. By Lemma 3.9, each of these direct summands is the R/J(R)-projective cover of a simple module. It is easily seen that the R/J(R)-projective cover of a simple module is a simple module. Thus PR /PR J(R) is the direct sum of n simple R-modules, that is, PR /PR J(R) is a finitely generated semisimple right R-module. Hence PR is the projective cover of a finitely generated semisimple right R-module. Conversely, if PR is the projective cover of a finitely generated semisimple right R-module SR , then SR ∼ = PR /K for some superfluous submodule K of PR . Thus K ⊆ PR J(R). But the semisimple module SR is annihilated by J(R), so that PR J(R) ⊆ K. Hence K = PR J(R), so SR ∼ = PR /PR J(R). Since SR is semisimple and finitely generated, its endomorphism ring End(SR ) is semisimple Artinian. But End(SR ) ∼ = End(PR /PR J(R)) ∼ = End(PR )/J(End(PR )). Hence  End(PR )/J(End(PR )) is semisimple Artinian, i.e., End(PR ) is semilocal. Let R be a ring, SR the full subcategory of Mod-R whose objects are all finitely generated projective R-modules PR with a semilocal endomorphism ring End(PR ), that is, projective covers of finitely generated semisimple right R-modules. For this category also, the canonical projection π : R → R/J(R) induces a monoid morphism V (Sπ ) : V (SR ) → V (SR/J(R) ). Theorem 4.68 ([Facchini and Wiegand, Theorem 2.1]). Let k be a field, M an additive monoid, and I a set. Let f : M → N(I) be a divisor homomorphism that is an injective mapping. Then there are a k-algebra R and two monoid isomorphisms (I) M → V (SR ) and N0 → V (SR/J(R) ) such that if V (Sπ ) : V (SR ) → V (SR/J(R) ) is the homomorphism induced by the canonical projection π : R → R/J(R), then the

Chapter 4. Additive Categories

156

diagram M

f

∼ =

 V (SR )

/ N(I) 0 ∼ =

V (Sπ )

 / V (SR/J(R) )

commutes. Notice that the ring R whose existence is guaranteed by Theorem 4.68 is not semilocal in general. Hence the ring R/J(R) is not necessarily semisimple. However, the indecomposable modules in SR/J(R) , i.e., the finitely generated indecomposable projective R/J(R)-modules with semilocal endomorphism rings, are simple modules. To see this, let P be a finitely generated indecomposable projective R/J(R)-module with a semilocal endomorphism ring. By Corollary 2.25, J(EndR/J(R) (P )) = HomR/J(R) (P, rad(P )) = 0. Therefore the ring EndR/J(R) (P ) is a semisimple ring. Since P is indecomposable, the ring EndR/J(R) (P ) does not have nontrivial idempotents. In particular, EndR/J(R) (P ) is a division ring. Since EndR/J(R) (P ) is a local ring, P is the projective cover of a simple module by [Anderson and Fuller, Proposition 17.19]. But rad(P ) = 0, so that P itself is a simple module. The realization Theorem 4.65 has also been extended to the case of infinitely generated projective modules [Herbera and Pˇr´ıhoda 2009]. By Theorem 2.17, every projective module is a direct sum of countably generated projective modules. This reduces the study of arbitrary projective modules to that of the countably generated ones. Instead of considering the full subcategory proj-R of Mod-R whose objects are all finitely generated projective right R-modules, we can consider the full subcategory proj∗ -R of Mod-R whose objects are all countably generated projective right R-modules. Notice that proj-R is a full subcategory of proj∗ -R. Now we can construct the monoid V (proj∗ -R) as in Section 1.3. That is, we fix a skeleton V (proj∗ -R) of proj∗ -R, i.e., a set of representatives of countably generated projective right R-modules up to isomorphism. Denote this set of representatives V (proj∗ -R), for coherence of notation, by V ∗ (RR ). For any countably generated projective right module PR , the unique module in V ∗ (RR ) isomorphic to PR will be denoted by PR . The set V ∗ (RR ) is a commutative monoid with respect to the addition defined by PR  + QR  = PR ⊕ QR  for every PR , QR  ∈ V ∗ (RR ). Notice that the element RR  of the reduced monoid V ∗ (RR ) is not an order-unit in V ∗ (RR ) in general. For instance, if k is a division ring, then V ∗ (kk ) is the monoid [0, ℵ0 ] consisting of all cardinals ≤ ℵ0 . The operation is addition between cardinals. The element kk  corresponds to the element 1 of [0, ℵ0 ] and is not an order-unit in [0, ℵ0 ]. More generally, if R is a semisimple Artinian ring with exactly n simple right modules up to isomorphism, then V ∗ (RR ) ∼ = [0, ℵ0 ]n , the direct sum of n copies of [0, ℵ0 ], and RR  corresponds to an n-tuple (m1 , . . . , mn ) with each mi different both from 0 and ℵ0 .

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157

Since direct summands of finitely generated projective modules are finitely generated projective modules, V (R) is a divisor closed submonoid of V ∗ (RR ). Like V (−), V ∗ (−) also turns out to be a functor from the category of rings to the category of commutative monoids. The duality Hom(−, R) : proj-R → R-proj between finitely generated projective right and left modules (Proposition 2.32) does not extend to a duality between the countably generated ones, so that if instead of the category proj∗ -R we consider the full subcategory R-proj∗ of R -Mod whose objects are all countably generated projective left R-modules, we get a monoid V ∗ (R R) different from the monoid V ∗ (RR ). Cf. [Herbera and Pˇr´ıhoda 2010, Example 8.5]. If R is a semilocal ring, the canonical projection π : R → R/J(R) induces a monoid homomorphism V ∗ (π) : V ∗ (RR ) → V ∗ (R/J(R)R/J(R) ) that is a monomorphism, that is, an injective mapping (Theorem 2.20). We have just seen that ∼ [0, ℵ0 ]n , where n is the number of simple right R-modules V ∗ (R/J(R)R/J(R) ) = up to isomorphism. The monoid monomorphism V ∗ (π) : V ∗ (RR ) → V ∗ (R/J(R)R/J(R) ) is not necessarily a divisor homomorphism, as remarked by Prihoda [Pˇr´ıhoda 2007, p. 830]. Namely, Gerasimov and Sakhajev gave in [Gerasimov and Sakhaev] an example of a semilocal ring R with an infinitely generated projective module PR with PR /PR J(R) finitely generated. Assume by contradiction that V ∗ (π) : V ∗ (RR ) → V ∗ (R/J(R)R/J(R) ) is a n divisor homomorphism. Then V ∗ (π)(PR ) = PR /PR J(R) ≤ nRR  = RR  for n some integer n ≥ 1, so that PR would be a direct summand of RR . So PR would be finitely generated, a contradiction. Thus V ∗ (π) : V ∗ (RR ) → V ∗ (R/J(R)R/J(R) ) is injective but not a divisor homomorphism in general, while its restriction V (π), viewed as a homomorphism of V (R) into V (R/J(R)), is a divisor homomorphism (Proposition 3.29). If R is a semilocal ring, the monoid monomorphism V ∗ (π) : V ∗ (RR ) → V ∗ (R/J(R)R/J(R) ) ∼ [0, ℵ0 ]n . allows us to identify V ∗ (RR ) with a submonoid of V ∗ (R/J(R)R/J(R) ) = ∗ n ∼ This submonoid of V (R/J(R)R/J(R) ) = [0, ℵ0 ] is closed not only for finite direct sums, but also for countable direct sums (in [0, ℵ0 ] the addition is the sum of cardinals, and therefore the sum of countably many summands is defined in [0, ℵ0 ]). [Herbera and Pˇr´ıhoda 2009] proved that if the semilocal ring R is also Noetherian, then V ∗ (π) : V ∗ (RR ) → V ∗ (R/J(R)R/J(R) ) is a divisor homomorphism. They also characterized the embeddings M → [0, ℵ0 ]n that can be obtained as the divisor homomorphism V ∗ (π) : V ∗ (RR ) → V ∗ (R/J(R)R/J(R) ) for some semilocal Noetherian ring R. The last two results we present in this section are due to [Wiegand 2001]. Recall that if M is a module over a commutative ring R and M ∗ = HomR (M, R)

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denotes the dual module, then M is reflexive if the canonical mapping M → M ∗∗ is an isomorphism.  If R is a commutative Noetherian local ring with maximal ideal m, we let R denote the m-adic completion of R. Consider the full subcategories mod-R (and  of Mod-R (and Mod-R,  respectively) whose objects are all finitely genermod-R)  ated R-modules (all finitely generated R-modules, respectively). There is a monoid  ⊗R M ], which  homomorphism ι : V (mod-R) → V (mod-R) defined by ι([M ]) = [R is a divisor homomorphism [Wiegand and Wiegand, Proposition 4.1]. The monoid homomorphism ι is injective, because for every pair of R-modules R M, R M  one  has that the R-modules R M,R M  are isomorphic if and only if the R-modules    R⊗R M, R⊗R M are isomorphic. (More generally, one can prove that if R is a com is the m-adic complemutative Noetherian local ring with maximal ideal m and R  tion of R, then R is a faithfully flat R-module. See [Matsumura]. By definition, an R-module R M is faithfully flat if an exact sequence R A → R B → R C of R-modules  ⊗R A → R  ⊗R B → R  ⊗R C is exact.) is exact if and only if the exact sequence R Recall that a local commutative ring R with maximal ideal M is Henselian if the following condition holds: if f, g0 , h0 ∈ R[x] are monic polynomials such that (1) f ≡ g0 h0 (mod M R[x]); and (2) g0 R[x] + h0 R[x] + M R[x] = R[x], then there exist monic polynomials g, h ∈ R[x] such that f = gh, g ≡ g0 (mod M R[x]), and h ≡ h0 (mod M R[x]). For instance, it is possible to prove (Hensel’s lemma) that complete local commutative rings, that is, the local rings R that are complete in the M -adic topology, where M is the maximal ideal of R, are Henselian rings. The link between the Krull–Schmidt theorem and Henselian rings was first pointed out by Swan and Evans [Evans]. P. V´amos [V´amos 1990, Lemma 13] and M. Siddoway [Siddoway] independently proved that every finitely generated indecomposable module MR over a Henselian local commutative ring R has a local endomorphism ring. Thus If R is a Henselian local commutative ring, the Krull– Schmidt theorem holds for the class of all finitely generated right R-modules. More is a free commutative monoid for every commutative over, the monoid V (mod-R) Noetherian local ring R. The injective divisor homomorphism ι : V (mod-R) →  now shows that V (mod-R) is a Krull monoid. V (mod-R) Theorem 4.69 ([Wiegand 2001, Main Theorem 3.1], [Wiegand and Wiegand, Theorem 4.4]). Let (M, u) be a commutative additive monoid with order-unit, n a positive integer, let (d1 , . . . , dn ) be an order-unit in the finitely generated free commutative monoid Nn0 , and let f : (M, u) → (Nn0 , (d1 , . . . , dn )) be a divisor homomorphism that is both injective and a morphism of monoids with order-units. Then there are a semilocal ring R and isomorphisms of monoids with order-unit (M, u) → (V (R), R) and (N0n , v) → (V (R/J(R)), R/J(R)) with the following properties: (a) R is the endomorphism ring of a finitely generated reflexive module over a commutative Noetherian local unique factorization domain of Krull dimension 2.

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(b) If π : R → R/J(R) denotes the canonical projection, then the diagram of monoids with order-unit f

(M, u) ∼ =

 (V (R), R)

/ (Nn0 , v) ∼ =

V (π)

 / (V (R/J(R)), R/J(R))

is commutative. The following is a very nice corollary obtained by Roger Wiegand. It is the last result in [Wiegand 2001]. Corollary 4.70. The following conditions on a commutative monoid M are equivalent: (a) M is a reduced finitely generated Krull monoid. (b) M ∼ = V (add(AR )) for some (cyclic) Artinian module AR over a suitable ring R. (c) M ∼ = V (S) for some semilocal ring S. (d) M ∼ = V (add(NT )) for a suitable finitely generated module NT over a commutative semilocal Noetherian ring T . From Corollary 4.70, we immediately find another proof that the Krull– Schmidt theorem does not hold for Artinian modules. In order to see it, fix any finitely generated reduced Krull monoid M that is not free, for instance the monoid M := M (2, 2) of Example 1.23. The monoid M is the submonoid of N20 generated by (1, 1), (2, 0), and (0, 2). We have seen in Example 1.23 that the three elements (1, 1), (2, 0), and (0, 2) are all the atoms of M , and that M is a Krull monoid, clearly finitely generated and reduced. The monoid is not free because the identity (2, 0) + (0, 2) = 2(1, 1) holds in M . From Corollary 4.70, we find that there exists an Artinian module AR over a suitable ring R with M ∼ = V (add(AR )). This isomorphism of M onto V (add(AR )) associates to any element (x, y) ∈ M an object A(x,y) of add(AR ), uniquely determined up to isomorphism. In particular, the modules A(2,0) , A(1,1) , and A(0,2) are three indecomposable pairwise nonisomorphic Artinian right modules over R with A(2,0) ⊕ A(0,2) ∼ = A2(1,1) . Thus Krull–Schmidt does not hold for Artinian modules. This is a solution of the problem posed by Krull and mentioned in the preface (p. xii). As we have already said there, the first example of Artinian modules with two nonisomorphic indecomposable decompositions was given by [Facchini, Herbera, Levy, and V´ amos]. Other examples were given in [Yakovlev] and [Ringel]. In this last paper, Ringel was able to construct examples of Artinian R-modules with two nonisomorphic indecomposable decompositions in which the base ring R was a local ring. The examples of the failure of the Krull–Schmidt theorem for Artinian modules constructed by [Facchini, Herbera, Levy, and V´ amos] are obtained from examples of suitable Noetherian modules for which the Krull–Schmidt theorem fails,

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making use of a construction, due to Camps and Menal [Camps and Menal], that allows one to transfer direct-sum decompositions. Camps and Menal’s construction derives from an example of [Armendariz, Fisher, and Snider, Example 3.2]. To the best of my knowledge, all the examples of the failure of the Krull–Schmidt theorem for Artinian modules constructed until now are built following the same pattern: not only the first examples of Artinian modules for which Krull–Schmidt fails in [Facchini, Herbera, Levy, and V´amos], but also the results of Wiegand and the examples of Yakovlev [Yakovlev] and Ringel [Ringel] are essentially constructed starting from other classes of modules for which Krull–Schmidt is known to fail, and then transferring this to Artinian modules via some construction. For instance, in the examples of Yakovlev and Ringel, the technique that allows them to transfer the direct-sum decompositions to Artinian modules is that developed later in [Pimenov and Yakovlev]. This partially explains why Corollary 4.70 holds: it is a corollary of Theorems 4.65 and 4.69. Roger Wiegand and his school extensively extended the theory, applying it to the study of modules over commutative rings. For instance, he used it to study finitely generated torsion-free modules over one-dimensional local Noetherian domains [Wiegand 2000], finitely generated modules over one-dimensional Noetherian Cohen–Macaulay local rings [Facchini, Hassler, Klingler, and Wiegand], and finitely generated modules that are free on the punctured spectrum [Hassler, Karr, Klingler, Wiegand]. For this direction of research, see the wonderful book [Leuschke and Wiegand] about maximal Cohen–Macaulay modules over local rings and the survey [Wiegand and Wiegand].

4.16 Notes on Chapter 4 Notice that in an arbitrary category C, the sets EndC (A), A ∈ Ob(C), are multiplicative monoids. If C has a zero object, they are multiplicative monoids with zero. If C is preadditive, they can be given the structure of a ring with identity. If C is additive, they have a canonical structure of a ring with identity. Most of the results in Sections 4.1–4.9 are standard, or can be found more or less explicitly in [Mitchell]. Proposition 4.17 is taken from [Facchini 2007, Lemma 2.1]. The Jacobson radical of an additive category was defined by [Kelly, p. 303]. Theorem 4.34 was generalized to the infinite case by [Walker and Warfield] (cf. [Arnold, Hunter, and Richman], where the case in which the endomorphism rings of the indecomposable direct summands are principal ideal domains is also considered). I am grateful to Wolfgang Rump for some remarks concerning the concept of coproduct of categories. Usually, in a category, there is no canonical morphism between the coproduct and the product of two objects. (For instance, look at what occurs in the category ˙ and X × Y .) It Set, where there is no canonical mapping between the sets X ∪Y would be easy to show that if a category C has a zero object, then for any pair

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161

  A, B of objects of C, there is a unique morphism h : A B → A B such that πB hεA = ζA,B , πA hεA = 1A , πA hεB = ζB,A , πB hεB = 1B . When the category C is additive, this morphism h is always an isomorphism. This can be employed, for instance, to prove that the category Grp is not preadditive, because the product and the coproduct in Grp of two groups are not isomorphic (Example 4.9). Notice that in Proposition 4.20, when we extend an additive functor F : C → A of a preadditive category C into an additive category A to an additive functor F  : C  → D, where C  is additive, then the functor F  depends on how we fix the biproducts in A of the objects F (U ), U ∈ Ob(C). It is clear that we can fix the biproducts in A a priori. This is the point of view followed in [Facchini and Fern´andez-Alonso]. Our presentation of additive categories here is extremely far from being exhaustive. For instance, we have skipped all the fundamental results about abelian categories. One of the nicest results of abelian category theory is the so-called Freyd–Mitchell theorem. Its precise statement is the following. Theorem 4.71 (Freyd–Mitchell theorem). If A is a small abelian category, then there exist a ring R and a full, faithful, and exact functor F : A → Mod-R. Thus the functor F is an equivalence between A and a full subcategory of Mod-R (that is, essentially, a class of modules) in such a way that kernels and cokernels computed in A correspond to the usual kernels and cokernels computed in Mod-R. That is, small abelian categories are essentially categories of modules. Theorem 4.58 is due to Warfield [Warfield 1980]. The results of Section 4.10 are taken from [Facchini 2006b]. Theorem 4.38 can be extended to prove that if A is a skeletally small additive category in which idempotents have a kernel, then the monoid V (A) is a Krull monoid if and only if there exist a family { Aj | j ∈ J } of IBN categories Aj and a direct-summand reflecting functor A → j∈J Aj . See [Facchini 2007]. We conclude this chapter with some history of the Krull–Schmidt theorem and in particular how it has evolved in other algebraic structures. The idea of uniqueness of decomposition in algebra goes back to factorization of a natural number as a product of primes. I would date the origins of the Krull–Schmidt theorem to 1879, when [Frobenius and Stickelberger] proved that any finite abelian group is a direct sum of cyclic groups whose orders are powers of primes, and these powers of primes are uniquely determined by the group. We will now see how this result was then generalized to the various algebraic structures. Groups. A first generalization to finite noncommutative groups of the result by Frobenius and Stickelberger is due to the Scottish mathematician J.H. Wedderburn [Wedderburn] (he mentions that some credit is due to G.A. Miller). Wedderburn stated that if a finite group G has two direct-product decompositions G = G1 × G2 × · · · × Gt = H1 × H2 × · · · × Hs with all the groups Gi and Hj indecomposable, then t = s and there exists an automorphism ϕ of G such that

Chapter 4. Additive Categories

162

ϕ(Gi ) = Hσ(i) for all i. But Wedderburn’s proof is not entirely convincing. The first convincing proof was given by Robert Erich Remak. Remak, in his doctoral ¨ die Zerlegung der endlichen Gruppen in indirekte unzerlegbare dissertation “Uber Faktoren” (“On the decomposition of finite groups into indirect indecomposable factors,” 1911), also proved that the automorphism ϕ could be chosen to be a central automorphism. The Soviet mathematician Otto Yulyevich Schmidt [Schmidt 1912, Schmidt 1913, Schmidt 1929], founder of the Department of Higher Algebra at Moscow State University in 1930, simplified and improved Remark’s results. The German name of Otto Schmidt derives from the fact that his ancestors were German settlers in Courland, a region of Latvia. Schmidt’s mother was a Latvian. The Krull–Schmidt theorem in group theory is the following. Theorem 4.72. Let G be a group satisfying both the ascending chain condition and the descending chain condition on normal subgroups. Then G is the direct product G1 × G2 × · · · × Gt of finitely many indecomposable groups, which is essentially unique in the following sense. If G = G1 × G2 × · · · × Gt = H1 × H2 × · · · × Hs , where G1 , G2 , . . . , Gt , H1 , H2 , . . . , Hs are indecomposable groups, then t = s and ∼ Hσ(i) for every there is a permutation σ of the indices 1, 2, . . . , t such that Gi = i = 1, 2, . . . , t. [Ore] extended Wedderburn’s result to modular lattices with the ascending and descending chain conditions. Modules. Wolfgang Krull [Krull 1925] extended the results from the case of groups to the case of “abelian operator groups” (i.e., modules) with the ascending and descending chain conditions. The theory was subsequently further deepened by Schmidt [Schmidt 1929] and Goro Azumaya [Azumaya 1950], who extended the Krull–Schmidt theorem from the case of modules of finite composition length to the case of possibly infinite direct sums of modules with local endomorphism rings. Notice that every indecomposable module of finite composition length has a local endomorphism ring (Fitting’s lemma, [Facchini 1998, Lemmas 2.20 and 2.21]). Theorem 4.73 (Krull–Schmidt–Remak–Azumaya theorem). Let  Mi M= i∈I

be a module that is a direct sum of modules Mi with local endomorphism rings End(Mi ). Then M is a direct sum of indecomposable modules in an essentially unique way in the following sense. If   Pk , Nj = M= j∈J

k∈K

where all the modules Nj (j ∈ J) and Pk (k ∈ K) are indecomposable, then there exists a bijection ϕ : J → K such that Nj ∼ = Pϕ(j) for every j ∈ J.

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163

This theorem can be extended to the categorical setting, getting hereby several applications. For instance, here is a nice application due to [Warfield 1969, Corollary 4.2]. Proposition 4.74. Let M = E( i∈I Mi ) be an injective module that is an injective envelope of a direct sum of indecomposable injective submodules Mi , i ∈ I. Then any two such decompositions of M are isomorphic. Furthermore, if N is an in= M M of jective submodule N ⊕ E( that ⊆ such a subset , there is I J i∈J Mi ) ∼ E( and N = i∈I\J Mi ). From Theorem 4.73, it follows that every module of finite composition length is a direct sum of indecomposables in an essentially unique way. Notice that a module has finite composition length if and only if it satisfies both the ascending chain condition and the descending chain condition. Krull knew that the ascending chain condition was not sufficient for the direct-sum decomposition into indecomposables to be essentially unique. He knew that Noetherian modules decompose as a direct sum of indecomposable modules, but that their direct-sum decompositions are not essentially unique. That is, he was able to construct Noetherian modules with nonisomorphic direct-sum decompositions into indecomposables. Thus, in 1932, [Krull 1932a, pp. 37–38] asked whether Artinian modules decompose into indecomposables in an essentially unique way. That is, it is easily seen that any Artinian module is a direct sum of indecomposable modules, but is the direct-sum decomposition of an Artinian module into indecomposables essentially unique? The negative answer to this question was given in 1995 in [Facchini, Herbera, Levy, and V´ amos]. In that paper, examples of the failure of the Krull–Schmidt theorem for Artinian modules were constructed. The idea was to start from known examples of suitable Noetherian modules for which the Krull–Schmidt theorem fails, and to use them to construct examples of Artinian modules for which Krull–Schmidt fails making use of a technique due to [Camps and Menal]. For more information about the Krull–Schmidt theorem for Noetherian modules, see Corollary 4.70, [Brookfield 1997] and [Brookfield 2002, Theorem 2.8]. We have already presented the Krull–Schmidt theorem in arbitrary additive categories in which idempotents split in Theorem 4.34. In Chapters 8 and 11, we will present the behavior of uniserial modules relative to direct-sum decompositions. Other classes of modules with a similar behavior will be presented in Section 8.13. More generally, we will consider modules whose endomorphism ring has at most two maximal right ideals in Chapter 10. Nowadays the name “Krull–Schmidt” is given to any theorem concerning uniqueness of direct-sum decompositions into indecomposables.

Chapter 5

Spectral Category and Dual Construction 5.1 The Spectral Category of a Grothendieck Category Now we will present a construction due to [Gabriel and Oberst]. Recall that a Grothendieck category is an abelian category with arbitrary coproducts, with exact direct limits and with a generator. Every Grothendieck category is a category with injective envelopes (see, for instance, [Popescu, Theorem 3.10.10]). Let A be a Grothendieck category. We will now construct a Grothendieck category Spec(A) obtained from A by formally inverting all essential monomorphisms of A. Recall that a monomorphism ε : A → B in A is essential if for every subobject B  of B, ε(A)∩B  = 0 implies B  = 0. By “formally inverting the essential monomorphisms of A,” we mean that we construct the category Spec(A) in a way similar to the way in which we construct the ring of fractions RS −1 = { f s−1 | f ∈ R, s ∈ S } from a commutative ring R formally inverting the elements of a multiplicatively closed subset S of R. If we have a commutative ring R and a multiplicatively closed subset S of R, we can first construct the Cartesian product R × S = { (f, s) | f ∈ R, s ∈ S }, then take the equivalence relation ∼ on R × S defined by (f, s) ∼ (f  , s ) if there is an element t ∈ S with f s t = f  st, and finally construct the factor set RS −1 := R × S/∼. Its elements, that is, the equivalence classes of the pairs (f, s), are denoted by f s−1 . Then RS −1 can be given a ring structure, in which the elements of S have become invertible. (One should say everything with more precision, because the new ring RS −1 does not contain R necessarily, there is only a ring morphism R → RS −1 , which is the smallest morphism in the sense that it has a suitable universal property.) The category Spec(A) will be obtained via a similar construction, formally inverting the class of all essential monomorphisms in A. We could construct the category Spec(A) with the same objects as A and with morphisms A → B (for A, B ∈ Ob(A)) the “fractions” f s−1 , where f : A → B and s : A → A are morphisms in A for some object A of A and s is an essential monomorphism. © Springer Nature Switzerland AG 2019 A. Facchini, Semilocal Categories and Modules with Semilocal Endomorphism Rings, Progress in Mathematics 331, https://doi.org/10.1007/978-3-030-23284-9_5

165

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The construction of the spectral category Spec(A) we will present now is equivalent to that of the previous paragraph, but easier to handle. Let A be a Grothendieck category. For any fixed object A in A, the set of all the essential subobjects of A is downward directed, because the intersection of two essential subobjects of A is an essential subobject of A. We will write A ≤e A for “A is an essential subobject of A.” If we fix another object B of A and apply the contravariant functor HomA (−, B) to the essential subobjects A of A and to the embeddings A → A , where A ≤e A, A ≤e A, and A ⊆ A , we get an upward directed family of abelian groups HomA (A , B) and abelian group morphisms HomA (A , B) → HomA (A , B). Take the direct limit lim HomA (A , B), −→ where A ranges over the set of all essential subobjects of A. The spectral category Spec(A) of A has the same objects as A and, for objects A and B of A, HomSpec(A) (A, B) := lim HomA (A , B), −→ where the direct limit is taken over all essential subobjects A of A. The composition in Spec(A) is defined as follows. If f ∈ HomSpec(A) (A, B) and g ∈ HomSpec(A) (B, C), then f is represented by a morphism f  : A → B for some essential subobject A of A and g is represented by a morphism g  : B  → C for some essential subobject B  of B. Then f −1 (B  ) is essential in A. The composite morphism gf in Spec(A) is the image in HomSpec(A) (A, C) := lim HomA (A , C) of −→ the composite morphism f −1 (B  ) → C in A of the restriction of f  to f −1 (B  ) and g  : B  → C. There is a canonical functor P : A → Spec(A) that is the identity on objects and maps any morphism f ∈ HomA (A, B) to its canonical image in HomSpec(A) (A, B). Notice that if A is an object of A and P (A) = 0, then A = 0 (because if P (A) = 0, then P (1A ) = 0, so that 1A is zero on an essential subobject of A; but 1A is zero only on the zero subobject, so that the zero subobject is essential in A, whence A = 0). For a morphism f : A → B in A, one has that P (f ) = 0 if and only if the kernel of f in A is an essential subobject of A. Proposition 5.1. Let A be a Grothendieck category. Then Spec(A) is a Grothendieck category in which every object is injective. For the proof, see [Gabriel and Oberst, Satz 1.3]. We just underline a couple of facts of the proof. If f ∈ HomSpec(A) (A, B) is a morphism in Spec(A), then f is represented by a morphism f  : A → B in A for some essential subobject A of A. Let k : K → A be a kernel of f  in A. Then a kernel of f in Spec(A) is given by the image in Spec(A) of the composite morphism of k and the inclusion A → A. From this, we get the following: Proposition 5.2 ([Gabriel and Oberst, p. 391]). The functor P : A → Spec(A) preserves kernels, that is, is left exact. For a more detailed proof, see [Stenstr¨om, Proposition V.7.2].

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167

Proposition 5.3. Let f : A → B be a morphism in a Grothendieck category A. The morphism P (f ) : P (A) → P (B) is an isomorphism in Spec(A) if and only if f is an essential monomorphism in A. Proof. Assume that g : P (B) → P (A) is the inverse of the isomorphism P (f ) : P (A) → P (B). Then there exist an essential subobject B  of B and a morphism g : B  → A such that g is the image of g in Spec(A). As g ◦ P (f ) = 1P (A) , there exists an essential subobject A of A with f (A ) ⊆ B  , and the composite morphism of the restriction f  : A → B  of f and g : B  → A is the embedding A → A. It follows that f  : A → B  is a monomorphism. Since A is essential in A, we get that f : A → B is a monomorphism. Now P (f ) ◦ g = 1P (B) . Thus there exists an essential subobject B  of B contained in B  such that if g  : B  → A denotes the restriction of g  to B  , then f ◦ g  is the embedding of B  into B. From this, it follows that the image of f contains B  . Thus the image of f is essential in B. Conversely, suppose that f : A → B is an essential monomorphism. Let f1 : f (A) → A be the inverse of the isomorphism f0 : A → f (A) induced by f . Let f1 : P (B) → P (A) be the image of f1 in Spec(A). Then f ◦ f1 is the embedding of the essential subobject f (A) in B, so that P (f ) ◦ f1 = 1P (B) . Moreover, f1 ◦ f0 = 1A , so that f1 ◦ P (f ) = 1P (A) . Thus f1 is the inverse of P (f ), and P (f ) is an isomorphism.  A second fact from the proof of Proposition 5.1 we want to stress is that if A is any subobject of an object B and C is a complement of A in B (that is, A∩C = 0 and C is maximal with respect to this property), then the inclusion A ⊕ C → B is an essential monomorphism in A, so that it becomes an isomorphism in Spec(A). It follows that P (A) ⊕ P (C) ∼ = P (B) in Spec(A), that is, every subobject P (A) of P (B) is a direct summand. This proves that every object P (A) of Spec(A) is injective.

5.2 Nonsingular Modules In the previous section, we have presented the construction of the spectral category Spec(A) of a Grothendieck category A. More generally, a category is a spectral category if it is a Grothendieck category in which every object is injective (equivalently, a Grothendieck category in which every object is projective). Spectral categories can be characterized in terms of nonsingular modules. For any right module AR , the singular submodule of AR is the submodule Z(AR ) := { x ∈ AR | annR (x) is essential in RR }

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of AR . Then f (Z(AR )) ⊆ Z(BR ) for every morphism f : AR → BR . The functor Z : Mod-R → Mod-R is an idempotent left exact functor. The module AR is said to be a singular module provided Z(AR ) = AR , and is a nonsingular module provided Z(AR ) = 0. The pair consisting of the class of all singular R-modules and the class of all nonsingular R-modules is not a torsion theory, because Z(AR /Z(AR )) is not necessarily equal to 0. In order to define a torsion theory, instead of Z(AR ) we need Z2 (AR ), which is the submodule of AR defined by Z2 (AR )/Z(AR ) := Z(AR /Z(AR )). Then there is a torsion theory whose class of torsion modules consists of all R-modules AR with Z2 (AR ) = AR , and whose class of torsion-free modules consists of all nonsingular modules. If R is any ring, it is easily seen that the right ideal Z(RR ) of R is a twosided ideal of R, called the right singular ideal of R. Similarly for the left analogue Z(R R). It is possible to construct examples for which the two ideals Z(RR ) and Z(R R) are distinct. Proposition 5.4. A module AR is singular if and only if there exists a module BR with an essential submodule CR such that AR ∼ = BR /CR . Proof. (⇐) We can assume that AR = BR /CR with CR essential in BR , and we must prove that AR is singular. Let a be an element in AR , and let b ∈ BR with a = b + CR . Consider the morphism λb : RR → BR , defined by λb (r) = br for −1 every r ∈ RR . By Proposition 2.28(d), λ−1 b (CR ) ≤e RR . Thus I := λb (CR ) is an essential right ideal of R and bI ⊆ CR . Hence I annihilates a and AR is a singular module. (⇒) Let AR be a singular module. There exist a free module FR and an epimorphism ϕ : FR → AR . It suffices to show that ker ϕ is essential in FR . Let { xi | i ∈ I } be a free set of generators of FR . For every i ∈ I, the element ϕ(xi ) of AR has an essential annihilator Ai := r.annR (ϕ(xi )). Thus Ai ≤e RR , so that ⊕i∈I xi Ai ≤e ⊕i∈I xi R = FR . But ϕ(xi )Ai = 0 implies xi Ai ⊆ ker ϕ, so that ⊕i∈I xi Ai ⊆ ker ϕ. Thus ker ϕ ≤e FR , as desired.  Corollary 5.5. Let A be a submodule of a nonsingular injective module BR . Then BR contains a unique injective envelope of A, that is, there is a unique direct summand C of BR with A ≤e C. Proof. Since injective envelopes are the minimal injective extensions, it is clear that BR contains an injective envelope of A, that is, there is a direct summand C of BR with A ≤e C. Let C, C  be two direct summands of BR with A ≤e C and A ≤e C  . Let f : C → BR /C  be the restriction to C of the canonical projection BR → BR /C  . Then f (A) = 0, so that f induces a morphism f : C/A → BR /C  . Now C/A is singular by Proposition 5.4 and BR /C  is nonsingular. Thus f = 0, so that f = 0, i.e., C ⊆ C  . By symmetry, C  ⊆ C. Therefore the direct summand  C of BR with A ≤e C is unique. Proposition 5.6. Injective envelopes of nonsingular modules are nonsingular modules.

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169

Proof. Assume the contrary, that is, assume that there exists a nonsingular module AR with Z(E(AR )) = 0. Then Z(E(AR )) ∩ AR = 0 is a submodule of AR whose elements have essential annihilators. Hence Z(AR ) ⊇ Z(E(AR )) ∩ AR . In particular, AR cannot be nonsingular, contradiction.  Proposition 5.7. Let f : AR → BR be a homomorphism. Assume BR nonsingular. Then there exists a unique homomorphism E(f ) : E(AR ) → E(BR ) that extends f . Proof. By Proposition 5.4, the module E(A)/A is singular. By Proposition 5.6, the module E(B) is nonsingular. Thus Hom(E(A)/A, E(B)) = 0. Applying the exact functor Hom(−, E(B)) to the exact sequence 0 → A → E(A) → E(A)/A → 0, one gets the exact sequence 0 → Hom(E(A), E(B)) → Hom(A, E(B)) → 0. That is, every homomorphism A → E(B) extends to a homomorphism E(A) → E(B) in a unique way. In particular, this holds for every homomorphism A → B.  We give a proof of only one of the two implications in the following nice result. Theorem 5.8 ([Gabriel and Oberst, Satz 2.1], [Goodearl and Boyle, Theorem 1.14]). For any ring R, the full subcategory NSI(R) of Mod-R whose objects are all nonsingular injective right R-modules is a spectral category. Conversely, if A is a spectral category, let U be a generator of A and R := HomA (U, U ) the endomorphism ring of U . Then R is a von Neumann regular, right self-injective ring, and A is equivalent to NSI(R). Proof. Let R be a ring. Clearly, NSI(R) is an additive full subcategory of Mod-R. If f : AR → BR is a morphism in NSI(R), then the injective module AR contains a unique injective envelope E(ker f ) of the kernel ker f of f in Mod-R by Corollary 5.5. Thus AR = E(ker f ) ⊕ CR . Hence f induces an injective morphism of E(ker f )/ ker f into BR . But E(ker f )/ ker f is singular by Proposition 5.4, and BR is nonsingular. It follows that E(ker f ) = ker f is an injective module. Thus ker f is a direct summand of AR ; hence it is an object of NSI(R). It is now clear that the kernel ker f of f in Mod-R is also a kernel of f in NSI(R). From the direct-sum decomposition AR = ker f ⊕ CR , we get that the image of f in Mod-R is isomorphic to CR , hence is an object of NSI(R). Thus the cokernels of f in Mod-R and NSI(R) coincide. It is now clear that images and coimages in Mod-R and NSI(R) also coincide. In particular, NSI(R) is an abelian category. The monomorphisms in NSI(R) are the injective morphisms. Similarly, the epimorphisms in NSI(R) are the surjective morphisms. The coproduct of a family { Ai | i ∈ I } of objects in NSI(R) is E(⊕i∈I Ai ). In fact, ⊕i∈I Ai is nonsingular, so that E(⊕i∈I Ai ) is nonsingular injective by Proposition 5.6. Every family of morphisms { fi : Ai → BR | i ∈ I } corresponds

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to a unique morphism ⊕i∈I Ai → BR , which extends to a unique morphism E(⊕i∈I Ai ) → BR . Thus NSI(R) is a cocomplete abelian category. Notice that the subobjects of an object AR of NSI(R) are the direct summands of AR in The product of a family { Ai | i ∈ I } of objects of NSI(R) is the usual Mod-R.  product i∈I Ai in Mod-R. A generator of NSI(R) is given by the injective envelope of the direct sum of all the cyclic modules RR /I, where I ranges over the set of all right ideals of R with RR /I a nonsingular module. In order to show that direct limits are exact in NSI(R), it suffices to show that NSI(R) satisfies condition AB 5 [Stenstr¨om, Proposition V.1.1], that is,     Ai ∩ B = (Ai ∩ B) i∈I

i∈I

for any direct family { Ai | i ∈ I } of subobjects of A and for any subobject B of A. Now i∈I Ai in NSI(R) is the unique direct summand of A containing the sums of the modules Ai in Mod-R. The intersection of two subobjects B, B  of A in NSI(R) is the kernel of the product A → A/B ×A/B  of the canonical projections,  so that the intersection B∩ with the intersection B ∩ B  coincides  B in NSI(R) A ∩ B ⊇ i∈I (Ai ∩ B) is easily in Mod-R. The inclusion  If the  verified. i∈I i A inclusion is proper, then i∈I (Ai ∩ B) is a proper subobject of i∈I i ∩ B in a complement of be C NSI(R), that is, a proper direct summand. Let i∈I (Ai ∩B)   in i∈I Ai ∩ B. Then C = 0, C ⊆ B, C ⊆ i∈I Ai and i∈I (Ai ∩ B) ∩ C = 0. Now C ⊆ i∈I Ai means that C is contained in an essential extension of the sum in Mod-R of the Ai . Thus the intersection of C with the sum in Mod-R of all the modules Ai is nonzero. It follows that the intersection of C with a finite sum in Mod-R of Ai is nonzero. Since the family of submodules Ai is direct, it follows that there exists j ∈ I with C ∩ Aj = 0. Thus C ∩ (Aj ∩ B) = 0, so that i∈I (Ai ∩ B) ∩ C = 0, a contradiction. This concludes the proof of one of the implications of the statement. For the proof of the converse, see [Stenstr¨om, Section XII.1] or [Goodearl and Boyle, Theorem 1.14]. 

5.3 The Functor P and Its Right Derived Functors For any object A of a Grothendieck category A, the injective envelope of A in A will be denoted by E(A). Lemma 5.9. Let A be a Grothendieck category and P : A → SpecA the canonical functor. (a) If A is an object of A, then P (A) ∼ = P (E(A)) in SpecA via the isomorphism P (ε) : P (A) → P (E(A)), where ε : A → E(A) is the embedding.

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171

∼ P (E  ) in SpecA if and only (b) If E, E  are injective objects of A, then P (E) =  if E ∼ = E in A. Proof. (a) Apply Proposition 5.3 to the essential monomorphism ε : A → E(A). (b) Assume P (E) ∼ = P (E  ) in SpecA. Let f : A → E  with A ≤e E be a morphism that induces an isomorphism P (E) → P (E  ). Extend f to a morphism g : E → E  . Then P (g) : P (E) → P (E  ) is the morphism induced by f , so that P (g) is an isomorphism. By Proposition 5.3, g : E → E  is an essential monomorphism. Now E is injective, so that g : E → E  is an isomorphism. The converse holds for any functor P .  Proposition 5.10. Let A be a Grothendieck category and P : A → SpecA the canonical functor of A into its spectral category. The following conditions are equivalent for two objects A and B of A: (a) The objects P (A) and P (B) of SpecA are isomorphic. (b) There exist an object C in A and two essential monomorphisms f : C → A and g : C → B. (c) The injective envelopes of A and B in A are isomorphic. ∼ P (E(B)) by Lemma 5.9(a), so Proof. (a) ⇒ (b) If P (A) ∼ = P (B), then P (E(A)) = ∼ that E(A) = E(B) by Lemma 5.9(b). Let f : E(A) → E(B) and g : E(B) → E(A) be mutually inverse isomorphisms. Then f −1 (B) ∩ A ≤e A, because f −1 (B) ∩ A is the inverse image of B via the composite morphism of the embedding A → E(A) and f . Similarly, g −1 (A) ∩ B ≤e B. Now gf = 1E(A) ; hence gf (A) = A implies f (A) = g −1 (A), so that f (f −1 (B) ∩ A) ⊆ B ∩ f (A) = g −1 (A) ∩ B. Similarly, g(g −1 (A) ∩ B) ⊆ f −1 (B) ∩ A ≤e A. Thus f and g induce an isomorphism of f −1 (B) ∩ A onto g −1 (A) ∩ B. Since they are essential subobjects of A and B respectively, statement (b) follows immediately.  (b) ⇒ (c) is trivial, and (c) ⇒ (a) follows from Lemma 5.9. Example 5.11. Let A be a Grothendieck category and A an object of A. Re∼ P (E(A)) by Lemma 5.9(a), and that the endomorphism ring call that P (A) = of an injective object in an abelian category is right self-injective and von Neumann regular modulo the Jacobson radical, and that idempotents lift modulo the Jacobson radical. Since E(A) is injective, every homomorphism of an essential subobject of E(A) into E(A) extends to E(A). It follows that the ring morphism End(E(A)) → EndSpec(A) (P (E(A))), induced by the functor P : A → Spec(A), is a surjective mapping. The kernel of this ring morphism consists of all the endomorphisms of E(A) with essential kernel. It follows that ∼ End(E(A))/J(End(E(A))). EndSpec(A) (P (A)) ∼ = EndSpec(A) (P (E(A))) = In particular, EndSpec(A) (P (A)) is right self-injective and von Neumann regular. In a Grothendieck category A, an object A of A is simple if it is nonzero and has no other subobjects than A and 0. Since in the spectral category Spec(A)

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every object is injective, an object P (A) of Spec(A) is simple if and only if it is indecomposable. Let A be an object of the Grothendieck category A. Let us prove that the object P (A) of Spec(A) is simple if and only if the object A of A is uniform, that is, of Goldie dimension 1. If P (A) is simple, its endomorphism ring EndSpec(A) (P (A)) is a division ring. But we have seen in the previous paragraph that EndSpec(A) (P (A)) ∼ = End(E(A))/J(End(E(A)),

(5.1)

so that End(E(A)) must be a local ring. Thus E(A) is an indecomposable object, that is, the object A of A is uniform. Conversely, A uniform implies E(A) indecomposable. By the isomorphism (5.1), EndSpec(A) (P (A)) is a division ring. Since division rings do not contain nontrivial idempotents, the object P (A) of the Grothendieck category Spec(A) is indecomposable, hence simple. Let G be the full subcategory of A whose objects are all the objects of A of finite Goldie dimension, and let S be the full subcategory of Spec(A) with Ob S := { P (A) | A ∈ Ob G }. Both G and S are additive categories with splitting idempotents, and the functor P : A → Spec(A) restricts to a functor P : G → S. The image via P of every uniform object is a simple object of S. Every object of S is a semisimple object of finite composition length of Spec(A). The full subcategory S of Spec(A) turns out to be an amenable semisimple category. The image via P of every object of finite Goldie dimension n is a semisimple object of S of finite composition length n. Now semisimple objects of finite composition length in a Grothendieck category are finite direct sums of objects in an essentially unique way. In our case, we can fix a set { Eλ | λ ∈ Λ } of representatives of the indecomposable injective objects of A up to isomorphism. The injective envelope of any object of A of finite Goldie dimension is isomorphic to ⊕λ∈Λ Eλnλ for suitable integers nλ ≥ 0 almost all zero. Thus every object of S is isomorphic to an object ⊕λ∈Λ P (Eλ )nλ . If kλ denotes the division ring EndSpec(A) w (P (Eλ )), vect-kλ , the category S turns out to be isomorphic to the weak coproduct λ∈Λ where vect-kλ denotes the category of all finite-dimensional right vector spaces over kλ . Cf. [Stenstr¨om, Example 2 on page 130]. Notice that in Theorem 5.8 and Example 5.11, we have seen that ∼ EndA (E(A))/J(EndA (E(A))) EndSpecA (A) = is a von Neumann regular right self-injective ring for every object A of a Grothendieck category A. Now, if A is a Grothendieck category, we saw in Proposition 5.2 that the covariant additive functor P is left exact. Let us compute its right derived functors. (Recall that in Grothendieck categories, it is possible to define right derived functors. See, for instance, [Grothendieck, p. 143].) Since in Grothendieck categories,

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173

every object has an injective envelope, every object A of A also has a minimal injective resolution δ−1 δ1 δ2 δ0 E1 −→ E2 −→ ··· . 0 → A −→ E0 −→

(5.2)

It is the exact sequence defined inductively as follows. Set E−1 := A. Let δ−1 : E−1 → E0 be an injective envelope of E−1 . If δn−1 : En−1 → En has been defined for some n ≥ 0, the morphism δn : En → En+1 is defined to be the composite morphism of the cokernel morphism En → coker δn−1 and an injective envelope coker δn−1 → En+1 of coker δn−1 . Notice that the image of δn is essential in En+1 for every n ≥ −1. Proposition 5.12. Let A be an object of a Grothendieck category A, 0 → A → E0 → E1 → E2 → · · · a minimal injective resolution of A, n ≥ 0 an integer, and P (n) : A → Spec(A) the nth right derived functor of P for some n ≥ 0. Then P (n) (A) = P (En ). Proof. Since the sequence (5.2) is exact, the kernel of δn+1 : En+1 → En+2 , which is equal to the image of δn , is essential in En+1 , so that P (δn+1 ) : P (En+1 ) → P (En+2 ) is the zero morphism for every n ≥ −1. The objects P (n) (A) are the 0 0 0 “cohomology groups” of the complex 0 −→ P (E0 ) −→ P (E1 ) −→ P (E2 ) −→ · · · , in which all morphisms are the zero morphisms. Thus P (n) (A) = P (En ) for all  n ≥ 0. Corollary 5.13. Let A be an object of a Grothendieck category A. Then P (n+1) (A) = 0 if and only if the injective dimension of A is ≤ n. Proof. If P (n+1) (A) = 0, then P (En+1 ) = 0 by the previous proposition, and we have remarked immediately before the statement of Proposition 5.1 that this implies En+1 = 0. Thus the injective dimension of A is ≤ n. The converse holds for the right derived functors of any covariant additive functor, as is immediately  seen.

5.4 First Applications of Spectral Categories In this section, we will present some applications of the functor P to the study of the objects A of a Grothendieck category A with endomorphism ring EndA (A) semilocal. Recall that an object A of a Grothendieck category A is directly finite if it is ∼ A implies not isomorphic to a proper direct summand of itself. That is, A ⊕ B = B = 0 for any object B of A.

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Any additive functor F of a preadditive category A into a preadditive category B induces a ring morphism EndA (A) → EndB (F (A)) for every object A of A. In particular, when A is a Grothendieck category, the canonical functor P : A → Spec(A) induces a ring morphism ϕA : EndA (A) → EndSpecA (P (A)) for every object A of A. Proposition 5.14 ([Facchini and Herbera 2006, Proposition 4.3]). Let A be a Grothendieck category. Let A be an object in A with the property that every monomorphism A → A is an automorphism. Then ϕA : EndA (A) → EndSpec(A) (P (A)) is a local morphism. Conversely, if ϕA is a local morphism and E(A) is directly finite, then every monomorphism A → A is an automorphism. Proof. Let A be an object such that every monomorphism A → A is an auto∼ P (E(A)) (Lemma 5.9(a)) and that morphism. We know that P (A) = EndSpec(A) (P (A)) ∼ = EndA (E(A))/J(EndA (E(A))) (Example 5.11). Thus if f ∈ EndA (A) with ϕA (f ) invertible and f : E(A) → E(A) is an extension of f to the injective envelope, then f +J(EndA (E(A))) is invertible in EndA (E(A))/J(EndA (E(A))), so that f is invertible in EndA (E(A)). Thus f is an automorphism of E(A), hence its restriction f to A is a monomorphism. By hypothesis, f is an automorphism of A. This proves that ϕA is a local morphism. Conversely, assume ϕA a local morphism and E(A) directly finite. Let f : A → A be a monomorphism. Let f : E(A) → E(A) be an extension of f to the injective envelope, so that f injective implies f injective. Since E(A) is directly finite, f must be an automorphism. It follows that ϕA (f ) is invertible. But ϕA is a local morphism, hence f is an automorphism.  Recall that dim denotes the Goldie dimension. Proposition 5.15. Let A be an object of a Grothendieck category A and let n be a nonnegative integer. The following conditions are equivalent: (a) dim(A) = n. (b) P (A) is a semisimple object of finite composition length n in the spectral category Spec(A). (c) The ring EndSpec(A) (P (A)) is a semisimple Artinian ring of Goldie dimension n. Proof. (a) ⇒ (b). Assume dim(A) = n. Then E(A) is the direct sum of n indecomposable injective objects E1 , . . . , En . Then P (A) ∼ = P (E(A)) ∼ = P (E1 ) ⊕ · · · ⊕ P (En ). It remains to show that each P (Ei ) is a simple object in Spec(A), that is, an indecomposable object in Spec(A). Its endomorphism ring EndSpec(A) (P (Ei )) ∼ =

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EndA (Ei )/J(EndA (Ei )) is a division ring; hence it has no nontrivial idempotents. Thus P (Ei ) has no nontrivial direct summands, that is, it is indecomposable. (b) ⇒ (c). In the spectral category Spec(A) every object is injective, so that every object of finite composition length is a finite direct sum of simple objects. It follows that its endomorphism ring is a finite product of rings of matrices over division rings, that is, the endomorphism ring of every object of finite composition length is a semisimple Artinian ring. (c) ⇒ (a). We know that ∼ EndA (E(A))/J(EndA (E(A))) EndSpec(A) (P (A)) = and that in the endomorphism ring of an injective object idempotents lift modulo the Jacobson radical (cf. Example 5.11). Thus, if EndSpec(A) (P (A)) is a semisimple Artinian ring of Goldie dimension n, then EndA (E(A)) is semiperfect (Example 3.13(9)) and E(A) decomposes as a direct sum of n subobjects with local endomorphism rings. Therefore E(A) is the direct sum of n indecomposable  injective objects. It follows that A has Goldie dimension n. We immediately obtain the following slight generalization of [Herbera and Shamsuddin, Theorem 3(1)]. Corollary 5.16. Let A be an object in a Grothendieck category A. Assume that A has finite Goldie dimension n and that every monomorphism A → A is an automorphism of A. Then EndA (A) is a semilocal ring of dual Goldie dimension ≤ n. Proof. The ring EndSpec(A) (P (A)) is semisimple Artinian of Goldie dimension n by Proposition 5.15. The ring morphism ϕA considered in Proposition 5.14 is a local morphism, as we saw in that proposition. The corollary now follows as an immediate application of Theorem 3.27 and Corollary 3.28.  From Corollary 5.16 and Lemma 3.18, applied to the category A = Mod-R, where R is an arbitrary ring, we get a second proof of Camps and Dicks’ Theorem 3.19 that every Artinian module MR has a semilocal endomorphism ring. Moreover, the dual Goldie dimension of End(MR ) is less than or equal to the Goldie dimension of the Artinian module MR , that is, codim(End(MR )) is less than or equal to the composition length of the socle soc(MR ) of MR . For a third proof of Camps and Dicks’ Theorem 3.19, it suffices to apply Theorem 3.27 and Corollary 3.28 to the ring morphism End(MR ) → End(soc(MR )) given by restriction to the socle. Cf. the weakly local functor soc in Example 4.48. Here is a different application. Recall that a commutative ring has Krull dimension zero if every prime ideal is maximal.

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Theorem 5.17 ([Vasconcelos 1970]). The following conditions are equivalent for a commutative ring R: (a) Every injective endomorphism of a finitely generated R-module is an automorphism. (b) The ring R has Krull dimension zero. Proof. (a) ⇒ (b). If R has Krull dimension greater than zero, there exist prime ideals P ⊂ Q. Let q be an element in Q \ P . It is easily seen that multiplication by q is an endomorphism of the finitely generated R-module R/P that is injective but is not an automorphism. (b) ⇒ (a). Let R be a commutative ring of Krull dimension 0, MR an Rmodule with a finite set of generators m1 , m2 , . . . , mn , and f : MR → MR an injective endomorphism. The R-module MR becomes a module over the ring R[x] of polynomials by setting mx = f (m) for every m ∈ M . The proof is now very similar to the proof of Proposition 4.56. For every index j = 1, . . . , n, there exist n elements a1j , . . . , anj ∈ R such that mj x = i=1 mi aij . We get n equalities n 

mi (xδij − aij ) = 0,

j = 1, 2, . . . , n,

(5.3)

i=1

where δij denotes the Kronecker delta. Let Dij denote n the cofactor of the (i, j) entry of the n × n matrix D = (xδij − aij )ij . Then j=1 (xδij − aij )Dkj = δik · det(D). Notice that det(D) is the characteristic polynomial of the matrix (aij )ij . In particular, det(D) is a monic polynomial belonging to R[x]. in (5.3) Dkj and sum the n equalities obtained Multiply the jth equality by n mi (xδij − aij )) Dkj = 0 for every k = in this way. One obtains that nj=1 ( i=1   n n = 0, so that mk det(D) = 0 for 1, . . . , n. Thus i=1 mi (xδ a − )D ij ij kj j=1 every k = 1, . . . , n. It follows that M det(D) = 0. Thus the monic polynomial det(D) belongs to the annihilator I of the R[x]-module MR[x] . Since the factor ring R[x]/(det(D)) is generated as an R-algebra by an element integral over R, it follows that R[x]/(det(D)) is an integral extension of R [Atiyah and Macdonald, Corollary 5.3]. Thus the ring R[x]/(det(D)) also has Krull dimension zero [Atiyah and Macdonald, Corollary 5.8]. Therefore the factor ring R[x]/I of R[x]/(det(D)) also has Krull dimension 0. Let us show that the element u := x + I ∈ R[x]/I is invertible in the ring S := R[x]/I. Assume the contrary. Then u is in a prime ideal P of the zerodimensional ring S. The nilradical of the localization SP is PP . Therefore the image of u in SP is nilpotent. Let t ≥ 1 be its degree of nilpotency. Then there exists s ∈ S\P with sut = 0 and sut−1 = 0 in S. Multiplication by u is the injective endomorphism f of M . Thus multiplication by ut is the injective endomorphism f t of M . Thus sut = 0 implies M sut = 0, that is, f t (M s) = 0, so that M s = 0 by the injectivity. Now M is a faithful S-module, so that s = 0, contradiction because sut−1 = 0. This contradiction shows that u is invertible in S. Multiplication by

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177

the inverse of u in S is an endomorphism of MR that is the inverse of f . Therefore  f is an automorphism of MR . Corollary 5.18. If R is a commutative ring of Krull dimension zero and MR is a finitely generated module of finite Goldie dimension n, then the endomorphism ring End(MR ) is a semilocal ring of dual Goldie dimension ≤ n. 

Proof. Corollary 5.16 and Theorem 5.17.

5.5 Finitely Copresented Objects We say that an object A of a Grothendieck category A is a finitely copresented object if there is an exact sequence 0 → A → L0 → L1 → 0 in A, where L0 is an injective object and both L0 and L1 are objects of finite Goldie dimension [Facchini and Herbera 2006]. Lemma 5.19. Let A be a Grothendieck category. The following conditions are equivalent for an object A of A. (a) A is finitely copresented. (b) Both the injective envelope E(A) of A and the factor object E(A)/A have finite Goldie dimension. (c) A is the kernel in A of a morphism between injective objects of finite Goldie dimension. Proof. (a) ⇒ (b). If (a) holds, there exists a short exact sequence 0 → A → L0 → L1 → 0 in A, where L0 is an injective object and both L0 and L1 are objects of finite Goldie dimensions dim(L0 ) and dim(L1 ), respectively. Then the injective envelope E(A) of A is a direct summand of L0 , and we obtain by restriction an exact sequence 0 → A → E(A) → L1 . Now dim(A) = dim(E(A)) ≤ dim(L0 ) and dim(E(A)/A) ≤ dim(L1 ). (b) ⇒ (c) A is the kernel of the canonical morphism E(A) → E(E(A)/A). (c) ⇒ (a) is trivial.  In the sequel, we will need the following result, called the snake lemma. It is a tool often used in homological algebra to construct long exact sequences. Theorem 5.20 (Snake lemma). Consider the following commutative diagram in an abelian category A: A

k

0

p

g

f

 /A

/ B

i

 /B

/ C h

l

 /C

/0

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Suppose that the two rows of the diagram are exact. Then there exists an exact sequence of the form δ

ker(f ) → ker(g) → ker(h) −→ coker(f ) → coker(g) → coker(h). Moreover, if k : A → B  is a monomorphism, then ker(f ) → ker(g) is a monomorphism, and if l : B → C is an epimorphism, then coker(g) → coker(h) is an epimorphism. We don’t give a proof of the snake lemma, which can be found in any textbook of homological algebra. The standard proof is via diagram chasing in the category of modules, and then one moves on to the general case of an arbitrary abelian category via the Freyd–Mitchell Theorem 4.71. To see the “snake,” draw the morphism δ as in the following diagram: ker(f )

k

εA

 A

/ ker(g)

p

εB  k

 / B

δ

ED

εC  p

g

f

/ ker(h)  / C h

/0 BC

GF  /A

0 @A

i

πf

δ

 / coker(f )

 /B

l

πg



i

 / coker(g)

 /C πh

l



 / coker(h)

Let us go back to spectral categories. Let A be a Grothendieck category. Consider the first right derived functor P (1) : A → Spec(A) of the canonical functor P : A → Spec(A). By Proposition 5.12, if A is an object of A and 0 → A → E0 → E1 → E2 → · · · is the minimal injective resolution of A, then P (1) (A) = P (E1 ). Equivalently, if E(A) is the injective envelope of A, then P (1) (A) = P (E(A)/A). Thus an object A is finitely copresented if and only if both objects P (A) and P (1) (A) of Spec(A) have finite composition length (equivalently, are finitely generated semisimple objects). If f : A → A is a morphism, f0 : E(A) → E(A ) is any extension of f to the injective envelopes, and f1 : E(A)/A → E(A )/A is the morphism induced by f0 , then P (1) (f ) = P (f1 ). Theorem 5.21 ([Facchini and Herbera 2006, Theorem 5.3]). If A is a Grothendieck category, the product functor P × P (1) : A → Spec(A) × Spec(A) is a local functor.

5.5. Finitely Copresented Objects

179

Proof. Let A, A be objects of A and f : A → A a morphism. Let f0 : E(A) → E(A ) be an extension of f to the injective envelopes, and let f1 : E(A)/A → E(A )/A be the morphism induced by f0 . We must prove that if both P (f ) and P (1) (f ) are isomorphisms in Spec(A), then f is an isomorphism in A. We have a commutative diagram with exact rows 0

/A f

0

 / A

/ E(A) f0

 / E(A )

/ E(A)/A

/0

f1

 / E(A )/A

/0

Assume P (f ) and P (1) (f ) isomorphisms. As we have remarked before the statement of the theorem, we have that P (1) (A) = P (E(A)/A), P (1) (A ) = P (E(A )/A ), and P (1) (f ) = P (f1 ). Thus P (f ) and P (f1 ) are isomorphisms. By Proposition 5.3, f and f1 are essential monomorphisms. Therefore the extension f0 : E(A) → E(A ) of f is also an essential monomorphism. Essential monomorphisms with an injective domain are isomorphisms. Thus f0 is an isomorphism in A. By the snake lemma (Theorem 5.20), we get an exact sequence 0 = ker f1 → cokerf → cokerf0 = 0. Thus cokerf = 0; hence f is also an epimorphism.



Notice that in general, the functor P : A → Spec(A) is not a local functor and is not an almost local functor. As an example, consider the endomorphism λ2 : Z → Z, given by multiplication by 2, in the category A := Ab. The endomorphism P (λ2 ) of P (Z) is an automorphism. Theorem 5.22. The endomorphism ring of any finitely copresented object A of a Grothendieck category A is a semilocal ring whose dual Goldie dimension is ≤ dim(A) + dim(E(A)/A). Proof. By Theorem 5.21, the functor P × P (1) : A → Spec(A) × Spec(A) induces a local morphism of EndA (A) into EndSpec(A) (P (A)) × EndSpec(A) (P (E(A)/A)), which is a semisimple Artinian ring of Goldie dimension dim(A) + dim(E(A)/A) (Proposition 5.15). Now apply Theorem 3.27 and Corollary 3.28.  In the rest of the section, we will apply the previous results to the case in which the Grothendieck category A is the category Mod-R of right modules over a ring R. An R-module M is quotient finite-dimensional if every homomorphic image of M has finite Goldie dimension. For instance, Noetherian modules and Artinian modules are clearly quotient finite-dimensional. Every submodule and every homomorphic image of a quotient finite-dimensional module is quotient finite-dimensional. Lemma 5.23. If M is a module, N is a submodule of M , and both N and M/N are quotient finite-dimensional modules, then so is M .

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Proof. We must show that M/P has finite Goldie dimension for any submodule P of M . Now M/(N +P ) has finite Goldie dimension because M/N is quotient finitedimensional, so that there exist finitely many indecomposable injective modules E1 , . . . , En and a module morphism f : M → E1 ⊕ · · · ⊕ En with kernel N + P . The module (N + P )/P ∼ = N/(N ∩ P ) has finite Goldie dimension because N is quotient finite-dimensional. Thus there exist finitely many indecomposable injective modules En+1 , . . . , Em and a module morphism g : N + P → En+1 ⊕ · · · ⊕ Em with kernel P . The morphism g extends to a morphism h : M → En+1 ⊕ · · · ⊕ Em ; hence (N + P ) ∩ ker h = ker g = P . The kernel of the morphism (f, h) : M → E1 ⊕· · ·⊕En ⊕En+1 ⊕· · ·⊕Em is ker(f, h) = ker f ∩ker h =  (N + P ) ∩ ker h = P . Therefore M/P has finite Goldie dimension ≤ m. In particular, we get the following corollary. Corollary 5.24. The direct sum of finitely many quotient finite-dimensional modules is a quotient finite-dimensional module. Proposition 5.25. A module M is quotient finite-dimensional if and only if the socle of every homomorphic image of M is of finite Goldie dimension. Proof. If M is quotient finite-dimensional, then every homomorphic image N of M has finite Goldie dimension, so that in particular its socle soc(N ) has finite Goldie dimension. Conversely, if M is not quotient finite-dimensional, then there exists a submodule N of M with M/N of infinite Goldie dimension. Thus M/N ⊇ N i≥0 i /N for suitable nonzero submodules Ni /N of M/N . Every nonzero module has a homomorphic image with simple socle. Thus, for every i ≥ 0, there We leave to exists a submodule Ni of Ni containing N with soc(Ni /Ni ) simple. the reader to show that the socle of the homomorphic image M/ i≥0 Ni has  infinite Goldie dimension. A module MR over a ring R is said to be ℵ0 -small if for every family { Mi | i ∈ I } of R-modules and any homomorphism ϕ : MR → ⊕i∈I Mi , there is a subset F ⊆ I with |F | ≤ ℵ0 such that ϕ(M ) ⊆ ⊕i∈F Mi . It is possible to prove the following result, though we will not need it here: Proposition 5.26 ([Facchini and Herbera 2004, Proposition 3.2]). Every quotient finite-dimensional module is ℵ0 -small. Proposition 5.27. The endomorphism ring of any submodule of a quotient finitedimensional injective module is a semilocal ring. Proof. Let N be a submodule of a quotient finite-dimensional injective module M . Then M and M/N have finite Goldie dimension, so that N is a finitely copresented module (Lemma 5.19). Now apply Theorem 5.22. 

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181

Lemma 5.28. Let R be a commutative integral domain with field of fractions Q. If Q is a quotient finite-dimensional R-module, S is an R-algebra, and MS is an Smodule that is torsion-free of finite rank as an R-module, then MS has a semilocal endomorphism ring. Proof. If Q is a quotient finite-dimensional R-module, then all the direct sums Qn are quotient finite-dimensional injective R-modules (Corollary 5.24). The submodules NR of the R-modules Qn , that is, torsion-free R-modules of finite rank, have semilocal endomorphism rings End(NR ) by Proposition 5.27. If MS is an S-module that is torsion-free of finite rank as an R-module, then MS has an endomorphism  ring End(MS ) that is semilocal by Proposition 3.35. The last results of this section generalize some results proved by Warfield. He proved in [Warfield 1980, Theorem 5.2] that if R is a semilocal commutative principal ideal domain and S is an R-algebra that is torsion-free and of finite rank as an R-module, then S is a semilocal ring. The following corollary generalizes that theorem. Corollary 5.29. Let S be an algebra over a commutative Noetherian semilocal domain R of Krull dimension 1. If MS is an S-module that is torsion-free of finite rank as an R-module, then End(MS ) is a semilocal ring. Proof. It is possible to prove that if Q is the field of fractions of a commutative Noetherian semilocal domain R of Krull dimension 1, then Q/R is an Artinian R-module [Matlis 1961, Theorem 1 p. 571]. It easily follows that Q is a quotient finite-dimensional injective R-module (Proposition 5.25). Thus we can apply  Lemma 5.28. The following corollary is proved in [Warfield 1980, Theorem 5.4] in the case MS = S. Corollary 5.30. If R is a commutative valuation domain, S is an R-algebra, and MS is an S-module that is torsion-free of finite rank as an R-module, then MS has a semilocal endomorphism ring. Proof. If R is a valuation domain, its field of fractions is clearly quotient finite dimensional. Thus we can apply Lemma 5.28. We now give a further extension of Corollary 5.30. We introduce a class of modules to which a large portion of this monograph will be devoted. A right module M over a not necessarily commutative ring is uniserial if its lattice of submodules is linearly ordered under set inclusion. That is, if for any submodules N and P of M either N ⊆ P or P ⊆ N . A module is said to be a serial module if it is a direct sum of uniserial submodules. Thus a module is serial of finite Goldie dimension if and only if it is a direct sum of finitely many uniserial modules, and a commutative integral domain R is a valuation domain if and only if RR is a

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uniserial module. The following corollary applies to modules over not necessarily commutative rings. Corollary 5.31. If E is an injective serial right module of finite Goldie dimension, then the endomorphism ring of any submodule of E is a semilocal ring. Proof. Since E is a direct sum of finitely many uniserial submodules, E is quotient finite-dimensional (Corollary 5.24). We can conclude by Proposition 5.27. 

5.6 The Dual Construction to the Construction of the Spectral Category Now we dualize the construction of the spectral category. Let A be an arbitrary Grothendieck category. Let A be the category with the same objects as A and, for all objects A, B ∈ Ob(A), with HomA (A, B) = lim HomA (A, B/B  ). −→ Here the direct limit is taken over the upward directed family of superfluous subobjects B  of B. Let F : A → A be the canonical functor, which is the identity on objects. We will denote the image F (A) of an object A of A by A, and the image F (f ) of a morphism f in A by f . (For a more formal construction, showing that the system of all the superfluous epimorphisms of a Grothendieck category A is a left-calculable multiplicative system of morphisms in A and that the category A = AS of additive fractions of A relative to S exists, see [Facchini and Herbera 2006, Section 6].) In the rest of this section, we will apply the construction of the category A only in the case that A is a category of modules Mod-R. First of all, let us describe in detail how the composition in (Mod-R) is defined in this case. Let f : A → B and g : B → C be morphisms in (Mod-R) . The morphisms f and g have representatives f : A → B/B  and g : B → C/C  in Mod-R for suitable superfluous submodules B  ≤s B and C  ≤s C. By Lemma 2.12, we get that g(B  ) ≤s C/C  . From Lemma 2.13, it follows that g(B  ) = C  /C  for a superfluous submodule C  of C with C  ⊇ C  . In particular, the morphism g : B → C/C  induces a morphism g  : B/B  → (C/C  )/(C  /C  ), and there is a canonical isomorphism h : (C/C  )/(C  /C  ) → C/C  . The composite mapping h ◦ g  ◦ f : A → C/C  is a morphism in A = Mod-R whose image in (Mod-R) is by definition the composite morphism g ◦ f : A → C in (Mod-R) . Let us determine when a morphism in Mod-R becomes the zero morphism in (Mod-R) . Lemma 5.32. Let A and B be R-modules, B  a superfluous submodule of B, and f : A → B/B  a morphism in Mod-R. Then f : A → B is the zero morphism in (Mod-R) if and only if f (A) is a superfluous submodule of B/B  .

5.6. The Dual Construction to the Construction of the Spectral Category

183

Proof. If f is the zero morphism, then f : A → B/B  and the zero morphism 0 : A → B/B  are mapped to the same element of the direct limit lim HomA (A, B/B  ). −→ Thus there is a superfluous submodule B  ≤s B containing B  such that the composition of f with the canonical projection B/B  → B/B  is the zero morphism. It follows that f (A) ⊆ B  /B  . Now B  /B  is superfluous in B/B  by Lemma 2.13. Therefore f (A) is superfluous in B/B  . Conversely, suppose f (A) superfluous in B/B  . Then f (A) = B  /B  for a suitable superfluous submodule B  of B (Lemma 2.13). Therefore f is also the f

image of the composite mapping A−→B/B  → B/B  , which is the zero morphism. It follows that f = 0.  Corollary 5.33. If A is a right R-module, then A is a zero object in (Mod-R) if and only if A is the zero module. Proof. Since (Mod-R) is an additive category, an object A is a zero object if and only if its identity morphism 1A is the zero endomorphism in (Mod-R) . Since 1A = 1A , this happens if and only if A is a superfluous submodule of A (Lemma 5.32), that is, if and only if A = 0.  Proposition 5.34. Let A and B be R-modules, B  a superfluous submodule of B, and f : A → B/B  a morphism in Mod-R. Then: (a) The morphism f : A → B in (Mod-R) is a monomorphism if and only if the inverse image via f of every superfluous submodule of B/B  is a superfluous submodule of A. (b) The morphism f : A → B is an epimorphism in (Mod-R) if and only if f is an epimorphism in Mod-R. (c) The morphism f : A → B is an isomorphism in (Mod-R) if and only if f is a superfluous epimorphism in Mod-R. Proof. (a) Let f : A → B be a monomorphism in (Mod-R) . Let C/B  be a superfluous submodule of B/B  . Let D = f −1 (C/B  ) be the inverse image of C/B  via f . Consider the embedding ε : D → A. The image of the composite mapping f ε is contained in C/B  , so that it is a superfluous submodule of B/B  . Thus we have that f ε = f ε = 0 in (Mod-R) by Lemma 5.32. Now f is a monomorphism, so that ε = 0. By Lemma 5.32 again, the image of ε is superfluous. Thus D = f −1 (C/B  ) is a superfluous submodule of A. For the converse, assume that inverse images via f of superfluous submodules are superfluous submodules. In order to prove that f is a monomorphism, fix an R-module morphism g : X → A/A with A superfluous in A, and assume f g = 0. The morphism f g is the image in A of the composite morphism g

f

X −→A/A −→(B/B  )/f (A ). From f g = 0, it follows that the image f  g(X) is superfluous in (B/B  )/f (A ) (Lemma 5.32). Now g(X) = C/A for a suitable

184

Chapter 5. Spectral Category and Dual Construction

submodule C of A containing A . Thus f  g(X) = f  (C/A ) = f (C)/f (A ). From f (C)/f (A ) ≤s (B/B  )/f (A ), we get that f (C) ≤s B/B  (Lemma 2.13). So f −1 (f (C)) ≤s A. Now C ⊆ f −1 (f (C)), so that C is superfluous in A. Thus g(X) = C/A is superfluous in A/A , from which g = 0 by Lemma 5.32. This proves that f is a monomorphism. (b) Suppose f : A → B is an epimorphism. Let p : B/B  → (B/B  )/f (A) denote the canonical projection. Clearly, pf = 0, and so pf = 0. Since f is an epimorphism, we have that p = 0. By Lemma 5.32, the image of p is a superfluous submodule of (B/B  )/f (A). But p is onto. Since the only module for which the improper submodule is a superfluous submodule is the zero module, we get that (B/B  )/f (A) = 0, and thus f is onto. For the converse, let f : A → B/B  be an epimorphism. Consider an Rmodule morphism g : B → X/X  with X  ≤s X, and assume gf = 0. The image of g  f : A → (X/X )/g(B  ), where g  : B/B  → (X/X  )/g(B  ) is the morphism induced by g, is a superfluous submodule of (X/X )/g(B  ). Since the image of g  f : A → (X/X  )/g(B  ) is g(B)/g(B  ), we get that g(B)/g(B  ) ≤s (X/X )/g(B  ), so that g(B) ≤s X/X  . Thus g = 0, which proves that f is an epimorphism in A . (c) Let f be an isomorphism. Then f is both a monomorphism and an epimorphism. From (a) we get that f has a superfluous kernel, and from (b) we get that f is an epimorphism. For the converse, let f be an epimorphism with a superfluous kernel K. Then f induces an isomorphism f  : A/K → B/B  . It is easily checked that if p : B → B/B  denotes the canonical projection and (f  )−1 p : B → A/K is the composite morphism, then (f  )−1 p : B → A is the inverse of f .  Proposition 5.35. Let NR be a finitely generated right R-module with a projective cover. Then EndA (N ) ∼ = EndR (N/N J(R)). Proof. Let p : PR → NR be a projective cover of NR . By Proposition 5.34(c), p is an isomorphism. Thus EndA (N ) ∼ = EndA (P ). Moreover, NR ∼ = PR / ker p, so ∼ that N J(R) = (P J(R) + ker p)/ ker p. Now P J(R) is the radical of PR (Proposition 2.16), and the radical contains the superfluous submodule ker p (Lemma 2.15). Thus NR /N J(R) ∼ = PR /P J(R). Finally, if NR is finitely generated, then its projective cover PR is also finitely generated (Theorem 2.27(a)). Thus, without loss of generality, we can suppose NR projective. If NR is a finitely generated projective module, every element of EndA (N ) is represented by a morphism f : NR → NR /N  for some superfluous submodule N  of NR , and f can be lifted to an endomorphism of NR . This proves that the canonical mapping Φ : EndR (NR ) → EndA (N ) is onto. An endomorphism f of NR is in the kernel of Φ if and only if f (NR ) ≤s NR (Lemma 5.32), if and only if f ∈ J(End(NR )) (Proposition 2.24). Thus ker Φ = J(End(NR )) and End(NR )/J(End(NR )) ∼ = EndA (N ).  Finally, End(NR )/J(End(NR )) ∼ = End(NR /N J(R)) by Corollary 2.25.

5.6. The Dual Construction to the Construction of the Spectral Category

185

For instance, if Ab is the category of abelian groups, then EndAb (Z) ∼ = Z by the previous proposition. In particular, EndAb (Z) is not a von Neumann regular ring, so that Ab is not a spectral category (Example 5.11). Also notice that in a spectral category, every epimorphism has a right inverse. The dual property, i.e., that every monomorphism has a left inverse, does not hold in the category Ab . For instance, multiplication by 2 is an element of EndAb (Z) ∼ = Z, which is a monomorphism by Proposition 5.34(a), because the inverse image of the unique superfluous submodule of Z is zero. But 2 does not have an inverse in Z. In the next proposition, we show that every morphism has a cokernel in the category (Mod-R) . Proposition 5.36. Let R be a ring. If f : A → B/B  is a morphism in Mod-R with B  ≤s B, and p : B → (B/B  )/f (A) is the canonical projection defined by p(b) = (b + B  ) + f (A) for every b ∈ B, then the image p : B → (B/B  )/f (A) of p in (Mod-R) is the cokernel of the image f : A → B of f in (Mod-R) . Proof. The mapping p can be written as the composite mapping of the two canonical mappings p : B → B/B  and p : B/B  → (B/B  )/f (A). Then p f = 0 implies that pf = p f = 0. Suppose that g : B → X/X  is any R-module morphism with X  ≤s X and gf = 0, and let g  : B/B  → (X/X  )/g(B  ) be the morphism induced by g. The image of the morphism g  f : A → (X/X  )/g(B  ) is a superfluous submodule of (X/X  )/g(B  ) (Lemma 5.32). Now f (A) = B  /B  for a suitable submodule B  of B. Then the image of g  f is g(B  )/g(B  ). It follows that g(B  ) is superfluous in X/X  . The composite morphism of g  and the canonical mapping (X/X  )/g(B  ) → (X/X  )/g(B  ) is a morphism g  : B/B  → (X/X )/g(B  ) such that g  f = 0. Since p is the cokernel of f , there is a unique morphism h : (B/B  )/f (A) → (X/X  )/g(B  ) in Mod-R such that g  = hp . Then g  p = hp, so that g = g  p = hp. From Proposition 5.34(b), we know that p is an epimorphism, which implies that h is the unique morphism in (Mod-R) with the property that g = hp.  In the next example, we show that the category (Mod-R) does not have kernels in general. Example 5.37. Let R = Z be the ring of integers and let Z(N) = ⊕n≥1 Zxn be a free abelian group of countable rank. Fix a prime p. Let S be the subgroup of Z(N) generated by the elements px1 and pxn+1 − xn for all n ≥ 1. The quotient group Z(N) /S is the Pr¨ ufer group Z(p∞ ). We will show that if π : Z(N) → Z(N) /S denotes the canonical projection, then the morphism π does not have a kernel in (Mod-Z) . The abelian group Z(N) has no nonzero superfluous subgroup, and in Z(p∞ ) all proper subgroups are superfluous. It follows that if f : G → Z(N) is an abelian group morphism, then πf = 0 if and only if the subgroup πf (G)

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of Z(N) /S is finite. We will say that a morphism f : G → Z(N) has rank n if πf (G) has pn elements, so that every morphism f : G → Z(N) with πf = 0 has finite rank n ≥ 0. Assume by way of contradiction that π has a kernel k in (Mod-Z) . Since Z(N) has no nonzero superfluous subgroup, k is the image in (Mod-Z) of a suitable group morphism k : G → Z(N) . Let n ≥ 0 be the rank of the morphism k. Now there exists a morphism f : H → Z(N) with πf = 0 and rank n + 1. Since k is the kernel of π, there exists a unique morphism g : H → G such that f = kg. The morphism g is the image in (Mod-Z) of a morphism g : H → G/G for a suitable superfluous subgroup G of G. Now Z(N) has no nonzero superfluous subgroup and k(G ) must be superfluous in Z(N) , so that k(G ) = 0. Thus k induces a morphism k  : G/G → Z(N) , and f = kg = k  g. Again, Z(N) does not have nonzero superfluous subgroups, and thus f = k  g. It follows that πf (H) = πk  g(H) ⊆ πk  (G/G ) = πk(G). Hence the rank of f must be less than or equal to the rank of k, which is a contradiction because the rank of f is n + 1 and the rank of k is n. Proposition 5.38. If R is a right perfect ring and J(R) is its Jacobson radical, then the factor ring R/J(R) is semisimple Artinian and the category (Mod-R) is equivalent to the category Mod-(R/J(R)). Proof. Every right perfect ring R is semilocal, i.e., the factor ring R/J(R) is semisimple Artinian [Anderson and Fuller, Theorem 28.4]. Let A be the full subcategory of Mod-R whose objects are all right R-modules annihilated by J(R). It is well known that A is canonically isomorphic to the category Mod-(R/J(R)). Moreover, the objects of A are exactly the semisimple right R-modules, because R/J(R) is semisimple Artinian and thus every R-module annihilated by J(R) is a direct sum of simple R-modules. Consider the restriction F |A : A → (Mod-R) of the canonical functor F : Mod-R → (Mod-R) . In order to conclude the proof of the proposition, it suffices to show that F |A is a category equivalence. If BR is an object of A, then BR is a semisimple R-module, so that rad(BR ) = 0, and therefore BR has no nonzero superfluous submodule (Lemma 2.15). Thus, for AR , BR objects of A, we have that Hom(Mod-R) (AR , BR ) = lim HomR (AR , BR /B  ) = HomR (AR , BR ). −→ This proves that the functor F |A : A → (Mod-R) is full and faithful. If MR is an arbitrary object of (Mod-R) , then the R-module MR has a projective cover p : PR → MR ; hence F (p) : F (PR ) → F (MR ) is an isomorphism in (Mod-R) by Proposition 5.34(c). Since R is a right perfect ring, its Jacobson radical J(R) is right T -nilpotent (Bass’s Theorem P, [Anderson and Fuller, Lemma 28.3]), so that PR J(R) is a superfluous submodule of PR by [Anderson and Fuller, Lemma 28.3] again. Thus the canonical projection PR → PR /PR J(R) is also a superfluous epimorphism in Mod-R. Thus F (PR ) ∼ = F (PR /PR J(R)) in (Mod-R)

5.7. Applications of the Category (Mod-R)

187

∼ F (PR /PR J(R)) = by Proposition 5.34(c). It follows that F (MR ) ∼ = F (PR ) =  F |A (PR /PR J(R)). This concludes the proof that F |A is an equivalence. The proof of Proposition 5.38 shows that for a right perfect ring R, the canonical functor F : Mod-R → (Mod-R) is the composite functor of the functor G : Mod-R → A that sends a right R-module MR to the right R-module MR /MR J(R) and the category equivalence F |A : A → (Mod-R) . The functor G is naturally isomorphic to the functor − ⊗R R/J(R) : Mod-R → A. Thus G and hence F are additive, covariant, right exact functors, and we can compute their derived functors G(n) and F(n) . Every object MR of Mod-R has a minimal projective resolution · · · → P2 → P1 → P0 → M → 0 because R is right perfect, and F(n) (M ) = F (Pn ) for every n ≥ 0. Thus, for a right perfect ring R, F(n) : Mod-R → (Mod-R) is the composite functor of the functor TorR n (−, R/J(R)) : Mod-R → A and the category equivalence F |A : A → (Mod-R) . The proofs are the duals of the proofs of Proposition 5.12 and Corollary 5.13.

5.7 Applications of the Category (Mod-R) Let R be any ring. The category (Mod-R) turns out to be an additive category with cokernels (but in general without kernels, Example 5.37), and the canonical functor F : Mod-R → (Mod-R) is right exact in the sense that it preserves cokernels. In this section, we will present some results of [Facchini and Herbera 2006]. Lemma 5.39. Let R be a ring. (a) The endomorphism ring End(Mod-R) (F (U )) of a couniform right R-module U is a division ring. (b) If U and V are couniform right R-modules, then Hom(Mod-R) (F (U ), F (V )) = 0 if and only if there exist proper submodules U  of U and V  of V with U/U  and V /V  isomorphic R-modules, if and only if F (U ) and F (V ) are isomorphic objects in the category (Mod-R) . Proof. (a) Any endomorphism f of F (U ) is represented by a morphism f : U → U/U  for a suitable superfluous (=proper, because U is couniform) submodule U  of U . By Lemma 5.32, f is the zero endomorphism if and only if f (U ) is a superfluous submodule of U/U  . Thus f nonzero implies f (U ) is nonsuperfluous in U/U  , that is, f is onto. In particular, the kernel of f is superfluous, so that f is an automorphism of F (U ) by Proposition 5.34(c). Thus f is invertible in End(Mod-R) (F (U )), and End(Mod-R) (F (U )) is a division ring. (b) By Lemma 5.32, Hom(Mod-R) (F (U ), F (V )) = 0 if and only if there exist a superfluous (=proper) submodule V  of V and a morphism f : U → V /V  with f (U ) nonsuperfluous in V /V  , that is, with f onto. If such an f exists, then

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∼ V /V  . Conversely, if there exist proper submodules U  < U and V  < V U/ ker f = ∼ V /V  , then there is an epimorphism f : U → V /V  . Thus the first with U/U  = two conditions in (b) are equivalent. Now if there exist proper submodules U  ∼ ∼ F (V /V  ) = of U and V  of V with U/U  ∼ = V /V  , then F (U ) ∼ = F (U/U  ) = F (V ). Conversely, an isomorphism F (U ) → F (V ) is represented by a superfluous epimorphism f : U → V /V  , which yields an isomorphism U/ ker f ∼ = V /V  with   ker f and V superfluous (=proper) submodules. Proposition 5.40. Let R be a ring and AR a right R-module of finite dual Goldie dimension n. Then: (a) The ring End(Mod-R) (F (AR )) is a semisimple Artinian ring of Goldie dimension n. (b) If f ∈ End(AR ), then F (f ) is invertible in End(Mod-R) (F (AR )) if and only if f is an R-module epimorphism. (c) If all R-module epimorphisms AR → AR are isomorphisms, then the ring morphism φAR : End(AR ) → End(Mod-R) (F (AR )), induced by the functor F , is a local morphism. Proof. (a) Since codim(AR ) = n, we know from Theorem 3.7 that there exists a finite coindependent set {N1 , N2 , . . . , Nn } of proper submodules of AR with AR /Ni couniform for every i = 1, 2, . . . , n and N1 ∩ N2 ∩ · · · ∩ Nn a superfluous submodule of AR . Since this set is coindependent, we have that AR /N1 ∩ ∼ F (AR /N1 ∩ N2 ∩ · · · ∩ Nn ) by N2 ∩ · · · ∩ Nn ∼ = ⊕ni=1 AR /Ni , and F (AR ) = n ∼ Proposition 5.34. Thus F (AR ) = ⊕i=1 F (AR /Ni ). By Lemma 5.39(a), the endomorphism rings End(Mod-R) (F (AR /Ni )) are division rings, and by Lemma 5.39(b), Hom(Mod-R) (F (AR /Ni ), F (AR /Nj )) = 0 if F (AR /Ni ) and F (AR /Nj ) are not isomorphic in (Mod-R) . Statement (a) now follows easily. (b) If f : AR → AR is an R-module morphism, F (f ) is invertible in (Mod-R) if and only if f is a superfluous epimorphism in Mod-R (Proposition 5.34(c)). Now AR has finite dual Goldie dimension n, so that all epimorphisms AR → AR have a superfluous kernel (Proposition 3.8(d)). (c) If f : AR → AR is a morphism and F (f ) is invertible in (Mod-R) , then f is an R-module epimorphism by (b), so that f is an isomorphism by hypothesis.  From Proposition 5.40 and Theorem 3.27, we get that if R is a ring, AR is a right R-module of finite dual Goldie dimension n, and all R-module epimorphisms AR → AR are isomorphisms, then the endomorphism ring of AR is semilocal and codim(End(AR )) ≤ codim(AR ) [Herbera and Shamsuddin, Theorem 3(2)]. Proposition 5.41. Let R be a ring and A the full subcategory of (Mod-R) whose objects are all right R-modules of finite dual Goldie dimension. Then A is an amenable semisimple category. Proof. The category A is additive, and the endomorphism ring EndA (A) of every object A of A is semisimple Artinian by Proposition 5.40(a). Let us prove that

5.7. Applications of the Category (Mod-R)

189

idempotents split in A. If A is any R-module of finite dual Goldie dimension, then A contains a superfluous submodule A with A/A a direct sum of finitely many couniform modules, and A ∼ = A/A in (Mod-R) (Proposition 5.34(c)), so that A is equivalent to its full subcategory whose objects are all direct sums of finitely many couniform modules. Hence it suffices to show that if V1 , . . . , Vn (n ≥ 1) are couniform R-modules, every endomorphism of B := V1 ⊕ · · · ⊕ Vn in A splits. Set E := EndA (B) and consider the additive, full and faithful functor F := HomA (B, −) : add(B) → proj-E of Lemma 4.18. Since E is semisimple Artinian of Goldie dimension n (Proposition 5.40(a)), every object of proj-E is a direct sum of the ≤ n simple E-modules, so that every object of add(B) is a direct sum of copies of V1 , . . . , Vn . Thus the functor F is also essentially surjective, that is, it is an equivalence between add(B) and proj-E. Since idempotents split in proj-E, it follows that idempotents split in add(B). It follows that every  idempotent of B in A splits. Now consider the functors P : Mod-R → Spec(Mod-R) and F : Mod-R → (Mod-R) . Here, again, R is any ring. The kernel of P is the ideal I of Mod-R consisting of all morphisms with an essential kernel. The kernel of F is the ideal K of Mod-R consisting of all morphisms with a superfluous image. Proposition 5.42. The product functor P × F : Mod-R → Spec(Mod-R) × (Mod-R) is a local functor. The kernel of the product functor is the ideal I ∩ K, which is contained in the Jacobson radical of Mod-R. Proof. Let f : AR → BR be an R-module morphism with P (f ) and F (f ) invertible. Then P (f ) invertible implies that f is an essential monomorphism by Proposition 5.3, and F (f ) invertible implies that f is a superfluous epimorphism (Proposition 5.34(c)). Now f mono and epi implies f invertible.  Corollary 5.43 ([Herbera and Shamsuddin, Theorem 3(3)]). If AR is a module with dim(AR ) = n and codim(AR ) = m, then its endomorphism ring End(AR ) is a semilocal ring of dual Goldie dimension ≤ n + m. Proof. The functor P × F : Mod-R → Spec(Mod-R) × (Mod-R) is a local functor (Proposition 5.42). Hence it induces a local morphism End(AR ) → EndSpec(Mod-R) (P (AR )) × End(Mod-R) (F (AR )). The ring EndSpec(Mod-R) (P (AR )) is a semisimple Artinian ring of Goldie dimension n (Proposition 5.15). The ring End(Mod-R) (F (AR )) is a semisimple Artinian ring of Goldie dimension m (Proposition 5.40(a)). Thus End(AR ) is semilocal by Corollary 3.28, and codim(End(AR )) ≤ n + m by Theorem 3.27. 

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5.8 Finitely Presented Modules over a Semilocal Ring Let R be a ring and consider the covariant additive functor F : Mod-R → (Mod-R) . The category (Mod-R) is not an abelian category, for instance it does not have kernels (Example 5.37). Hence we cannot define the left derived functors of the functor F : Mod-R → (Mod-R) . Notice that in the category (Mod-R) , every morphism has a cokernel and that the functor F preserves cokernels (Proposition 5.36); hence F can be considered, in some sense, a “right exact” functor. In order to overcome these difficulties, we will consider the full subcategory C of Mod-R whose objects are all right R-modules with a projective cover. The category C is an additive category. We have already remarked that the kernel of the functor F : Mod-R → (Mod-R) is the ideal K of Mod-R consisting of all morphisms with a superfluous image. Thus F induces a faithful functor F : Mod-R/K → (Mod-R) (so that Mod-R/K can be viewed as a subcategory of (Mod-R) , which is not a full subcategory). Now define a functor K : C → Mod-R/K as follows. For every object MR of C, fix with the axiom of choice for classes a projective cover πM : P (M ) → MR , and let ker πM denote the kernel of πM in Mod-R. Let K map the object MR of C to the object ker πM of Mod-R/K. If f : MR → NR is a morphism in C, f lifts to a morphism f0 : P (M ) → P (N ), which restricts to a morphism f1 : ker πM → ker πN in Mod-R. Let K map the morphism f of C to the image of f1 in Mod-R/K. Lemma 5.44. The functor K : C → Mod-R/K is well defined. Proof. Assume we fix a different lifting f0 : P (M ) → P (N ) of f . Let f1 : ker πM → ker πN be the restriction of f0 . Since f0 and f0 induce the same morphism P (M )/ ker πM → P (N )/ ker πN , their difference f0 −

f0

induces the zero morphism P (M )/ ker πM → P (N )/ ker πN ,

that is, the image (f0 − f0 )(P (M )) of f0 − f0 is contained in ker πN . Thus we can restrict the codomain of f0 − f0 : P (M ) → P (N ), getting a morphism g : P (M ) → ker πN . That is, if εN : ker πN → P (N ) denotes the embedding, we have that f0 − f0 = εN g. Now, the image of a superfluous submodule is a superfluous submodule, so that g(ker πM ) is a superfluous submodule of ker πN . Now g(ker πM ) = (f0 − f0 )(ker πM ) = (f1 − f1 )(ker πM ). Thus the image of the difference f1 − f1 : ker πM → ker πN is a superfluous submodule of ker πN . That is, f1 − f1 is in the ideal K of Mod-R. This proves that the functor K is well defined.  Notice that if we fix a different choice of the projective covers πM : P (M ) → MR , we get a functor naturally isomorphic to K.

5.8. Finitely Presented Modules over a Semilocal Ring

191

Now let F(1) : C → (Mod-R) be the composite functor of the functors K : C → Mod-R/K and F : Mod-R/K → (Mod-R) . Theorem 5.45. The functor F × F(1) : C → (Mod-R) × (Mod-R) is local. Proof. Let f : MR → NR be a morphism in C with both F (f ) and F(1) (f ) isomorphisms in (Mod-R) . The morphism f lifts to a morphism f0 : P (M ) → P (N ), which restricts to a morphism f1 : ker πM → ker πN . Thus we get the commutative diagram / P (M ) πM / MR /0 / ker πM 0 f1

0



/ ker πN

f0

 / P (N )

f

πN

 / NR

/0

Now, both F (f ) and F(1) (f ) are isomorphisms in (Mod-R) . Thus f is a superfluous epimorphism in Mod-R by Proposition 5.34(c). Since F(1) (f ) = F (K(f )) = F (f1 ), we similarly get that f1 is a superfluous epimorphism in Mod-R. In order to conclude the proof, we must prove that f is a monomorphism. The commutativity of the diagram gives us that f πM = πN f0 , so that F (f )F (πM ) = F (πN )F (f0 ). Since πM and πN are superfluous epimorphisms, we get that F (πM ), F (πN ), and F (f ) are isomorphisms, so that F (f0 ) is an isomorphism as well; hence f0 is a superfluous epimorphism. A superfluous epimorphism onto a projective module is necessarily an isomorphism. Thus f0 is an isomorphism. From the commutativity of the diagram and the snake lemma, Theorem 5.20, we get an exact sequence / ker(f ) / coker(f1 ) = 0, so that ker(f ) = 0. This proves that 0 = ker(f0 ) f is an isomorphism.  Corollary 5.46. Let KR be a superfluous submodule of a projective module PR . If KR and PR have finite dual Goldie dimension, then End(PR /KR ) is a semilocal ring and codim(End(PR /KR )) ≤ codim(KR ) + codim(PR ). Proof. The functor F × F(1) : C → (Mod-R) × (Mod-R) of Theorem 5.45 is local. Hence it induces on the object PR /KR a local morphism End(PR /KR ) → End(Mod-R) (F (PR /KR )) × End(Mod-R) (F(1) (PR /KR )). Now codim(PR /KR ) = codim(PR ) is finite (Proposition 3.8(d)), so that the ring End(Mod-R) (F (PR /KR )) is a semisimple Artinian ring of Goldie dimension codim(PR ) (Proposition 5.40(a)). Similarly, F(1) (PR /KR ) ∼ = F (KR ) has finite dual Goldie dimension, so that the ring End(Mod-R) (F(1) (PR /KR )) is a semisimple Artinian ring of Goldie dimension codim(KR ). Now Theorem 3.27 allows us to conclude. 

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For the proof of Theorem 5.48, which is the main application of Corollary 5.46, we need a further preliminary result. Lemma 5.47. If S is a simple right module over a semilocal ring R, there exists a finitely presented right R-module N with S ∼ = N/N J(R). Proof. The simple module S is isomorphic to a direct summand of R/J(R). Thus ∼ R/J(R). Let ϕ : S ⊕ T → R/J(R) there exists an R/J(R)-module T with S ⊕ T = be an isomorphism. Now T is a cyclic R/J(R)-module. Let t be a generator of T , and let r be an element of R such that ϕ(t) = r + J(R). The right R-module N := R/rR has the property required in the statement, because N/N J(R) = (R/rR)/(R/rR)J(R) ∼ = (R/rR)/(J(R) + rR/rR) ∼ ∼ = R/(J(R) + rR) = (R/J(R))/(J(R) + rR/J(R)) ∼ ∼ S. = (R/J(R))/r(R/J(R)) = (R/J(R))/(r + J(R))(R/J(R)) =



Theorem 5.48 ([Facchini and Herbera 2006, Proposition 3.1]). The endomorphism ring of any finitely presented right module over a semilocal ring is a semilocal ring. Proof. Let MR be a finitely presented module over a semilocal ring R. The module MR /MR J(R) is a finitely generated module over the semisimple Artinian ring R/J(R), so that MR /MR J(R) is a finitely generated semisimple R/J(R)-module. Thus, we can “complete it” to a finitely generated free R/J(R)-module, that is, there exist simple R/J(R)-modules S1 , . . . , Sm such that MR /MR J(R) ⊕ S1 ⊕ · · · ⊕ Sm ∼ = (RR /J(R))n for some integer n ≥ 0. For every i = 1, 2, . . . , m, there exists a finitely presented R-module Ni with Si ∼ = Ni /Ni J(R) (Lemma 5.47). Hence NR := N1 ⊕ · · · ⊕ Nm is a finitely presented module and MR /MR J(R) ⊕ n NR /NR J(R) ∼ → MR /MR J(R)⊕ = (RR /J(R))n . Thus there is an epimorphism RR n NR /NR J(R) with superfluous kernel J(R)R , which can be lifted to an epimorphism n → MR ⊕ NR with a kernel that is superfluous because it is contained in RR n the superfluous submodule J(R)nR of RR . Therefore the finitely presented module n ∼ Rn /KR MR ⊕ NR has a projective cover isomorphic to RR , that is, MR ⊕ NR = R n for some finitely generated superfluous submodule KR of RR . Since R is semilocal, every finitely generated R-module has finite dual Goldie dimension, so that we can apply Corollary 5.46. Thus MR ⊕ NR has a semilocal endomorphism ring, and so MR has a semilocal endomorphism ring by Proposition 3.38.  Corollary 5.49. The endomorphism ring of any finitely generated right module over a semilocal right Noetherian ring is a semilocal ring. Notice that Theorem 4.58, according to which the endomorphism ring of every finitely generated module over a commutative semilocal ring is semilocal, cannot be extended to finitely generated modules over noncommutative semilocal rings: there exists a finitely generated module over a noncommutative semilocal ring whose endomorphism ring is not semilocal. See [Facchini and Herbera 2006, Example 3.5].

5.9. Notes on Chapter 5

193

5.9 Notes on Chapter 5 The injective envelope E(−) is not a functor of Mod-R to Mod-R, but it is very similar to a functor: it associates to any object AR of Mod-R the object E(AR ) of Mod-R and associates to any morphism f : AR → BR an extension E(f ) : E(AR ) → E(BR ). Notice that E(AR ) is defined only up to isomorphism. Moreover, the extension E(f ) : E(AR ) → E(BR ) of a morphism f : AR → BR is not unique. But the real difficulty is that for any two morphisms f : AR → BR and g : BR → CR , there is no guarantee that E(g)E(f ) = E(gf ). More precisely: Proposition 5.50 ([Goodearl 1976, Proposition 1.12]). There does not exist a functor F : Mod-Z → Mod-Z ∼ E(A) for all abelian groups AZ . such that F (A) = In Proposition 5.50, F need not even be an additive functor. For very special rings R, the injective envelope E(−) : Mod-R → Mod-R turns out to be a functor: Proposition 5.51 ([Goodearl 1976, Exercise 24, p. 48]). Let R be a ring. Suppose that all singular right R-modules are injective. Then there exists an additive functor F : Mod-R → Mod-R for which F (AR ) is an injective envelope of AR for all right R-modules AR . The functor F in the statement of Proposition 5.51 is defined by F (AR ) = Z(AR ) ⊕ S ◦ AR for every R-module AR and F (f ) = f |Z(AR ) ⊕ S ◦ f for every morphism f : AR → BR in Mod-R. Here S ◦ is the localization functor associated with the singular torsion theory, whose torsion modules are all R-modules AR with Z2 (AR ) = AR , and whose torsion-free modules are all nonsingular modules. In this case, in which all singular R-modules are injective, every module is the direct sum of a singular module and a nonsingular module, Z2 = Z and S ◦ A = E(A/Z(A)). These rings R, over which all singular right modules are injective, were studied and completely characterized in [Goodearl 1972]. For an example of a commutative ring R over which all singular modules are injective, fix a field F , let F N be the direct product of countably many copies of F , that is, the ring of all functions N → F , and R the subring of F N consisting of all functions N → F that are constant almost everywhere (that is, the functions α : N → F for which there exist a finite subset S of N and an element a ∈ F such that α(x) = a for every x ∈ N \ S). The ring R := 1F + F (N) ⊆ F N has the property that all singular modules are injective. There are other partial possible solutions to the problem of “trying to force E(−) to become a functor,” for instance specializing the modules, that is, restricting our attention to the full subcategory NS(R) of Mod-R whose objects are all nonsingular right R-modules. Then E(−) : NS(R) → Mod-R turns out to

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be a functor by Proposition 5.7. Another possible solution is changing the morphisms of Mod-R, that is, changing the category, as we have done in Section 5.1 in introducing the spectral category. The main sources for the results in this chapter as far as singular modules and injective modules are concerned are [Facchini and Herbera 2006] and [Goodearl 1976]. The construction of the spectral category is due to [Gabriel and Oberst]. Proposition 5.4 is taken from [Goodearl 1976, Proposition 1.20(b)]. The right derived functors of the functor P : A → Spec(A) for a Grothendieck category A (Proposition 5.12) were first studied in [Facchini 2008]. Theorem 5.48, which is probably the main result of [Facchini and Herbera 2006], extends the following three results: (1) Theorem 4.58 (The endomorphism ring of any finitely generated module over a commutative semilocal ring is semilocal.) (2) [Bj¨ork, Theorem 4.1] (The endomorphism ring of a finitely presented module over a semiprimary ring is semiprimary. A ring is semiprimary if it is semilocal and its Jacobson radical is nilpotent.) (3) [Schofield, Theorem 7.18] and Rowen [Rowen, Corollary 11] (The endomorphism ring of a finitely presented right module over a right (or left) perfect ring is a right (left, respectively) perfect ring.) For several other examples of modules with semilocal endomorphism rings, see [Herbera and Shamsuddin].

Chapter 6

Auslander–Bridger Transpose, Auslander–Bridger Modules 6.1 The Functor Auslander–Bridger Transpose The idea of the Auslander–Bridger transpose is the following. We have a finitely presented right R-module MR , defined by m generators and s relations say. Thus MR has a free presentation s RR

λA

/ Rm R

/ MR

/0.

s m and RR can be viewed as column vectors, that is, as Here the elements of RR s m s × 1 and m × 1 matrices respectively, so that the morphism λA : RR → RR is left multiplication by an m × s matrix A. Then we can apply to our free presentation the functor (−)∗ := Hom(−, R) : Mod-R → R -Mod, getting an exact sequence / M ∗ = Hom(MR , RR ) / R Rm ρA / R Rs . Here the elof left R-modules 0 m s ements of the left R-modules R Rm ∼ , RR ) and R Rs ∼ , RR ) = Hom(RR = Hom(RR are row vectors, that is, 1 × m and 1 × s matrices respectively, and the morphism ρA : R Rm → R Rs is right multiplication by the same m × s matrix A. We would like to call the cokernel of ρA , which is a finitely presented left R-module with a presentation with s generators and m relations, “the Auslander–Bridger transpose” of MR . The idea is interesting, but it does not work, because with this definition, “the Auslander–Bridger transpose” of MR would depend not only on MR , but also on the free presentation of MR we have originally fixed. For instance, let R := Z be the ring of integers and M the Z-module Z/2Z. We can take several presentations of M . For example, let n1 , . . . , ns be s ≥ 1 integers whose greatest common divisor is 2. Then n1 , . . . , ns generate the submodule 2Z λ(n1 ... ns ) of Z, that is, n1 Z + · · · + ns Z = 2Z, and there is a presentation Zs −→ Z→ Z/2Z → 0. When we dualize it, we get the exact sequence 0 → Hom(Z/2Z, Z) = ρ(n1 ... ns ) s 0 → ZZ −→ Z Z . Then “the Auslander–Bridger transpose” of Z/2Z would

© Springer Nature Switzerland AG 2019 A. Facchini, Semilocal Categories and Modules with Semilocal Endomorphism Rings, Progress in Mathematics 331, https://doi.org/10.1007/978-3-030-23284-9_6

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Chapter 6. Auslander–Bridger Transpose, Auslander–Bridger Modules

be Zs /Z(n1 . . . ns ). Since the greatest common divisor of n1 , . . . , ns is 2, one has that ni = 2ni for every i, where ni ∈ Z and the greatest common divisor of the s integers n1 , . . . , ns is 1. Then Z(n1 . . . ns ) ⊆ Z(n1 . . . ns ) ⊆ Zs , and we leave to the reader to show that Zs /Z(n1 . . . ns ) is torsion-free and finitely ∼ Zs−1 , the canongenerated, hence free, of rank s − 1, so that Zs /Z(n1 . . . ns ) = s s   ical epimorphism Z /Z(n1 . . . ns ) → Z /Z(n1 . . . ns ) splits, and its kernel is ∼ Zs−1 ⊕ Z/2Z. This Z(n1 . . . ns )/Z(n1 . . . ns ) ∼ = Z/Z2. Thus Zs /Z(n1 . . . ns ) = shows that “the Auslander–Bridger transpose” of Z/2Z depends on the presentation chosen, which is not good. There is a very similar situation in homological algebra, where the same problem with the choice of a presentation occurs when derived functors are defined. Let R be a ring, and let Mod-R be the category of all right R-modules (in fact, instead of Mod-R, we could take any abelian category A with enough projectives). A chain complex (or 0-sequence) C of right R-modules is a family (Cn )n∈Z of right R-modules with a family of R-module morphisms dn : Cn → Cn−1 , n ∈ Z, such that the composite mapping dn−1 ◦ dn : Cn → Cn−2 is the zero morphism for every n ∈ Z: / Cn+1 dn+1 / Cn dn / Cn−1 dn−1 / · · · ··· Chain complexes form a category Ch(Mod-R). A morphism f : C → C of chain complexes is a family of right R-module morphisms fn : Cn → Cn such that the following diagram commutes: / Cn+1

···

fn+1



 / Cn+1

···

dn+1

/ Cn

dn

/ Cn−1

fn

dn+1

 / Cn

dn



dn−1

/ ···

fn−1

 / Cn−1

dn−1

/ ···

That is, dn+1 ◦ fn+1 = fn ◦ dn+1 for every n ∈ Z. The idea in homological algebra is the following. Take a module MR and fix a projective resolution of MR . A projective resolution of MR is a chain complex P(M ) such that Pi = 0 for every i < 0, Pi is projective for every i ∈ Z, and there is an epimorphism ε : P0 → M such that the “augmented chain complex” ···

d3

/ P2

d2

/ P1

d1

/ P0

ε

/M

/0

is an exact sequence. The problem is that associating to any right R-module MR a projective resolution P(M ) is not a functor Mod-R → Ch(Mod-R) for a number of reasons, one of which is the fact that the projective resolution of a module MR is not unique. That is, the module MR has projective resolutions that are nonisomorphic objects of Ch(Mod-R). Nevertheless, the projective resolutions of a module MR become unique up to isomorphism if we replace the category Ch(Mod-R) with the factor

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6.1. The Functor Auslander–Bridger Transpose

category K(Mod-R) := Ch(Mod-R)/N H of Ch(Mod-R). Here N H is the ideal of Ch(Mod-R) consisting of all null homotopic morphisms of chain complexes. They are defined as follows. It is easily seen that if C, C are arbitrary chain  complexes, we fix a right R-module morphism sn : Cn → Cn+1 for every n ∈ Z,  and definefn : Cn → Cn to be fn := dn+1 ◦ sn + sn−1 ◦ dn for every n, then (fn )n∈Z is a morphism C → C of chain complexes. Morphisms C → C of this type are called null homotopic. Let N H be the ideal of Ch(Mod-R) such that N H(C, C ) is the set of all null homotopic morphisms C → C for every C, C ∈ Ob(Ch(Mod-R)). In other words, for any two complexes C, C , there is a group homomorphism    n∈Z Hom(Cn , Cn+1 ) → HomCh(Mod-R) (C, C ), (6.1) (sn )n∈Z

→ (fn )n∈Z , where fn = dn+1 ◦ sn + sn−1 ◦ dn , whose image is N H(C, C ). It is easily checked that N H is an ideal of Ch(Mod-R). Let K(Mod-R) := Ch(Mod-R)/N H be the factor category, so that the morphisms in K(Mod-R) are the equivalence classes of chain morphisms modulo null homotopic chain morphisms. Then any two projective resolutions of MR turns out to be isomorphic in K(Mod-R), and associating to any right R-module MR a projective resolution P(M ) of M is a functor Mod-R → K(Mod-R). This functor is then used in homological algebra to define derived functors. For instance, in ori (−, R) of the contravariant functor der to define the right derived functors ExtR ∗ (−) := Hom(−, RR ) : Mod-R → R -Mod, we fix, for any right R-module MR , a projective resolution P(MR )

=

d

d

d

· · · → P2 →2 P1 →1 P0 →0 0 → 0 → · · ·

of MR , which is an object of K(Mod-R). We then apply to the projective resolution the functor (−)∗ := Hom(−, RR ), getting a cochain complex of left R-modules Hom(P, R)) = P∗

=

···

/0

/0

d∗ 0

/ P0∗

d∗ 1

/ P1

d∗ 2

/ P2

d∗ 3

/ ··· ,

which is an object of K(R -Mod). Taking the cohomology modules H i (Hom(P(MR ), R)) = ker(d∗i+1 )/ im(d∗i ), we get the left R-modules ExtiR (M, R), that is, the right derived functors ExtiR (−, R). Hence the right derived functor ExtiR (−, R) is the composite functor of the functor Mod-R → K(Mod-R), the functor K(Mod-R) → K(R -Mod) induced by Hom(−, RR ) and the ith cohomology functor H i : K(R -Mod) → R -Mod.

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With the Auslander–Bridger transpose, we have a similar problem. That is, let MR be a finitely presented right R-module, and associate to it a projective resolution P(MR ) of MR with P1 and P0 finitely generated. This P(MR ) is an object of K(Mod-R). Now apply to P(MR ) the functor (−)∗ := Hom(−, RR ), getting the cochain complex of left R-modules Hom(P, R)) = P∗ , which is an object of K(R -Mod). Finally, apply the functor coker(d∗1 ) := P1∗ / im(d1∗ ). The problem now is that coker(d∗1 ) is a functor Ch(R -Mod) → R -Mod, but not a functor K(R -Mod) → R -Mod. In order to solve this problem, we notice that first of all, it suffices to consider the full subcategory K  of K(Mod-R) whose objects are chain complexes of modules C

=

/ Cn+1 dn+1 / Cn

···

dn

/ Cn−1 dn−1 / · · ·

with all Cn projective right R-modules and C1 , C0 finitely generated modules. Thus, when we apply coker(d∗1 ), we get a finitely presented left R-module, that is, an object of R-mod. In order to get a functor defined on the category K  , we replace the category R-mod with the stable category R-mod (Section 4.11), that is, the factor category of the full subcategory R-mod of finitely presented left modules modulo the ideal of all morphisms that can be factored through a projective module, that is, the factor category R-mod/R I. In fact, we have the following proposition. Proposition 6.1. The functor coker(d∗1 ) : Ch(Mod-R) → R -Mod induces a functor coker(d∗1 ) : K  → R-mod. That is, coker(d∗1 ) maps N H(C, C ) into R I(coker(d∗1 )(C), coker(d∗1 )(C )) for every C, C ∈ Ob(K  ). Proof. Let C, C be objects of K  , so that the modules Cn , Cn are all projective right modules and the modules C1 , C0 , C1 , C0 are finitely generated. Let f : C → C be a morphism of chain complexes in N H, so that there exist right R-module  morphisms sn : Cn → Cn+1 , n ∈ Z, such that fn : Cn → Cn is defined by fn :=  dn+1 ◦ sn + sn−1 ◦ dn . The diagram we are working on is the following: ...

...

/ Cn dn / Cn−1 dn−1 / Cn−2 z z ww zz zz f ww fn+1 zz fn−2 fn zz n−1 w z w s s zs  |zzdn+1n  |zz d n−1  {wwdn−1n−2  n    / Cn / Cn−1 / Cn−2 / Cn+1

/ Cn+1

dn+1

/ ...

/ ...

For n = 1, we have that f1 = d2 ◦ s1 + s0 ◦ d1 . Consider the mapping s1 : C1 → C2 . The projective module C2 is contained as a direct summand in a free R-module F  , and the image of s1 is finitely generated. Hence there is a finitely generated free direct summand F2 of F  that contains the image of s1 . Hence s1 : C1 → C2

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6.1. The Functor Auslander–Bridger Transpose

induces a morphism σ1 : C1 → F2 . Similarly, d2 : C2 → C1 can be first extended to a morphism F  → C1 and then restricted to a morphism δ2 : F2 → C1 . Clearly, d2 ◦ s1 = δ2 ◦ σ1 , so that f1 = δ2 ◦ σ1 + s0 ◦ d1 . Recall that we must prove that coker(d∗1 ) maps f into R I(coker(d∗1 )(C), coker(d∗1 )(C ). Now f1∗ := coker(d∗1 )(f ) : coker(d∗1 )(C∗ ) → coker(d∗1 )(C∗ ), that is, f1∗ : C1 ∗ / im(d1 ∗ ) → C1 ∗ / im(d1 ∗ ) and f1∗ = σ1 ∗ δ2 ∗ + d∗1 s∗0 , and we must prove that the map f1∗ : C1 ∗ / im(d1 ∗ ) → C1 ∗ / im(d1 ∗ ) induced by f1∗ : C1 ∗ → C1 ∗ factors through a projective module. But for every c1 ∗ ∈ C1 ∗ , one has that f1∗ (c1 ∗ + im(d1 ∗ )) = f1∗ (c1 ∗ ) + im(d1 ∗ ) = ∗ ∗ (σ1 ∗ δ2 + d∗1 s∗0 )(c1 ∗ ) + im(d1 ∗ ) = (σ1 ∗ δ2 ) + im(d1 ∗ ). Thus ∗

f1∗ = σ1 ∗ δ2 , ∗

∗

where δ2 : C1 ∗ / im(d1 ∗ ) → F2 ∗ is induced by δ2 : C1 ∗ → F2 ∗ and σ1 ∗ : F2 ∗ → C1∗ . Therefore f1∗ = coker(d∗1 )(f ) factors through the finitely generated projective left  R-module F2 ∗ , as we wanted to prove. Thus we have a functor coker(d∗1 ) : K  → R-mod. Composing it with the functor mod-R → K(Mod-R), we get a functor mod-R → R-mod. If we apply this composite functor mod-R → R-mod to any finitely generated projective module PR , which has ···

/0

/0

d1

/ PR

d0

/0

/ ···

as a projective resolution, we get the cochain complex ···

/0

d∗ 0

/ (PR )∗

d∗ 1

/0

/ · · ·,

in which the cokernel of d∗1 is zero. This proves that the functor mod-R → R-mod maps finitely generated projective right R-modules to zero. Hence it maps all morphisms that factor through a (finitely generated) projective right R-module to zero. It follows that the functor mod-R → R-mod induces a functor Tr : mod-R → R-mod. Similarly, one finds a functor Tr : R-mod → mod-R, and these two functors are clearly one the quasi-inverse of the other. We have thus proved the following theorem: Theorem 6.2. For any ring R, the Auslander–Bridger transpose is a duality Tr : mod-R → R-mod of the stable category mod-R of finitely presented right R-modules into the stable category R-mod of finitely presented left R-modules.

200

Chapter 6. Auslander–Bridger Transpose, Auslander–Bridger Modules

6.2 Projective Modules with a Semiperfect Endomorphism Ring As we have seen in the previous section, the Auslander–Bridger transpose of a finitely presented right R-module MR is not one left R-module, but a whole class of stably isomorphic finitely presented left R-modules (Theorem 6.2 and Proposition 4.41). It is therefore natural to ask whether, in this class of stably isomorphic finitely presented left R-modules, there is a canonical choice for one left R-module that should then be called “the” Auslander–Bridger transpose of MR . This occurs in some special cases. The most classical of these special cases is that in which the ring R is semiperfect. If MR is a finitely generated module over a semiperfect ring R, then MR = N ⊕ P , where P is a projective module and N has no nonzero projective summands. Moreover, this decomposition is unique up to isomorphism, in the sense that if M = N  ⊕ P  is another direct-sum decomposition with P  a projective module and N  without nonzero projective summands, then ∼ P  ([Warfield 1975, Theorem 1.4]; also see [Facchini 1998, The∼ N  and P = N= orem 3.15]). Moreover, when R is semiperfect, every finitely generated module has a projective cover, so that every finitely presented module MR has a miniq /P p /M / 0 . Here Q and P are finitely mal projective presentation Q generated projective modules and the image of q is superfluous in P . Thus every finitely presented module has a canonical projective presentation, the minimal one. Applying the dual (−)∗ := Hom(−, R) to the minimal presentation, one obtains q∗

/ Q∗ whose image is again superfluous in Q∗ . Thus the sita morphism P ∗ uation is symmetric, and the cokernel of q ∗ : P ∗ → Q∗ deserves to be called the Auslander–Bridger transpose of MR . In the rest of this chapter, we will present the theory of the Auslander– Bridger transpose from a different point of view, that adopted in [Facchini and Girardi 2012]. Instead of supposing R semiperfect, we assume R arbitrary, and we focus our attention on the modules for which the above condition holds, essentially all modules with a sufficiently good projective presentation, a minimal one, and no nonzero projective direct summands. In Lemma 3.9, we saw several characterizations of couniform projective modules. Every proper submodule of a couniform module is superfluous, so that every homomorphic image of a couniform projective module has a projective cover. In the next lemma, we consider the (not necessarily cyclic) projective modules with this property. Recall that the notation N ≤s MR indicates that N is a superfluous submodule of MR . Lemma 6.3. The following conditions are equivalent for a projective module PR : (a) Every homomorphic image of PR has a projective cover.

6.2. Projective Modules with a Semiperfect Endomorphism Ring

201

(b) For every submodule U of PR , there exists a direct summand K of PR with K ⊆ U and U/K ≤s P/K. (c) For every submodule U of PR , there is a direct-sum decomposition PR = H ⊕ K of PR with K ⊆ U and U ∩ H ≤s H. Proof. (a) ⇒ (b) Assume that (a) holds. Fix a submodule U of PR . Then there is a projective cover ϕ : QR → PR /U of the homomorphic image PR /U of PR . Let π : P → P/U denote the canonical projection. By Theorem 2.27, there is a direct∼ Q, K ⊆ U , and such that the sum decomposition PR = H ⊕ K of PR with H = restriction π|H : H → PR /U of π is a projective cover for the module PR /U . The composite mapping of π|H : H → PR /U and the canonical isomorphism PR /K → H is the canonical projection PR /K → PR /U , because the isomorphism PR /K → H sends an element h+ k + K of PR /K to h, and π|H maps h to h+ U = h+ k + U . Therefore we get that the canonical projection PR /K → PR /U is also a projective cover for PR /U . In particular, its kernel U/K is a superfluous submodule of PR /K. (b) ⇒ (c) If U is a submodule of PR and (b) holds, then there exists a direct-sum decomposition PR = H ⊕ K of PR with K ⊆ U and U/K ≤s PR /K. It follows that U = (U ∩ H) ⊕ K. (This is elementary. Otherwise, see [Facchini 1998, Lemma 2.1].) The canonical isomorphism H → PR /K, h → h+ K, maps the submodule U ∩ H of H onto the submodule ((U ∩ H) ⊕ K)/K = U/K of PR /K. Thus U/K ≤s PR /K implies U ∩ H ≤s H. (c) ⇒ (a) Let PR /U be any homomorphic image of PR . From (c), we get a direct-sum decomposition PR = H ⊕ K with K ⊆ U and U ∩ H ≤s H. In order to show that the homomorphism ϕ : H → PR /U , defined by ϕ(h) = h + U for every h ∈ H, is a projective cover for PR /U , notice that K ⊆ U , so that H + U = PR .  In particular, ϕ is onto. Moreover, ker ϕ = U ∩ H ≤s H. Remark 6.4. Let M be a (not necessarily projective) module satisfying condition (2) of the previous lemma, that is, such that for every submodule U of M there is a direct-sum decomposition M = H ⊕ K of M with K ⊆ U and U ∩ H ≤s H. Let us prove that a submodule of M is superfluous if and only if it contains no nonzero direct summands of M . If U is a superfluous submodule of M and N is a direct summand of M contained in U , then there is a direct-sum decomposition M = N ⊕ N  , so that M = U + N  . But U superfluous in M implies M = N  , so that N = 0. Conversely, let U be a submodule of M that does not contain nonzero direct summands of M . There is a direct-sum decomposition M = H ⊕ K of M such that K ⊆ U and U ∩ H ≤s H. Since U contains no nonzero direct summands of M , we must have K = 0, that is, H = M . Now U ∩ H ≤s H implies U ≤s M . In Lemma 3.9, we described couniform projective modules. We now consider direct sums of finitely many couniform projective modules. Proposition 6.5. The following conditions are equivalent for a nonzero projective right module PR over a ring R: (a) PR is a direct sum of finitely many couniform submodules.

202

Chapter 6. Auslander–Bridger Transpose, Auslander–Bridger Modules

(b) PR is a finitely generated module, the module PR /PR J(R) is semisimple, and every homomorphic image of PR has a projective cover. (c) The module PR is the projective cover of a finitely generated semisimple module TR , and all direct summands of TR have a projective cover. (d) The endomorphism ring End(PR ) of PR is semiperfect. Proof. (a) ⇒ (b) Assume that (a) holds, so that PR = P1 ⊕ · · · ⊕ Pn for suitable couniform modules P1 , . . . , Pn . Every Pi is local (Lemma 3.9((a)⇔(e)), hence ∼ P1 /P1 J(R)⊕ cyclic, so that PR is a finitely generated module. Now PR /PR J(R) = Pn /Pn J(R). Also, Pi local implies that rad(Pi ) is the unique maximal submodule of Pi , and rad(Pi ) = Pi J(R) (Proposition 2.16). Thus every Pi /Pi J(R) is a simple module. It follows that the direct sum PR /PR J(R) is semisimple. Now, every homomorphic image of PR /PR J(R) is isomorphic to a direct sum of some of the Pi /Pi J(R)’s. The direct sum of the corresponding modules Pi is a projective cover of the homomorphic image of PR /PR J(R). This proves that every homomorphic image of PR /PR J(R) has a projective cover. Let us prove that every homomorphic image of PR has a projective cover. If ϕ : PR → MR is an epimorphism, then ϕ induces an epimorphism PR /PR J(R) → MR /MR J(R). The homomorphic image MR /MR J(R) of PR /PR J(R) has a projective cover ψ : QR → MR /MR J(R), which lifts to a homomorphism ψ  : QR → MR . Now ψ surjective implies ψ  (QR ) + MR J(R) = MR . Thus ψ  (QR ) = MR by Nakayama’s lemma (Theorem 2.2), because the homomorphic image MR of PR is finitely generated. Thus ψ  is onto. As ker ψ  ⊆ ker ψ, the submodule ker ψ  of QR is superfluous. This proves that the homomorphic image MR of PR has a projective cover ψ  . (b) ⇒ (c) Assume that Condition (b) hold. As PR is finitely generated, its submodule PR J(R) is superfluous by Nakayama’s Lemma 2.2, so that the canonical projection PR → PR /PR J(R) is the projective cover of T := PR /PR J(R), which is a finitely generated semisimple module. Any direct summand of T is a homomorphic image of T , hence is a homomorphic image of PR , so it has a projective cover. This proves (c). (c) ⇒ (a) Suppose that the module PR is the projective cover of a finitely generated semisimple module TR . Let TR = S1 ⊕ · · · ⊕ Sn be a direct-sum decomposition of TR with the modules Si simple submodules of T . By (c), every Si has a projective cover Pi . Thus P1 ⊕ · · ·⊕ Pn is a projective cover of the semisimple module TR . By the uniqueness of the projective cover (Theorem 2.27(b)), we get ∼ P1 ⊕ · · · ⊕ Pn . The modules P1 , . . . , Pn are couniform by an isomorphism PR = Lemma 3.9((a)⇔(c)). This concludes the proof of (a). (a) ⇔ (d) We know that a module PR has a semiperfect endomorphism ring End(PR ) if and only if PR is a direct sum of finitely many right R-modules with local endomorphism ring (Example (9) on page 100). The conclusion follows from Lemma 3.9((a)⇔(b)).  In the rest of this chapter, PR (R P, respectively) will denote the full subcategory of Mod-R (R -Mod, respectively) whose objects are all projective right

6.3. Auslander–Bridger Modules

203

(resp. left) R-modules satisfying the equivalent conditions of Proposition 6.5. Notice that couniform projective modules have a local endomorphism ring (Lemma 3.9), so that by the Krull–Schmidt–Azumaya theorem (Theorem 4.34), every direct summand of a module satisfying the equivalent conditions of Proposition 6.5 also satisfies the equivalent conditions of Proposition 6.5. That is, the full subcategory PR of Mod-R is an additive category with split idempotents. By the last part of the statement of Lemma 3.9, the functors (−)∗ = Hom(−, R RR ) give an additive duality between the categories PR and R P.

6.3 Auslander–Bridger Modules For simplicity of notation, in what follows, we will consider right R-modules and denote by P the category PR . A module M is said to be P-finitely generated (or finitely generated by P [Anderson and Fuller, p. 105]) if there exists an onto module morphism P → M for some P ∈ Ob(P). We will say that M is P-finitely presented if there is an exact sequence Q

q

/P

p

/M

/0

(6.2)

with Q and P both in Ob(P). Lemma 6.6. Every P-finitely generated module M has a projective cover PM → M, and PM ∈ Ob(P). Proof. Let M be a P-finitely generated module, so that M is a homomorphic image of some Q ∈ Ob(P). By Proposition 6.5, every homomorphic image of Q has a projective cover, so that M has a projective cover PM → M . By the fundamental lemma of projective covers (Theorem 2.27(a)), the module PM must be isomorphic to a direct summand of Q, so that PM ∈ Ob(P).  Lemma 6.7. The following conditions are equivalent for a ring R: (a) (b) (c) (d)

The full subcategories proj-R and P of Mod-R coincide. Every finitely generated right R-module is P-finitely generated. Every finitely presented right R-module is P-finitely presented. The ring R is semiperfect.

Proof. (a) ⇒ (b) is trivial. f / Rm g / M / 0 be a presentation of a finitely (b) ⇒ (c) Let Rn presented module M . We have an exact sequence

0

/ f (Rn )

/ Rm

g

/M

/0.

204

Chapter 6. Auslander–Bridger Transpose, Auslander–Bridger Modules

By (b), M is P-finitely generated, and therefore there is an exact sequence /K

0

/P

/M

/ 0,

with P ∈ Ob(P) and K superfluous in P . By Schanuel’s lemma [Anderson and Fuller, Exercise 18.9], f (Rn ) ⊕ P ∼ = Rm ⊕ K. Now, P is a direct summand of Rm and can be canceled from direct sums (Proposition 3.36), so that K is a direct summand of f (Rn ). Thus K is finitely generated. The conclusion now follows by applying (b) to the finitely generated module K. (c) ⇒ (d) From (c) and Lemma 6.6, we get that every finitely presented module has a projective cover. (d) ⇒ (a) If R is semiperfect and P is a finitely generated module, then P is a direct summand of Rn . Now, RR is a direct sum of modules with local endomorphism rings. From the Krull–Schmidt–Azumaya theorem, we get that the projective module P is also a direct sum of modules with a local endomorphism rings. Thus P is in Ob(P).  In the next proposition, we will show that every P-finitely presented Rmodule M has a minimal projective presentation of the form (6.2). Moreover, in such a minimal projective presentation, both Q and P are necessarily objects of P. Proposition 6.8. Every P-finitely presented module M has a minimal projective presentation QM

/ PM

ϑM

πM

/M

/ 0,

(6.3)

and in such a minimal projective presentation, both QM and PM are modules in Ob(P). Proof. Fix a presentation of the form (6.2) of M . The module M has a projective cover pM : PM → M with PM ∈ Ob(P) by Lemma 6.6. By the fundamental lemma of projective covers (Theorem 2.27), there is a direct-sum decomposition P ∼ = PM ⊕ C. By Schanuel’s lemma [Anderson and Fuller, Exercise 18.9], P ⊕ker(pM ) ∼ = PM ⊕ ker(p). Now, PM has a semilocal endomorphism ring, hence cancels from direct sums, so that C ⊕ ker(pM ) ∼ = ker(p). Thus there is an epimorphism Q → ker(pM ), and therefore ker(pM ) is P-finitely generated. By Lemma 6.6 again, ker(pM ) has a projective cover QM with QM ∈ Ob(P).  /P /M / 0 with Q and P in Ob(P) is Clearly, a presentation Q minimal if and only if q(Q) is superfluous in P and ker(q) is superfluous in Q, that is, if and only if q(Q) ⊆ P J(R) and ker(q) ⊆ QJ(R). q

p

The class of P-finitely presented modules is clearly closed under finite direct sums. In the next lemma, we show that it is also closed under direct summands. Lemma 6.9. Direct summands of P-finitely presented modules are P-finitely presented modules.

6.3. Auslander–Bridger Modules

205

Proof. Assume that M = A ⊕ B is a P-finitely presented module. The modules A and B are P-finitely generated. Therefore A and B have projective covers π  : P  → A and π  : P  → B (Proposition 6.5) with P  , P  ∈ Ob(P). Since M is P-finitely presented, M has a minimal presentation QM

ϑM

/ PM

πM

/M

/ 0,

with QM and PM in P (Proposition 6.8). Thus we have two short exact sequences 0 → ker π  ⊕ ker π  → P  ⊕ P  → A ⊕ B = M → 0 and 0 → ker(πM ) → PM → M → 0. The two projective covers PM and P  ⊕P  of M are isomorphic, so that ker(πM ) ∼ = ker π  ⊕ ker π  . Therefore ker π  and ker π  are homomorphic images of QM .  Proposition 6.10. Every P-finitely generated module M decomposes as M = P ⊕N with P ∈ Ob(P) and N without nonzero projective direct summands. Moreover, if M = P  ⊕ N  is another decomposition of M with P  ∈ Ob(P) and N  without nonzero projective direct summands, then P ∼ = P  and N ∼ = N . Proof. Existence. Since every module in P has finite dual Goldie dimension, every P-finitely generated module M also has finite dual Goldie dimension, so that we can argue by induction on codim(M ). For codim(M ) = 0, one has that M = 0, and there is nothing to prove. Assume codim(M ) ≥ 1. If M has no nonzero projective direct summands, we are done again. If M has a nonzero projective direct summand P , we can write M = P ⊕ M  , for some module M  , necessarily P-finitely generated and of dual Goldie dimension < codim(M ). By the inductive hypothesis, M  = P  ⊕ N , where P  and N obviously have the required properties. Now, P is isomorphic to a direct summand of the projective cover of M , so that P ∈ Ob(P). Thus P ⊕ P  ∈ Ob(P), and the conclusion is immediate. Uniqueness. Let P, P  be modules in Ob(P), let N, N  be P-finitely generated modules with no nonzero projective summands, and assume P ⊕ N ∼ = P  ⊕ N . This second part of the proof is by induction on codim(P ). If codim(P ) = 0, then P  = 0, N ∼ = N  , and we are done. If codim(P ) ≥ 1, let P = C ⊕ Q be a direct-sum decomposition of P with C a projective couniform submodule. Then C has a local endomorphism ring, so that C must be isomorphic to a direct summand of P  or to a direct summand of N  . Thus C must be isomorphic to a direct summand of P  , that is, P  ∼ = C ⊕ Q for some Q ∈ Ob(P). Now modules with a (semi)local endomorphism ring cancel from direct sums (Proposition 3.36). Thus Q ⊕ N ∼ = Q ⊕ N  . The conclusion follows immediately from the inductive hypothesis.  An R-module M is an Auslander–Bridger module if it is a P-finitely presented module with no nonzero projective direct summands. We will denote by

206

Chapter 6. Auslander–Bridger Transpose, Auslander–Bridger Modules

AB R the full subcategory of Mod-R whose objects are all Auslander–Bridger right R-modules and by R AB the full subcategory of R -Mod whose objects are all Auslander–Bridger left R-modules. The class of Auslander–Bridger modules is closed under finite direct sums and direct summands (Lemma 6.9), so that AB R is an additive subcategory of Mod-R with splitting idempotents. Any Auslander–Bridger module is P-finitely generated, hence has finite dual Goldie dimension. It follows that every Auslander–Bridger module decomposes as a direct sum of finitely many indecomposable modules, each of which is necessarily an Auslander-Bridger module. We now apply the notions of Section 4.11 to the full subcategory A of Mod-R whose objects are all P-finitely presented right R-modules and S is Ob(P). Thus we can construct the factor category A/IS , where IS is the ideal of A of all morphisms that factor through an object of P. Lemma 6.11. A morphism f : M → N in A factors through a projective module if and only if f factors through an object of P. Proof. Let f : M → N be a morphism in A that factors through a projective module PR , so that there exist morphisms g : M → P and h : P → N with hg = f . Since N is an object of A, there is an epimorphism : Q → N with Q ∈ Ob(P). By the projectivity of P , there exists a morphism h : P → Q such that h = h . Then f = (h g) is a factorization of f through Q ∈ Ob(P).  Recall that two modules M and N are said to be stably isomorphic if they are isomorphic in the stable category Mod-R (Section 4.11), equivalently, if there exist projective modules P, Q ∈ Ob(P) such that M ⊕ P ∼ = N ⊕ Q (Proposition 4.41). Corollary 6.12. If M and N are P-finitely presented modules, then M and N are stably isomorphic if and only if there exist P, Q ∈ Ob(P) such that M ⊕P ∼ = N ⊕Q. Proof. The P-finitely presented modules M and N are stably isomorphic if and only if they are isomorphic in the category Mod-R = (Mod-R)/IS , where S is the class of all projective modules, if and only they are isomorphic in the category A/(IS ∩ A), where A is the full subcategory of Mod-R whose objects are all Pfinitely presented right R-modules. By Lemma 6.11, IS ∩ A is equal to the ideal IOb(P) of A. But A is an additive category in which idempotents split (Lemma 6.9), so that Proposition 4.41 allows us to conclude.  From Proposition 6.10 and Corollary 6.12, we obtain the following corollary: Corollary 6.13. (a) Any P-finitely presented module is a direct sum of an Auslander–Bridger module and a module in P in a unique way, up to isomorphism. (b) Any P-finitely presented module is stably isomorphic to an Auslander–Bridger module, unique up to isomorphism.

6.4. The Auslander–Bridger Transpose

207

(c) Two Auslander–Bridger modules are stably isomorphic if and only if they are isomorphic. It follows that in any stable isomorphism class of P-finitely presented modules there is a unique Auslander–Bridger module, up to isomorphism.

6.4 The Auslander–Bridger Transpose Let R be a ring. The functor (−)∗ := HomR (−, R) : Mod-R → R -Mod induces a duality proj-R → R-proj. For any P-finitely presented module M , we can fix a minimal projective presentation QM

ϑM

/ PM

πM

/M

/ 0,

of M and apply the contravariant functor (−)∗ , getting an exact sequence / M∗

0

∗ πM

/ P∗ M

ϑ∗ M

/ Q∗ , M

which we know to be unique up to isomorphism. Lemma 6.14. Let g : Q → P be a morphism in PR . Then g(Q) is a superfluous submodule of P if and only if g ∗ (P ∗ ) is a superfluous submodule of Q∗ . Proof. Suppose that g(Q) is not a superfluous submodule of P . In Remark 6.4, we saw that there is a nonzero direct summand A of P with A ⊆ g(Q). Thus there exists an epimorphism π : P → A that is the identity on the submodule A. Since A is projective and πg : Q → A is an epimorphism, there is a morphism α : A → Q with πgα = 1A . Applying the functor (−)∗ , one sees that 1A∗ = α∗ g ∗ π ∗ , from which one gets that g ∗ π ∗ (A∗ ) is a nonzero direct summand of Q∗ contained in g ∗ (P ∗ ). Thus g ∗ (P ∗ ) is not a superfluous submodule of Q∗ . For the converse, suppose g ∗ (P ∗ ) not superfluous in Q∗ . By what we have seen in the previous paragraph, g ∗∗ (Q∗∗ ) is not a superfluous submodule of P ∗∗ . Since (−)∗ is a duality, (−)∗∗ is naturally isomorphic to the identity; hence g(Q) is not superfluous in P .  Proposition 6.15. Let M be an Auslander–Bridger right module with a miniϑM / PM πM / M / 0. Then the cokernel mal projective presentation QM ∗ ∗ ∗ Tr(M ) of the morphism ϑM : PM → QM is an Auslander–Bridger left module with minimal projective presentation ∗ PM

ϑ∗ M

/ Q∗ M

/ Tr(M )

/ 0.

208

Chapter 6. Auslander–Bridger Transpose, Auslander–Bridger Modules

Proof. From Lemma 3.9 and the additivity of the functor (−)∗ , we get that the left ∗ ∗ ∗ (PM and Q∗M are in R P. Moreover, ϑM ) is a superfluous submodule modules PM ∗ of QM by Lemma 6.14. To conclude the proof, we must show that ker(ϑ∗M ) is ∗ superfluous in PM and that Tr(M ) has no nonzero projective direct summands. ∗ ∗ , then ker(ϑ∗M ) contains a nonzero direct If ker(ϑM ) is not superfluous in PM ∗ ∗ summand A of PM (Remark 6.4). Let e be an idempotent endomorphism of PM ∗ ∗∗ ∗ ∗ ∗ ∗∗ with e(PM ) = A. Then ϑM = ϑM (1 − e). It follows that ϑM = (1 − e) ϑM . In ∗∗ ∗ ∗ . (PM ) is contained in a proper direct summand of PM particular, the module ϑM ∗∗ It follows that coker(ϑM ) has a nonzero projective direct summand. This implies that M also has a nonzero projective summand, which is a contradiction. It remains to show that Tr(M ) has no nonzero projective direct summands. Assume the contrary, and let A be a nonzero projective direct summand of Tr(M ). Let πA : Tr(M ) → A be an epimorphism that is the identity on A, and let π be ∗ ∗ onto Tr(M ) (that is, the cokernel of ϑM ). Since the canonical projection of QM ∗ A is projective and πA π : QM → A is an epimorphism, there exists φ : A → Q∗M ∗ ∗ with πA πφ = 1A . Dualizing, one finds that φ∗ π ∗ πA = 0 implies = 1A∗ . Now πϑM ∗∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ϑM π = 0, so that π (Tr(M ) ) ⊆ ker(ϑM ). A fortiori, π πA (A ) ⊆ ker(ϑ∗∗ M ). But ∗ ∗∗ π ∗ πA (A∗ ) is a nonzero direct summand of QM . This is a contradiction, because ker(ϑM ) ≤s QM .  The left module Tr(M ) of Proposition 6.15 is called the Auslander–Bridger transpose of the Auslander–Bridger right module M . The following proposition, which inverts Proposition 6.15, explains why we must consider modules with no nonzero projective direct summands. Proposition 6.16. Let M be a P-finitely presented module with minimal projective ϑM / PM πM / M / 0. If π ∗ (M ∗ ) is superfluous in P ∗ , presentation QM M M then M has no nonzero projective direct summands. Proof. Assume the contrary, so that M has a nonzero projective direct summand P  , M = P  ⊕ N with P  ∈ Ob(P) say. Then P  is isomorphic to a direct summand of Rn for some n, so that P  /P  J(R) is isomorphic to a direct summand of (R/J(R))n . Now P  = 0 implies P  /P  J(R) = 0. Hence there is a nonzero morphism P  /P  J(R) → R/J(R). Thus there is a morphism P  → R whose image is not superfluous in R. It follows that there exists a morphism ϕ : M → R ∗ (ϕ) = ϕπM has its image whose image is not superfluous in R. So ϕ ∈ M ∗ and πM ∗ ϕπM (PM ) = ϕ(M ), which is not superfluous in R. But πM (M ∗ ) is superfluous ∗ ∗ ∗ ∗ ∗ ∗ in PM , so that πM (M ) ⊆ J(R)PM ; hence πM (ϕ) ∈ J(R)PM . Every element of ∗ J(R)PM is a morphism PM → R whose image is contained in J(R), hence is superfluous in R. This gives the required contradiction.  Theorem 6.17. If M, N are Auslander–Bridger modules, then: (a) M ∼ = N if and only if Tr(M ) ∼ = Tr(N ). (b) Tr(Tr(M )) ∼ = M.

6.5. The Functors − ⊗R R/J(R) and TorR 1 (−, R/J(R))

209

∼ Tr(M ) ⊕ Tr(N ). (c) Tr(M ⊕ N ) = Proof. (a) Let M and N be Auslander–Bridger right modules. By Corollary ∼ N if and only if M and N are isomorphic in mod-R. By The6.13(c), M = orem 6.2, this occurs if and only if Tr(M ) and Tr(N ) are isomorphic in R-mod. By Proposition 6.15, the left modules Tr(M ) and Tr(N ) are Auslander–Bridger modules. Thus Tr(M ) and Tr(N ) are isomorphic in the category R-mod if and only if Tr(M ) ∼ = Tr(N ) in R-mod (Corollary 6.13(c)). (b) By Theorem 6.2, the module Tr(Tr(M )) is stably isomorphic to M for every finitely presented module M . If M is an Auslander–Bridger module, then Tr(Tr(M )) is also an Auslander–Bridger module (Proposition 6.15). Thus Tr(Tr(M )) ∼ = M by Corollary 6.13(c). (c) Since Tr : mod-R → R-mod is an additive contravariant functor, the left modules Tr(M ⊕ N ) and Tr(M ) ⊕ Tr(N ) must be stably isomorphic. Both of them are Auslander–Bridger modules, so that Tr(M ⊕ N ) ∼  = Tr(M ) ⊕ Tr(N ) by Corollary 6.13(c). For every category C with skeleton V (C), let − : C → V (C) be the mapping that associates to each object X of C the unique object X of V (C) isomorphic to X. Recall that AB R denotes the full subcategory of Mod-R whose objects are all Auslander–Bridger right R-modules, and R AB denotes the full subcategory of R -Mod whose objects are all Auslander–Bridger left R-modules. Corollary 6.18. There is a monoid isomorphism η : V (AB R ) → V (R AB), defined by η(X) = Tr(X) for every X ∈ V (AB R ). Proof. By Theorem 6.17: η is injective (by (a)), surjective (by (b)), and a monoid morphism (by (c)). 

6.5 The Functors − ⊗R R/J(R) and TorR 1 (−, R/J(R)) There is a standard technique of homological algebra to lift a homomorphism between two modules to their projective resolutions. If MR and NR are two Auslander–Bridger modules, every morphism f : MR → NR lifts to a morphism f0 : PM → PN . With a second similar step, we can construct a commutative diagram / PM /M /0 (6.4) QM f1

 QN

f0



/ PN

f

 /N

/0

210

Chapter 6. Auslander–Bridger Transpose, Auslander–Bridger Modules

Lemma 6.19. Let f : MR → NR be a morphism between two Auslander–Bridger modules. Consider commutative diagram (6.4). Then: (a) f : MR → NR is surjective if and only if f0 : PM → PN is surjective, if and only if f ⊗R R/J(R) : M ⊗R R/J(R) → N ⊗R R/J(R) is surjective. (b) f1 : QM → QN is surjective if and only if R R TorR 1 (f, R/J(R)) : Tor1 (MR , R/J(R)) → Tor1 (NR , R/J(R))

is surjective. Proof. Let (6.3) be a minimal projective presentation of M . We have a short exact sequence 0

/ QM / ker(ϑM )

/ PM

ϑM

πM

/M

/ 0.

Applying the functor − ⊗R R/J(R), we get an exact sequence δ

M 0 −→ TorR 1 (MR , R/J(R)) −→ QM / ker(ϑM ) ⊗R R/J(R)

−→ PM ⊗R R/J(R)

ϑM ⊗R/J(R)

πM ⊗R/J(R)

−→

−→

M ⊗R R/J(R) −→ 0.

Now, ϑM ⊗ R/J(R) is the zero morphism, because the image of ϑM , which is equal to the image of ϑM , is superfluous in PM , hence is contained in PM J(R). It follows that both the natural morphisms δM and πM ⊗ R/J(R) are isomorphisms. Since these are two natural isomorphisms, the morphism f : MR → NR induces two commutative squares TorR 1 (MR , R/J(R))

δM

TorR 1 (f,R/J(R))

 TorR (N , R R/J(R)) 1

/ QM / ker(ϑM ) ⊗R R/J(R) f1 ⊗R/J(R)

δN

 / QN / ker(ϑN ) ⊗R R/J(R)

and PM ⊗R R/J(R)

πM ⊗R/J(R)

f0 ⊗R/J(R)

 PN ⊗R R/J(R)

This concludes the proof.

/ M ⊗R R/J(R) f ⊗R/J(R)

πN ⊗R/J(R)

 / N ⊗R R/J(R).



Notice that, as is well known in homological algebra, the two morphisms f0 and f1 that complete diagram (6.4) are not uniquely determined by f . By

6.5. The Functors − ⊗R R/J(R) and TorR 1 (−, R/J(R))

211

Lemma 6.19, if in diagram (6.4) we replace f0 and f1 with two morphisms f0 and f1 making the diagram analogous to diagram (6.4) commute, then f0 : PM → PN is surjective if and only if f0 : PM → PN is surjective, and f1 : QM → QN is surjective if and only if f1 : QM → QN is surjective. Let R be a ring with Jacobson radical J := J(R). Let mod-R be the category of all finitely presented right R-modules and Cp the full subcategory of mod-R whose objects are all finitely presented R-modules with a projective cover. Consider the two functors − ⊗R R/J and Tor1R (−, R/J), viewed as functors of Cp into mod-R/J. Lemma 6.20. If π : PR → AR is the projective cover of a finitely generated Rmodule AR , then PR is a finitely generated R-module and the induced mapping π : PR /PR J → AR /AR J is a projective cover of AR /AR J viewed as an R/Jmodule. Proof. PR is finitely generated by Theorem 2.27(a). For the rest, the only nontrivial thing is the proof that the kernel of the induced morphism π : PR /PR J → AR /AR J is a superfluous R/J-submodule of PR /PR J. Set K := ker(π), so that K = K  /PR J for a suitable R-submodule K  of PR containing PR J. Since π : PR → AR is a projective cover and π(K  ) ⊆ AR J, it follows that π(K  ) is a superfluous submodule of AR . Let C be an R/J-submodule of PR /PR J with C +K = PR /PR J. Then C = C  /PR J for a suitable R-submodule C  of PR containing PR J. Therefore C  + K  = PR , and thus π(C  ) + π(K  ) = AR . Hence π(C  ) = AR , so that C  + ker π = PR , from which C  = PR . This proves that C = PR /PR J, and K is superfluous in PR /PR J.  Theorem 6.21. Let mod-R denote the category of all finitely presented right Rmodules, Cp the full subcategory of mod-R whose objects are all finitely presented Rmodules with a projective cover, and mod-R/J the category of all finitely presented right R/J-modules. Then the product functor − ⊗R R/J × TorR 1 (−, R/J) : Cp → mod-R/J × mod-R/J is a local functor. Proof. Let f : AR → BR be a morphism in Cp . TorR 1 (f, R/J) are two isomorphisms in mod-R/J. sequences 0 → K → P → AR and 0 → K  → P  morphisms P → AR and P  → BR are projective construct a commutative diagram 0 → 0 →

K ↓ K

f1

→ →

P ↓ f0 P

→ →

Suppose that f ⊗ R/J and We can fix two short exact → BR in which the two epicovers. It is then possible to

AR ↓f BR

→

0

→

0.

(6.5)

212

Chapter 6. Auslander–Bridger Transpose, Auslander–Bridger Modules

Apply the functor − ⊗R R/J to diagram (6.5), getting the commutative diagram 0 →

Tor(A, R J) Tor(f, R J ) ↓ 0 → Tor(BR , R J)

→

K⊗R J f1 ⊗ R J ↓ → K ⊗ R J

→

P⊗R → J ↓ f0 ⊗ RJ P ⊗ R → J

→

A⊗ R → 0 J ↓ h⊗ RJ B⊗R → 0. J

Now f ⊗ R/J an isomorphism implies f surjective by Nakayama’s lemma. Since f ⊗ R/J is an isomorphism and P/P J → A/AJ and P  /P  J → B/BJ are two projective covers, it follows that f0 ⊗ R/J is an isomorphism by the uniqueness of projective covers (Theorem 2.27). Thus f0 is also an isomorphism by the uniqueness of the projective covers P → P/P J and P  → P  /P  J. Since f0 ⊗ R/J, f ⊗ R/J, and TorR 1 (f, R/J) are three isomorphisms, it follows that f1 ⊗ R/J is an isomorphism by the five lemma [Weibel, Exercise 1.3.3]. Thus f1 is an epimorphism by Nakayama’s lemma. By the five lemma, applied to diagram (6.5), the fact that f0 is an isomorphism and f1 is an epimorphism implies that f injective [Weibel, Exercise 1.3.3]. We can therefore conclude that f is an  automorphism of AR . If M is an Auslander–Bridger R-module and QM

ϑM

/ PM

πM

/M

/0

is its minimal presentation, then the modules PM and QN are projective modules of finite dual Goldie dimension, so that the modules ∼ M ⊗R R/J(R) PM ⊗R R/J(R) = and ∼ TorR 1 (MR , R/J(R)) = QM / ker(ϑM ) ⊗R R/J(R) are semisimple modules of finite length. Recall that a module is isomorphic to a simple module with a projective cover if and only if it is isomorphic to X/XJ(R) for some couniform projective module (Lemma 3.9). If M is an Auslander–Bridger module, then the right R-modules PM ⊗R R/J(R) ∼ = M ⊗R R/J(R) and TorR (MR , R/J(R)) 1

are the semisimple right R-modules with a projective cover YR with End(YR ) semiperfect, that is, semisimple right R-modules that are P-finitely generated. In particular, M has a projective cover. Thus − ⊗R R/J(R) and TorR 1 (−, R/J(R)) are functors of the category AB R of Auslander–Bridger modules into the category SimpR of all semisimple right R-modules. Thus, from Theorem 6.21, we get the following: Corollary 6.22. The additive functor − ⊗R R/J(R) × TorR 1 (−, R/J(R)) : AB R → SimpR × SimpR is local. We are ready for a description of the endomorphism ring of an Auslander– Bridger module. Recall that a ring R is a prime ring if aRb = 0 for every nonzero

6.6. Notes on Chapter 6

213

a, b ∈ R. Equivalently, a ring R is prime if IJ = 0 implies I = 0 or J = 0 for every pair of two-sided ideals I and J of R. A proper two-sided ideal I of R is prime if R/I is a prime ring. That is, a proper ideal I of R is prime if and only if for each a, b ∈ R, aRb ⊆ I implies a ∈ I or b ∈ I. For instance, simple rings are prime rings. Thus maximal two-sided ideals are prime ideals. Theorem 6.23. Let E := End(MR ) be the endomorphism ring of a nonzero Auslander–Bridger module MR . Set I := { f ∈ E | Tor1R (f, R/J(R)) = 0 } and K := { f ∈ E | f ⊗R R/J(R) = 0 }. Then I and K are two proper two-sided ideals of E, the canonical morphism E → E/I × E/K is a local morphism, and every maximal right ideal, every maximal left ideal, and every maximal two-sided ideal of E contains either the ideal I or the ideal K. Moreover, I ∩ K ⊆ J(E). Proof. It is easily seen that I and K turn out to be two-sided ideals of the ring E. Since PM and QM are both nonzero, both I and K are proper ideals of E. Let f be an element of E and suppose that both f +I ∈ E/I and f +K ∈ E/K are invertible elements. Then there exist g, h ∈ E such that f g − 1 ∈ I and f h−1 ∈ K. Thus (f1 g1 −1)(QM ) ⊆ QM J(R) and (f0 h0 −1)(PM ) ⊆ PM J(R). From this, it follows that QM ⊆ f1 g1 (QM )+QM J(R) and PM ⊆ f0 h0 (PM )+PM J(R). In particular, both f1 g1 and f0 h0 are onto, so that f1 , f0 are two surjective endomorphisms. Now QM and PM are projective modules of finite dual Goldie dimension, so that they are directly finite. Thus both f0 and f1 are automorphisms. Since diagram (6.4) is commutative, we get that f is an automorphism of MR , and so E → E/I × E/K is a local morphism. It follows that the kernel I ∩ K of this local morphism is contained in J(E) by Lemma 3.24. This proves the last part of the statement of the theorem. By Corollary 5.46, the ring E is semilocal. By Proposition 3.14, the Jacobson radical J(E) is the intersection of all the maximal two-sided ideals of E. In particular, if M is a maximal two-sided ideal of E, then IK ⊆ I ∩ K ⊆ J(E) ⊆ M .  Since maximal two-sided ideals are prime, either I ⊆ M or K ⊆ M . We will come back to Auslander–Bridger modules in Chapter 8.

6.6 Notes on Chapter 6 Consider the duality of Theorem 6.2, induced by the Auslander–Bridger transpose. For a semiperfect ring R, the transpose induces a bijection between isomorphism classes of finitely presented right R-modules with no nonzero projective direct summands and isomorphism classes of finitely presented left R-modules with no nonzero projective direct summands [Warfield 1975, Theorem 2.4]. Our main source for this chapter has been the paper [Facchini and Girardi 2010]. Proposition 6.10, Corollary 6.13, Proposition 6.15, and Theorem 6.17 generalize Theorem 1.4, Corollary 1.5, Lemma 2.3, and Theorem 2.4 of [Warfield 1975], respectively.

Chapter 7

Semilocal Categories and Their Maximal Ideals 7.1 Maximal Ideals We will now study the maximal ideals of a preadditive category C. We say that an ideal M of a preadditive category C is maximal if the improper ideal HomC of C is the unique ideal of the category C properly containing M. Obviously, if all objects of C are zero objects, maximal ideals do not exist in C. In this section, we will characterize maximal ideals in preadditive categories. For instance, we saw in Lemma 4.29 that if C is a preadditive category, Mat(C) is the corresponding free additive category, and C is the idempotent completion of C, then there is an order-preserving one-to-one correspondence between ideals of  It follows that there is a natural one-to-one C, ideals of Mat(C), and ideals of C.  correspondence between the maximal ideals of C, those of Mat(C), and those of C. We have already defined the ideal AI of the category C associated to an ideal I of the endomorphism ring of an object A of C (Example 4.30). If A is an object of C and I is a two-sided ideal of the ring EndC (A), then AI is an ideal of C that extends to the whole category C the ideal I. Recall that AI (A, A) = I, and AI is the greatest of the ideals I of the category C such that I(A, A) ⊆ I (Remark 4.31). The following lemma, due to Pˇr´ıhoda, considers the special case in which I is a maximal ideal of EndC (A). Notice that when we say “maximal ideal” in a ring, we mean “maximal two-sided ideal,” that is, “maximal in the set of all proper two-sided ideals.” Lemma 7.1. Let C be a preadditive category and let A, B be two nonzero objects of C. Let I be a maximal ideal of EndC (A) and AI the ideal of C associated to I. Suppose that the ideal I  := AI (B, B) of EndC (B) is a proper ideal of EndC (B). Then: (a) The ideal AI coincides with the ideal AI  of C associated to I  . © Springer Nature Switzerland AG 2019 A. Facchini, Semilocal Categories and Modules with Semilocal Endomorphism Rings, Progress in Mathematics 331, https://doi.org/10.1007/978-3-030-23284-9_7

215

216

Chapter 7. Semilocal Categories and Their Maximal Ideals

(b) If EndC (B) is a semilocal ring, then I  is a maximal ideal of EndC (B). Proof. (a) We must prove that AI (X, Y ) = AI  (X, Y ) for every pair (X, Y ) of objects. In order to show that AI (X, Y ) ⊆ AI  (X, Y ), fix a morphism f ∈ AI (X, Y ). Then βf α ∈ AI (B, B) = I  for every α : B → X and every β : Y → B. Thus f ∈ AI  (X, Y ). In order to prove the other inclusion, fix a morphism f : X → Y in the category C and assume that f ∈ / AI (X, Y ). There exist α : A → X and β : Y → A / I. Thus EndC (A)g EndC (A) + I = EndC (A). with g := βf α ∈ Assume g ∈ AI  (A, A). Then HomC (A, B) EndC (A) HomC (B, A) = HomC (A, B)(EndC (A)g EndC (A) + I) HomC (B, A). Now HomC (A, B) EndC (A)g EndC (A) HomC (B, A) ⊆ I  because g ∈ AI  (A, A) and AI  (B, B) = I  . Moreover, HomC (A, B)I HomC (B, A) = HomC (A, B)AI (A, A) HomC (B, A) ⊆ AI (B, B) = I  . Thus EndC (A) ⊆ AI  (A, A). Since the ideal I  of EndC (B) is proper, we know that 1B ∈ / I  = AI (B, B). Thus there exist morphisms ϕ : A → B and ψ : B → A such that ψϕ ∈ / I. Since maximal ideals are prime ideals, we get that ψϕ EndC (A)ψϕ ⊆ I. Now, EndC (A) ⊆ AI  (A, A) implies that ϕ EndC (A)ψ ⊆ AI  (B, B) = I  = AI (B, B), so that ψϕ EndC (A)ψϕ ⊆ I, which is a contradiction. This contradiction proves that g ∈ / AI  (A, A). Thus there exist two morphisms α : B → A and  β : A → B with β  βf αα ∈ / I  . In particular, f ∈ / AI  (X, Y ). (b) Now suppose the ring EndC (B) semilocal. Let J be the Jacobson radical of the category C, so that J (A, A) coincides with the Jacobson radical J(EndC (A)) of the ring EndC (A) for every nonzero object A of C. Every maximal ideal is right primitive, and J(EndC (A)) is the intersection of all right primitive ideals of EndC (A). Thus J (A, A) ⊆ I. Since AI is the greatest of the ideals of C with this property, we get that J ⊆ AI . It follows that J (B, B) ⊆ AI (B, B), i.e., J(EndC (B)) ⊆ I  . Now, I  is a proper ideal of EndC (B) and EndC (B) is a semilocal ring. Thus EndC (B)/I  is a semisimple Artinian ring. We will now show that I  is maximal. For this, it suffices to prove that EndC (B)/I  is a simple Artinian ring. If the semisimple Artinian ring EndC (B)/I  is not simple, there exist elements f, g ∈ EndC (B) with the property that f + I  , g + I  are nontrivial orthogonal central idempotents of the ring EndC (B)/I  . Since these idempotents are nonzero, we have that f, g ∈ / I  = AI (B, B). Thus   there exist α, α : A → B and β, β ∈ B → A with βf α ∈ / I and β  gα ∈ / I.   Since maximal ideals are prime, it follows that β gα EndC (A)βf α ⊆ I. Now f +

7.1. Maximal Ideals

217

I  and g + I  are orthogonal and central, so that g EndC (B)f ⊆ I  . Hence, a fortiori, gα EndC (A)βf ⊆ I  . Since I  = AI (B, B), we know that β  I  α ⊆ I. Thus β  gα EndC (A)βf α ⊆ I, which contradicts what we had previously proved. It follows that EndC (B)/I  is a simple Artinian ring and I  is a maximal ideal.  If X, Y , and A are objects of a preadditive category C, we will denote by HomC (A, Y ) HomC (X, A) the additive subgroup of HomC (X, Y ) generated by all the composite morphisms f g, where f ranges over HomC (A, Y ) and g ranges over HomC (X, A). Proposition 7.2. Let A, A be nonzero objects of a preadditive category C, let M (resp. M  ) be a maximal two-sided ideal of EndC (A) (resp. EndC (A )), and let AM (resp. AM  ) be the ideal of C associated to M (resp. M  ). The following conditions are equivalent: AM = AM  . AM (B, B) = AM  (B, B) for every B ∈ Ob(C). M = AM  (A, A). M  = AM (A , A ). There exists an object C ∈ Ob(C) such that AM (C, C) = AM  (C, C), and this is a proper ideal of the ring EndC (C). (e) HomC (A, A )M HomC (A , A) ⊆ M  and HomC (A, A ) HomC (A , A)  M  . (e ) HomC (A , A)M  HomC (A, A ) ⊆ M and HomC (A , A) HomC (A, A )  M . (f) There are two morphisms ϕ : A → A , ψ : A → A such that (a) (b) (c) (c ) (d)

ψ EndC (A )ϕ ⊆ M, ϕ EndC (A)ψ ⊆ M  , ψM  ϕ ⊆ M, and ϕM ψ ⊆ M  . Proof. It is clear that (a) ⇒ (b) ⇒ (c) ⇒(d) and (b) ⇒ (c ) ⇒ (d). (d) ⇒ (a) By Lemma 7.1(a), the ideal of C associated to AM (C, C) coincides with AM . Thus AM is the greatest of the ideals I of C such that I(C, C) ⊆ AM (C, C). From AM  (C, C) = AM (C, C), it follows that AM  ⊆ AM . By symmetry, AM ⊆ AM  . (c) ⇒ (e) Suppose (c) holds. Then HomC (A, A )M HomC (A , A) ⊆ M  . If HomC (A, A ) HomC (A , A) ⊆ M  , then EndC (A) ⊆ AM  (A, A) = M , which is a contradiction. Thus HomC (A, A ) HomC (A , A) ⊆ M  . (e) ⇒ (f) Assume that (e) holds. From HomC (A, A ) HomC (A , A)  M  , it / M . follows that there exist two morphisms ϕ : A → A and ψ : A → A with ϕψ ∈    Since M is a prime ideal, ϕψ ∈ / M implies that ϕψ EndC (A )ϕψ ⊆ M  . By the first condition in (e), it follows that ψ EndC (A )ϕ ⊆ M . Moreover, the condition ϕψ ∈ / M  implies the second condition in (f). For the third, notice that ϕ EndC (A)ψM  ϕ EndC (A)ψ ⊆ M  because M  is a two-sided ideal in EndC (A ). From the first condition in (e), it follows that ϕ(EndC (A)ψM  ϕ EndC (A) + M )ψ ⊆ M  . By (e), we get that EndC (A)ψM  ϕ EndC (A) ⊆ M . This gives the third condition in (f). The fourth one follows from the first condition in (e).

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(f) ⇒ (a) Now assume that condition (f) holds. In order to prove that AM ⊆ AM  , fix two objects X, Y ∈ Ob(C) and a morphism f : X → Y in AM (X, Y ). In order to prove that f belongs to AM  (X, Y ), fix two arbitrary morphisms α : A → X and β : Y → A . We have to show that βf α ∈ M  . From f ∈ AM (X, Y ), we know that (EndC (A)ψ EndC (A )β)f (α EndC (A )ϕ EndC (A)) ⊆ M. By the fourth condition in (f), we get that ϕ(EndC (A)ψ EndC (A )βf α EndC (A )ϕ EndC (A))ψ ⊆ M  . Thus EndC (A )(ϕ EndC (A)ψ EndC (A )βf α EndC (A )ϕ EndC (A)ψ) EndC (A ) ⊆ M  . This is the product of three ideals of the ring EndC (A ): EndC (A )ϕ EndC (A)ψ EndC (A ), and

EndC (A )βf α EndC (A ),

EndC (A )ϕ EndC (A)ψ EndC (A ).

Notice that EndC (A )ϕ EndC (A)ψ EndC (A ) ⊆ M  by the second condition in (f). Since the ideal M  is prime, we have that βf α ∈ M  . This proves that AM ⊆ AM  .  The proofs that AM  ⊆ AM and (c ) ⇒ (e ) ⇒ (f) are similar. Proposition 7.3. Let M be a proper ideal of a preadditive category C. Then M is a maximal ideal if and only if the following two conditions hold for every object A of C with M(A, A) = EndC (A) : (a) M(A, A) is a maximal ideal of EndC (A), and (b) M is the ideal of the category C associated to the ideal M(A, A) of the ring EndC (A). Proof. Let M be a maximal ideal of C and let A be an object of C with M(A, A) = EndC (A). If the two-sided ideal M(A, A) is not a maximal ideal of the ring EndC (A), and I is a maximal ideal of EndC (A) containing M(A, A), then the ideal AI of the category C associated to I has the property that M ⊂ AI , because M(A, A) ⊂ I. Since M is a maximal ideal, it follows that AI is the improper ideal HomC . This is a contradiction, because AI (A, A) = I. The contradiction proves that M(A, A) is a maximal ideal of EndC (A). In order to prove that (b) also holds, set M := M(A, A). We must show that AM = M. Now, M(A, A) = M implies that M ⊆ AM . Since M is maximal, it follows that either AM = M or AM is the improper ideal HomC . But AM (A, A) = M is a proper ideal, so that AM is a proper ideal. Therefore M = AM . For the converse, assume that M is a proper ideal of C for which the two conditions (a) and (b) hold for every object A of C with M(A, A) = EndC (A). Let I

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be any ideal containing M. Then I(A, A) ⊇ M(A, A) for every object A ∈ Ob(C). Assume that there exists an object B of C with I(B, B) = M(B, B) = EndC (B). By condition (b), M is the ideal associated to M(B, B), so that M is the greatest of the ideals I  with I  (B, B) ⊆ M(B, B). It follows that I = M in this case. Thus we can assume that for every A ∈ Ob(C), either I(A, A) ⊃ M(A, A) or I(A, A) = M(A, A) = EndC (A). By condition (a), we have in both cases that I(A, A) = EndC (A). Thus I is the improper ideal HomC in this second case,  which proves that M is maximal. In the following proposition, we give a good description of the maximal ideals of the category proj-R. Proposition 7.4. Let R be a ring. Then the maximal ideals of the category proj-R are exactly the ideals of proj-R associated to the maximal two-sided ideals of the ring R. Proof. If M is a maximal ideal of proj-R, then M(RR , RR ) is either equal to the ∼ R or is a maximal ideal of HomR (RR , RR ) = ∼R improper ideal HomR (RR , RR ) = (Lemma 7.3). In the first case, M(RR , RR ) = HomR (RR , RR ) implies that M(PR , PR ) = HomR (PR , PR ) for every finitely generated projective module PR . Thus M is the improper ideal, which is a contradiction. Hence M(RR , RR ) is a maximal ideal M of HomR (RR , RR ) ∼ = R, so that M is associated to M by Lemma 7.3(b). Conversely, let M be any maximal two-sided ideal of End(RR ). Let AM be the ideal of proj-R associated to M . Let I denote any ideal of proj-R containing ∼ End(RR ) containing AM (RR , RR ). If AM . Then I(RR , RR ) is an ideal of R = I(RR , RR ) = End(RR ), then I is the improper ideal of proj-R. If I(RR , RR ) = AM (RR , RR ), then AM = I, because as we have seen in Example 4.30, the associated ideal AM is maximal among the ideals I  with I  (RR , RR ) ⊆ AM (RR , RR ). 

7.2 Simple Additive Categories A preadditive category is simple if it has exactly two ideals, necessarily the trivial ones. Hence, a simple category has nonzero objects. It is easily seen that the dual of a simple category is a simple category. Let A be an object of a preadditive category C. Then A is a generator if for every nonzero morphism f : B → C in C, there exists a morphism α : A → B such that f α = 0. The object A is a cogenerator if it is a generator in the dual category. Theorem 7.5. Let C be a preadditive category. The following conditions are equivalent: (a) The category C is simple.

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(b) The category C has a nonzero object, the endomorphism ring of every nonzero object of C is a simple ring, and for every A, B, C ∈ Ob(C) such that A = 0 and every f : B → C, if βf α = 0 for every α : A → B and every β : C → A, then f = 0. (c) The category C has a nonzero object, every nonzero object of C is a generator and a cogenerator for C, and every nonzero object of C has a simple endomorphism ring. (d) The category C has a nonzero object and there exists a simple ring R such that C is equivalent to a full subcategory of the category proj-R. Proof. The implication (a) ⇒ (b) follows immediately from Lemma 7.3, because if C is a simple category, then for every nonzero object A, the zero ideal of C is the ideal associated to the zero ideal of EndC (A). (b) ⇒ (c) Suppose that (b) holds. Let A be a nonzero object. In order to show that A is a generator, fix a nonzero morphism f : B → C in C. By condition (b), there exist morphisms α : A → B and β : C → A such that βf α = 0. In particular, f α = 0 and βf = 0. Thus A is a generator and a cogenerator. (c) ⇒ (d) Suppose that (c) holds. Fix a nonzero object A of C, so that the endomorphism ring R := EndC (A) of A is a simple ring. We will prove that add(A) = C. Let B = 0 be an object of C. Since B is a generator and 1A = 0A , there is a morphism α : B → A with 1A α = 0A α, that is, α = 0. Since B is a cogenerator, there is a morphism β : A → B with βα = 0. But EndC (B) is a simple ring, so that the two-sided ideal of EndC (B) generated by βα is the whole n ring, that is, there exist γ1 , . . . , γn , γ1 , . . . , γn ∈ EndC (B) with 1B = i=1 γi βαγi . This proves that B is an object of add(A), and add(A) = C. From Lemma 4.18(a), we can conclude that the additive functor F := HomC (A, −) : C → Mod-R is both full and faithful. (d) ⇒ (a) Let I be a nonzero ideal of a full subcategory C of proj-R for some simple ring R. We must show that I is the improper ideal HomC of C. Fix a nonzero morphism f : A → B in I. We must prove that any morphism g : X → Y in C is in I. There exist an epimorphism πA : Rn → A and a monomorphism εB : B → Rm . Thus the morphism εB f πA : Rn → Rm is a nonzero R-module morphism, so that there exist two morphisms εR : R → Rn and πR : Rm → R simple ring, so that there exist enwith πR εB f πA εR : R → R nonzero. But R is a n domorphisms f1 , . . . , fn , g1 , . . . , gn of RR with i=1 fi πR εB f πA εR gi = 1R . Since t t t Y is a direct summand of RR , there exist α : Y → RR and β : RR → Y with t t → , . . . , ε : π , . . . , π R . If R and ε → R are morphisms such that : R = 1 βα 1 t 1 t Y t t , then ε = 1 π j j R j=1 t

t βεj πj αg = βεj 1R πj αg j=1 j=1 n t (βεj fi πR εB )f (πA εR gi πj αg) =

g = βαg = β1Rt αg =

i=1

is in I(X, Y ), as desired.

j=1



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Notice that by condition (d) of Theorem 7.5, every simple preadditive category is necessarily skeletally small. Also, by condition (d) again, every full subcategory of a simple preadditive category containing a nonzero object is a simple category. Proposition 7.6. An additive category C in which idempotents split is simple if and only if it is equivalent to the category proj-R for some simple ring R. Proof. We already know that if R is a simple ring, the category proj-R is simple (Theorem 7.5, (d) ⇒ (a)). Conversely, if C is a simple additive category with splitting idempotents, then every nonzero object of C has a simple endomorphism ring (Theorem 7.5). In the proof of Theorem 7.5, (c) ⇒ (d), we have seen that for every fixed nonzero object A, one has add(A) = C. By Lemma 4.18(b), the functor F := HomC (A, −) : C →  proj-R is a category equivalence. Remark 7.7. Ideals of a preadditive category C coincide with kernels of additive functors F : C → D for some preadditive category D. Maximal ideals of a preadditive category C coincide with kernels of nonzero functors F : C → proj-R for some simple ring R. Let us show that in this case it suffices to consider the simple rings R of the form EndC (A)/M , where A is a nonzero object of C and M is a maximal ideal of the endomorphism ring EndC (A). As we have seen in Lemma 7.3, if M is a maximal ideal of a category C, then M is associated to a maximal ideal M of the endomorphism ring EndC (A) for some nonzero object A of C. The object A, viewed as an object of the factor category C/M, is a nonzero object whose endomorphism ring is the simple ring EndC (A)/M . As we have seen in the proof of (c) ⇒ (d) in Theorem 7.5, the functor F = HomC/M (A, −) : C/M → proj-(EndC (A)/M ) is full and faithful. It follows that the kernel of the functor HomC (A, −) : C → proj-(EndC (A)/M ) is exactly the maximal ideal M.

7.3 Maximal Ideals Exist in Semilocal Categories We begin this section showing that maximal ideals do not necessarily exist in a preadditive category C. The first trivial example is that all objects of C are zero objects. Here are two more interesting examples. Examples 7.8. (1) Let k be a division ring. Recall that for any k-linear mapping f : Vk → Wk , the rank ρ(f ) of f is the dimension of the image im(f ) of f . For any infinite cardinal c, consider the ideal Ic of Mod-R defined, for every Vk , Wk ∈ Ob(Mod-R), by Ic (Vk , Wk ) := { f ∈ Hom(Vk , Wk ) | ρ(f ) < c }. It is easy to check that for f : Vk → Wk and f  : Vk → Wk , there exist α : Vk → Vk and β : Wk → Wk such that f = βf  α if and only if ρ(f ) ≤ ρ(f  ). It follows that the ideals of Mod-R are the zero ideal, the improper ideal Hom, and the ideals Ic for every infinite cardinal c. Thus maximal ideals do not exist in the category Mod-k.

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(2) One may have the impression that the lack of maximal ideals in the category Mod-k in the example of the previous paragraph depends on set-theoretical problems. The reason is not this. The reason is that preadditive categories have a behavior that looks very much like the behavior of rings without identity. For instance, it is easy to construct rings without identity without maximal ideals ufer group with zero multiplication). It is easy to modify the (consider the Pr¨ example of the previous paragraph and get an example of a small category without maximal ideals. As before, let k be a division ring and V k a right vector space of infinite dimension d. Let C be the full subcategory of Mod-k whose objects are all vector subspaces of V k of dimension < d. Consider the ideal Ic defined in the previous paragraph for each infinite cardinal c, and let Ic also denote the restriction of that ideal to C. Again, from the fact that for every f : Vk → Wk and f  : Vk → Wk , there exist α : Vk → Vk and β : Wk → Wk such that f = βf  α if and only if ρ(f ) ≤ ρ(f  ), it follows that the ideals of C are the zero ideal, the improper ideal HomC , and the ideals Ic for every infinite cardinal c also in this case, but clearly, the improper ideal and all the ideals Ic with c ≥ d coincide. An object Vk of C becomes the zero object in the factor category C/Ic if and only if dim(Vk ) < c. Now, maximal ideals exist in C if and only if d is the successor of some cardinal d (and in this case C has a unique maximal ideal, which is the ideal Id ). For instance, if d = ℵω , then d is not the successor of a cardinal, the ideals of the small abelian category C are the two trivial ideals and the ideals Iℵn with n a finite ordinal, and so maximal ideals do not exist in C. If d = ℵ1 , then d is the successor of the cardinal ℵ0 , and C has Iℵ0 as its unique maximal ideal. In this case, the canonical functor C → C/Iℵ0 is not isomorphism-reflecting, because all finite-dimensional objects of C become zero objects in the factor category C/Iℵ0 . The previous examples also show that though every maximal ideal of a category C is the ideal associated to a maximal ideal of the endomorphism ring of a nonzero object of C, as we have seen in Lemma 7.3, the converse is not always true, even for small categories. In the category Mod-R, if Vk is any vector space of infinite dimension c, the ideal Ic is the ideal associated to the maximal ideal of End(Vk ), which consists of all the endomorphisms of Vk of rank < c. If Vk = 0 is a vector space of finite dimension, the zero ideal of Mod-R is the ideal associated to the zero ideal of End(Vk ), which is the unique maximal ideal of End(Vk ), because End(Vk ) is a simple ring. Now let us move on to consider maximal ideals of semilocal categories. Proposition 7.9. If C is a semilocal category, then: (a) The ideals of C associated to the maximal ideals of the endomorphism rings of nonzero objects of C are exactly the maximal ideals of C. (b) Every proper ideal of C is contained in a maximal ideal of C. (c) Maximal ideals exist in C.

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Proof. (a) We already know that every maximal ideal of C is the ideal associated to a maximal ideal of the endomorphism ring of a nonzero object of C (Proposition 7.3). Conversely, let M be a maximal two-sided ideal of the endomorphism ring EndC (A) for some nonzero object A ∈ Ob(C). We must prove that the ideal AM associated to M is maximal. Let I be an ideal in C that properly contains AM . By Lemma 7.1(b), for any nonzero object B ∈ Ob(C), the ideal AM (B, B) of EndC (B) is always either EndC (B) or a maximal ideal of EndC (B). Since I properly contains AM , the ideal I(B, B) also must be either the improper ideal EndC (B) or the maximal ideal AM (B, B) of EndC (B). But if I(B, B) = AM (B, B) is a maximal ideal of EndC (B), we obtain from Lemma 7.1(a) that I ⊆ AM . This is a contradiction, because I properly contains AM . The contradiction shows that I(B, B) = AM (B, B) cannot be a maximal ideal of EndC (B). Thus I is the improper ideal HomC , and so AM is maximal. (b) If I is a proper ideal of C, then there exists a nonzero object A of C with I(A, A) = EndC (A). Let M be a maximal ideal of EndC (A) containing the proper ideal I(A, A). Then, by (a), the ideal associated to M is a maximal ideal of C and contains I by the maximality of the associated ideal (Remark 4.31).  (c) follows by applying (b) to the zero ideal of C. Recall that if R is a simple Artinian ring, then R has a unique simple right module SR up to isomorphism, and all finitely generated right R-modules MM are n semisimple and isomorphic to SR , where n = dim(MR ), the Goldie dimension of MR . Corollary 7.10. Let C be a semilocal category and M a maximal ideal of C. Then there exist a simple Artinian ring R and a full and faithful functor F : C/M → fgss-R of the factor category C/M into the full subcategory fgss-R of Mod-R whose objects are all finitely generated semisimple right modules over the simple Artinian ring R. Moreover, for every object B of C, the Goldie dimension dim(F (B) of the semisimple right R-module F (B) coincides with the dual Goldie dimension codim(EndC (B)/M(B, B)) of the ring EndC (B)/M(B, B). Proof. We will argue as in the proof of Theorem 7.5((c) ⇒ (d)), applying Lemma 4.18(a). Let A be a nonzero object of the factor category C/M. Set R := EndC (A)/M(A, A), which is simple Artinian because M(A, A) is a maximal ideal in the semilocal ring EndC (A). Since R is simple artinian, every right R-module is semisimple and projective. For every finitely generated semisimple module M , the Goldie dimension of M coincides with the dual Goldie dimension of its endomorphism ring. Thus, for every object B of C, we have that dim(F (B)) = codim(EndR (F (B))) = codim(EndC/M (B)) = codim(EndC (B)/M(B, B)).



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As we have already done previously, for a category C we will denote by V (C) the skeleton of C, and A ∈ V (C) will be the unique object of V (C) isomorphic ∼ B in C if and to A for every object A of C, so that for every A, B ∈ Ob(C), A = only if A = B. Let C be a semilocal category. Consider the class P of all pairs (A, M ), where A ranges over the class of all nonzero objects of C and M is a maximal ideal in the endomorphism ring EndC (A). It is possible to define an equivalence relation ∼ on this class by setting (A, M ) ∼ (A , M  ) if AM = AM  . (We saw equivalent formulations of this equivalence relation in Proposition 7.2.) Let Max(C) be a class of representatives of the pairs in P modulo ∼. We will call Max(C) the maximal spectrum of the semilocal category C. Example 7.11. Let C be a preadditive category in which EndC (A) is a local ring for every A ∈ Ob(C). This implies, in particular, that C has no zero object. We will now show that there is a bijection f between Max(C) and V (C), defined by f (A, M ) = A for every (A, M ) ∈ Max(C). In order to see that f is injective, fix (A, M ), (A , M  ) ∈ Max(C) with A ∼ = A . If g : A → A is an isomorphism, then for every morphism h : X → Y in C, h ∈ AM (X, Y ) if and only if the endomorphism βhα of A is not an automorphism of A for every α : X → A and every β : Y → A, that is, if and only if the endomorphism gβhαg −1 of A is not an automorphism of A for every α : X → A and every β : Y → A . This occurs if and only if h ∈ AM  (X, Y ). Thus AM = AM  and (A, M ) ∼ (A , M  ), so that (A, M ) = (A , M  ). In order to prove that f is onto, fix an object A ∈ V (C). Then A is nonzero, so that the local ring EndC (A) has a maximal ideal M . Let (A , M  ) ∈ Max(C) be such that (A, M ) ∼ (A , M  ). Then, from AM = AM  , it follows that there exist g : A → A , h : A → A, and α : A → A with hαg an automorphism of A (Proposition 7.2, (a) ⇒ (f)). Thus there exists g  : A → A with hαgg  = 1A . Hence h is right invertible and αgg  h is a nonzero idempotent of EndC (B). Now, EndC (B) is a local ring, so that its only nonzero idempotent is the identity. Thus h is also left invertible, hence an isomorphism. It follows that A  = A and f (A , M  ) = A. Here is a further example of a maximal spectrum of a semilocal category. Let R be a ring and SR the full subcategory of proj-R whose objects are finitely generated projective right R-modules with a semilocal endomorphism ring. We had already met this full subcategory in Theorem 4.68. As we had noticed before the statement of that theorem, the objects of SR are the projective covers of all finitely generated semisimple modules. Let H be the full subcategory of Mod-R whose objects are all the modules AR /B, where AR ∈ SR and B is a maximal submodule of AR . Let V (H) be a skeleton of H. Thus, for every AR ∈ SR there exist integers nS ≥ 0, almost all zero, with AR /AR J(R) ∼ = ⊕S∈V (H) S nS . For every S ∈ V (H), there is a unique R-submodule AS of AR such that AS ⊇ AR J(R) and AS /AR J(R) is the S-socle of AR /AR J(R), that is, AS /AR J(R) ∼ = S nS . Then

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225

AR /AR J(R) = ⊕S∈V (H) AS /AR J(R). Now End(AR )/J(End(AR )) ∼ = EndR/J(R) (AR /AR J(R))  ∼ End(AS /AR J(R)), = S∈V (H)

where each End(AS /AR J(R)) is the zero ring if nS = 0 and is a simple Artinian ring if nS > 0. It follows that the maximal ideals of End(AR ) are the ideals MS = { f ∈ End(AR ) | f (AS ) ⊆ AR J(R) }, where S ranges over the objects of V (H) with nS > 0. Theorem 7.12. In the notation of the previous paragraph, there is a bijection Max(SR ) → V (H). Proof. Let (AR , M ) ∈ Max(SR ), so that M is a maximal ideal of the semilocal endomorphism ring of the finitely generated projective module AR . Consider the correspondence Φ : Max(SR ) → V (H) that associates to every (AR , M ) the unique S ∈ V (H) with nS > 0 and M = MS . To show that the correspondence Φ is onto, fix S ∈ V (H). The module S is  , M  ) be the unique element a homomorphic image of an object AR of SR . Let (AR   in Max(SR ) with (AR , M ) ∼ (AR , MS ). This, by Proposition 7.2, (a) ⇔ (c),  is equivalent to M  = AMS (AR , AR ) = {f ∈ End(AR ) | βf α ∈ MS for every    α : AR → AR and β : AR → AR } = {f ∈ End(AR ) | βf α(AS ) ⊆ AR J(R) for every α : AR → AR and β : AR → AR }. Since every morphism g : AR /AR J(R) →  /AR J(R) can be lifted to a morphism g : AR → AR by the projectivity of AR , AR we have that M  = {f ∈ End(AR ) | βf (AS ) ⊆ AR J(R) for every β : AR → AR }. Similarly, every morphism h : AR /AR J(R) → AR /AR J(R) can be lifted to a morphism h : AR → AR by the projectivity of AR , and hence M  = {f ∈ End(AR ) | f (AS ) ⊆ AR J(R)} = MS  . This concludes the proof that Φ is onto. To show that Φ is injective, fix two finitely generated projective modules AR and AR with semilocal endomorphism rings, let M and M  be two maximal ideals of End(AR ) and End(AR ) respectively, and suppose Φ(AR , M ) = Φ(AR , M  ). Then there exists S ∈ V (H) such that M = MS = { f ∈ End(AR ) | f (AS ) ⊆ AR J(R) } and M  = MS = { f  ∈ End(AR ) | f  (AS ) ⊆ AR J(R) }. We have to show that AM = AM  . By Propositions 7.2 and 7.9, it suffices to prove that AM (AR , AR ) = { f  ∈ End(AR ) | βf  α(AS ) ⊆ AR J(R) for every α : AR → AR , β : AR → AR } is contained in M  . If f  is a morphism in End(AR ) \ M  , then f  (AS ) ⊆ AR J(R); hence f  induces a nonzero endomorphism f  of AS /AR J(R). Now, both AS /AR J(R) and AS /AR J(R) are nonzero modules that are direct sums of finitely many copies of S, so that there exist a morphism α : AS /AR J(R) → AS /AR J(R) and a morphism β : AS /AR J(R) → AS /AR J(R) with βf  α = 0.

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Since both AR and AR are projective, the morphisms α and β can be lifted to morphisms α : AR → AR and β : AR → AR , and βf  α(AS ) ⊆ AR J(R). It follows that f  ∈ / AM (AR , AR ), as desired.  Let Cλ be a preadditive category for every λ in a class Λ, and let ⊕λ∈Λ Cλ be the weak direct sum (Section 4.8). For every λ0 ∈ Λ, there is a full and faithful canonical functor Eλ0 : Cλ0 → ⊕λ∈Λ Cλ . If D is an additive category with a zero object and Gλ : Cλ → D is an additive functor for every λ ∈ Λ, there exists an additive functor G : ⊕λ∈Λ Cλ → D with GEλ naturally isomorphic to Gλ for every λ ∈ Λ. Moreover, the additive functor G with this property is unique up to natural isomorphism. Let D be a preadditive category. Let Fλ : D → Cλ , where λ ranges over the class Λ, be an additive functor. Assume that for every object A ∈ Ob(D), the object Fλ (A) is a zero object of Cλ for almost all λ ∈ Λ. Then it is possible to define an additive functor F : D → ⊕λ∈Λ Cλ as follows. Consider the composite functors Eλ Fλ : D → ⊕λ∈Λ Cλ , λ ∈ Λ. If A is an object of D and λ1 , . . . , λn are the elements λ ∈ Λ such that Fλ (A) is a nonzero object of Cλ , let F (A) be the coproduct in ⊕λ∈Λ Cλ of the objects Eλ1 Fλ1 (A), . . . , Eλn Fλn (A). If f : A → B is a morphism in D and μ1 , . . . , μm are the elements μ ∈ Λ with Fμ (B) a nonzero object of Cμ , then F maps f to the m × n matrix having the (i, j)-entry equal to Eμi Fμi (f ) for μi = λj , and all the other entries equal to zero. We say that the functor F : D → ⊕λ∈Λ Cλ is the functor induced by the collection of functors Fλ : D → Cλ , λ ∈ Λ. Theorem 7.13. If C is a semilocal category, then: (a) The Jacobson radical of C is the intersection of all maximal ideals of C. Moreover, for every object A in C, there exist finitely many maximal ideals M1 , . . . , Mn (n ≥ 0) such that for every maximal ideal M in C, A is a nonzero object in the factor category C/M if and only if M = Mi for some i = 1, . . . , n. (b) The canonical functor F : C → ⊕M∈Max(C) C/M, induced by the collection of canonical functors C → C/M, M ∈ Max(C), is isomorphism reflecting. (c) If C is additive with splitting idempotents, the functor F is also direct-summand reflecting. Proof. (a) We first show that J is contained in all the maximal ideals of C. Let A, B be objects of C, and f a morphism in J (A, B). Let M be any maximal ideal of C. We must prove that f ∈ M(A, B). By Lemma 7.3, either (1) M(A, A) = EndC (A), or (2) M(A, A) is a maximal ideal of EndC (A) and M is the ideal of C associated to M(A, A). If M(A, A) = EndC (A), then M(A, B) = HomC (A, B), and thus f ∈ M(A, B) and we are done. Therefore we can assume that M := M(A, A) is a maximal ideal of EndC (A) and M is the ideal of C associated to M . Since f ∈ J (A, B), we have that βf α ∈ J (A, A) = J(EndC (A)) ⊆ M for every α : A → A and every β : B → A. Thus f belongs to the maximal ideal M associated to M .

7.3. Maximal Ideals Exist in Semilocal Categories

227

Conversely, suppose f ∈ M(A, B) for every maximal ideal M of C. If A is a zero object in C, then f = 0 ∈ J (A, B). If A is not a zero object in C, fix any maximal ideal M of the ring EndC (A). Let AM be the ideal in C associated to M . Then AM is maximal by Proposition 7.9(a). Since f ∈ AM (A, B), we have that gf ∈ AM (A, A) = M for every g : B → A. This is true for every maximal ideal M of EndC (A), which is a semilocal ring (Proposition 3.14). It follows that gf ∈ J (A, A) for every g : B → A. Thus 1A − gf has a left inverse, and f ∈ J (A, B). This proves that J is the intersection of all the maximal ideals of C. Now let A be an object of C. If A = 0 in C, then A = 0 in C/M for every maximal ideal M. Suppose A nonzero. Then EndC (A) is a semilocal ring. Let M1 , . . . , Mn be the n maximal two-sided ideals of the ring EndC (A), and AM1 , . . . , AMn the corresponding associated ideals in the category. Let M be a maximal ideal of C different from all the ideals AMi (i = 1, . . . , n). Since M is not associated to M1 , . . . , Mn , we get that M(A, A) = EndC (A) by Lemma 7.3. Thus the object A is the zero object in C/M. This concludes the proof of (a). Notice that (a) allows us to say that the canonical functors C → C/M, M ∈ Max(C), induce a functor F into the weak direct sum category. It sends the object A of C into the finite set {(M1 , A1 ), . . . , (Mn , An )}, where Ai is the image of A in the factor category C/Mi . (b) and (c). Let B be a nonzero object of C. By (a), there are only finitely many maximal ideals M with B nonzero in the factor category C/M. Let M1 , . . . , Mn be these finitely many distinct maximal ideals. Then Mi (B, B) = EndC (B) for every i, so that Mi is the ideal associated to the maximal ideal Mi (B, B) of EndC (B) (Lemma 7.3). By Proposition 7.9(a), the maximal ideals of the ring EndC (B) are exactly the n ideals Mi (B, B). Since EndC (B) is a semilocal ring, we have a canonical isomorphism ∼ EndC (B)/J(EndC (B)) =

n 

EndC (B)/Mi (B, B).

i=1

Thus, for every i = 1, . . . , n, there exists an endomorphism δi ∈ EndC (B) with δi ≡ 1B (mod Mi (B, B)), δi ≡ 0B (mod Mj (B, B)) for every j = i, and δi δj ∈ J(EndC (B)) for every i = j. To prove (b), fix an object A of C with A ∼ = B in C/M for every maximal ideal M of C. Then A is nonzero in C; otherwise, B = 0 in C/M for every maximal ideal M of C, so B = 0 in C/J , hence B = 0 in C, which is a contradiction. Thus A and B are both nonzero in C. Now, for every M, A = 0 in C/M if and only if B = 0 in C/M. Hence we can apply the argument of the previous paragraph to A also, and find that there exist endomorphisms δ1 , . . . , δn ∈ EndC (A) with δi ≡ 1A (mod Mi (A, A)), δi ≡ 0A (mod Mj (A, A)) for j = i, and δi δj ∈ J(EndC (A)) for i = j. Now, for every index i = 1, . . . , n, let fi : A → B be a morphism in C that becomes an isomorphism in C/Mi and let gi : B → A be a morphism that lifts to

Chapter 7. Semilocal Categories and Their Maximal Ideals

228

n n C the inverse of fi in C/Mi . Set f := i=1 δi fi δi and g := i=1 δi gi δi . Then  δi gi δi δj fj δj ≡ δi gi δi fi δi ≡ gi fi ≡ 1A (mod Mi ) gf = i,j

for all indices i, hence modulo J . it follows that f is left invertible in C. Similarly one proves that the composite morphism f g is invertible in C, so that f is right ∼ B in C. invertible. Therefore f is an isomorphism in C, and A = To prove (c), suppose C additive with splitting idempotents. Let A be an object of C with F (A) a direct summand of F (B). Then we have morphisms fi : A → B and gi : B → A such that gi = 1, . . . , n. fi ≡ 1A (mod Mi ) for every index i Define f := ni=1 δi fi and g := ni=1 gi δi . Then gf = i,j gi δi δj fj ≡ ni=1 gi δi fi (mod J ), so that gf ≡ gi fi (mod Mi ), that is, gf ≡ 1A (mod Mi ) for all indices i = 1, . . . , n. Since F (A) is a direct summand of F (B) and M(B, B) = EndC (B) for every maximal ideal M of C different from M1 , . . . , Mn , it follows that M(A, A) = EndC (A) for every maximal ideal M of C different from M1 , . . . , Mn , so gf ≡ 1A (mod M) for every maximal ideal M of C. By (a), gf ≡ 1A (mod J ). Thus gf is left invertible in EndC (A). Hence g  f = 1A for a suitable morphism g  . As we have seen in the second paragraph of Section 4.4, A turns out to be isomorphic to a direct summand of B.  Notice that it is necessary that the category be semilocal for the theorem to hold. For instance, if C is the category of all vector spaces of dimension ≤ ℵ1 over a fixed division ring, then C has a unique maximal ideal M consisting of all morphisms of rank ≤ ℵ0 , and all vector spaces of dimension ≤ ℵ0 become isomorphic modulo M.

7.4 The Monoid V (C) for a Semilocal Category C We are ready to prove the following theorem. Theorem 7.14. Let C be a semilocal additive category in which idempotents split. Then the monoid V (C) is a Krull monoid. Notice that here C is not necessarily skeletally small, in which case the monoid V (C) will be a large monoid. Proof. Let C be an additive semilocal category in which idempotents split. Then the direct-summand reflecting functor F : C → ⊕M∈Max(C) C/M of Theorem 7.13(c) induces a monoid homomorphism V (F ) : V (C) → V (⊕M∈Max(C) C/M). The monoid homomorphism V (F ) is a divisor homomorphism. Moreover, (Max(C)) V (⊕M∈Max(C) C/M) ∼ = N0 = ⊕M∈Max(C) V (C/M) ∼

7.4. The Monoid V (C) for a Semilocal Category C

229 (Max(C))

(Corollary 7.10). Here the (possibly large) free monoid N0 is defined as on page 130. This shows that there is a divisor homomorphism of V (C) into a free commutative monoid, so that the monoid V (C) turns out to be a Krull monoid.  Thus, if C is a semilocal additive category in which idempotents split, the objects of C have direct-sum decompositions with regular patterns in the sense of Section 1.3 (Figure 1.3). Notice that by Theorem 7.14, for every semilocal category C the monoid (I) V (C) can be embedded in a suitable free monoid N0 . Therefore every object of a semilocal category can be described, up to isomorphism, by only finitely many nonzero positive integers. For example, Artinian modules can be described, up to isomorphism, by mappings with values in N0 . This is a nontrivial important result. Let us show that the argument of Theorem 7.14 can be inverted. We present it here because we think that it helps clarify the point. To avoid set-theoretic difficulties, we restrict ourselves to the case of skeletally small categories. Let I be (I) a set. Let N0 be the free commutative monoid with free set I of generators and let Z(I) be the free abelian group with free set I of generators. We will denote the (I) elements of N0 as functions s : I → N0 with s(x) = 0 for almost all x ∈ I. The (I) support of any element s ∈ N0 is the finite set supp(s) := { x ∈ I | s(x) = 0 }. For  denote the additive category every preadditive category C, Mat(C) and (Mat(C)) generated by C and the additive category with splitting idempotents generated by C, respectively. As we saw in Example 4.28, the maximal ideals of C, Mat(C), and  coincide. (Mat(C)) (I)

Theorem 7.15. Let I be a set and let S be a subset of the monoid N0 such that

(I) generated by S and let ZS s∈S supp(s) = I. Let N0 S be the submonoid of N0 (I) be the subgroup of Z generated by S. Then there exists a preadditive category  induce a C such that the full and faithful embeddings C → Mat(C) → Mat(C) commutative diagram of sets and mappings N0 S S → ↓∼ ↓∼ = = V (C) → V (Mat(C))

(I)

ZS ∩ N0 ↓∼ =  → V (Mat(C)) →

(I)

N0 ∼ ↓=

→ →

(Max(C))

N0

.

In this diagram, the vertical arrows represent bijections, and the squares in the middle and on the right are commutative squares of monoids and monoid homomorphisms. (I)

(I)

Proof. Since the embedding ZS ∩ N0 → N0 is a divisor homomorphism, the (I) monoid ZS ∩ N0 is a reduced Krull monoid. AS in Theorem 4.68, for any ring R, let SR denote the full subcategory of Mod-R whose objects are all finitely generated projective right R-modules with a semilocal endomorphism ring. Let k be a field. (I) By Theorem 4.68, there exist a k-algebra R and two isomorphisms g : ZS ∩ N0 →

Chapter 7. Semilocal Categories and Their Maximal Ideals

230 (I)

V (SR ) and h : N0 → V (SR/J(R) ) such that if τ : V (SR ) → V (SR/J(R) ) is the monoid morphism induced by the canonical projection π : R → R/J(R), then the diagram of monoids and monoid homomorphisms (I)

ZS ∩ N0 g ↓ ∼ = V (SR ) commutes. We claim that

(I)

N0 ∼ h↓ = τ −→ V (SR/J(R) ) →

∼ N(Max(SR )) . V (SR/J(R) ) = 0

(I) To prove the claim, notice that V (SR/J(R) ) ∼ = N0 . So it suffices to show that there is a bijection between Max(SR ) and the class of atoms of V (SR/J(R) ). From ∼ V (H) (notation as in that theorem). In Theorem 7.12, we know that Max(SR ) = the paragraph before the statement of Theorem 4.68, we saw that for any ring R, the objects of SR are the projective envelopes of the finitely generated semisimple right R-modules. In particular, for the ring R/J(R), the objects of SR/J(R) are the projective envelopes in Mod-R/J(R) of the finitely generated semisimple right R/J(R)-modules. But over R/J(R) free modules have zero radical, so that projective R/J(R)-modules have no nonzero superfluous submodules; hence an R/J(R)module has a projective envelope if and only if it is projective. Thus the objects of SR/J(R) are the projective finitely generated semisimple right R/J(R)-modules. In particular, the class of atoms of V (SR/J(R) ) consists of a class of representatives of the simple projective R/J(R)-modules. Thus we must prove that a simple R-module M is a homomorphic image of a finitely generated projective R-module with semilocal endomorphism ring if and only if M is a projective R/J(R)-module. Now, if the simple R-module M is a homomorphic image of a finitely generated projective R-module AR with End(AR ) semilocal, then M ⊗R/J(R) ∼ = M/M J(R) is a homomorphic image of AR ⊗ R/J(R) ∼ = AR /AR J(R), and AR /AR J(R) is a direct sum of finitely many simple modules, as we have seen in the paragraph before the statement of Theorem 7.12. Now AR /AR J(R) is a projective R/J(R)-module, and so the R/J(R)-module M is also a projective module. Conversely, let M be a simple projective R/J(R)-module. Then M corresponds to an element x ∈ I via the isomorphism h, so that x ∈ supp(s) for some s ∈ S. The element s corresponds to a projective module AR ∈ V (SR ) via g, and M is a homomorphic image of AR . This shows that M ∈ H, and we have proved the claim. Now let C be the full subcategory of SR whose class of objects is g(S).  is equivalent to SR and there is a monoid isomorphism N0 S → Then (Mat(C)) V (Mat(C)). 

7.5. Comparing Ideals of Endomorphism Rings of Distinct Objects

231

7.5 Comparing Ideals of Endomorphism Rings of Distinct Objects In this final section of the chapter, we compare the ideals of the endomorphism ring of two objects A and B of a preadditive category C. The most natural way of associating to each ideal of EndC (A) an ideal of EndC (B) is to associate to the ideal I of EndC (A) the ideal AI (B, B) of EndC (B), where AI indicates the ideal in the category C associated to I. Similarly, we can associate to each ideal K of EndC (B) the ideal AK (A, A) of EndC (A). We will now compare the ideals of EndC (A) and the ideals of EndC (B), making use of these correspondences. The lattice of all two-sided ideals of the endomorphism ring EndC (A) of an object A will be denoted by L(EndC (A)). Lemma 7.16. Let A and B be nonzero objects of a preadditive category C. The following conditions are equivalent: (a) The mapping α : L(EndC (A)) → L(EndC (B)), I ∈ L(EndC (A)) → AI (B, B) ∈ L(EndC (B)), is a left inverse of the mapping β : L(EndC (B)) → L(EndC (A)), K ∈ L(EndC (B)) → AK (A, A) ∈ L(EndC (A)), where AI and AK are the ideals in C associated to I and K respectively. (b) HomC (A, B) HomC (B, A) = EndC (B). (c) add(B) ⊆ add(A). Moreover, if the category C is additive with splitting idempotents, the previous three conditions are also equivalent to: (d) The object B is isomorphic to a direct summand of An for some positive integer n. Proof. (a) ⇒ (b) Set K := HomC (A, B) HomC (B, A). The correspondence β sends K to β(K) = EndC (A) and α sends EndC (A) to α(EndC (A)) = EndC (B). Thus HomC (A, B) HomC (B, A) = EndC (B) by (a). (b) ⇒ (c) Suppose D ∈ add(B). Then there are morphisms f1 , . . . , fn : B → D

and g1 , . . . , gn : D → B

n : A → B and with 1D = i=1 fi gi . By (b), there exist m morphisms h1 , . . . , hm n

1 , . . . , m : B → A such that 1B = i=1 fi 1B gi = j=1 hj j . Thus 1D = i,j fi hj lj gi , so that D ∈ add(A). (c) ⇒ (d) The composition αβ sends an ideal K ∈ L(EndC (B)) to the ideal αβ(K) = { g ∈ EndC (B) | αδgγβ ∈ K for every α : A → B, γ : A →

Chapter 7. Semilocal Categories and Their Maximal Ideals

232

n B, β : B → A, δ : B → A }. Clearly, K ⊆ αβ(K). Then 1B = i=1 fi hi for suitable fi ∈ HomC (A, B) and hi ∈ HomC (B, A) by (c). Thus g ∈ αβ(K) implies that i,j fi hi gfj hj ∈ K, that is, g ∈ K.  (b) ⇔ (d) is a trivial logical equivalence. We have already mentioned Morita equivalence in the case of rings with enough idempotents in Section 4.4. Let us briefly recall the main concepts about Morita equivalence in the classical case of rings with identity (for the details, see [Anderson and Fuller, Sections 21 and 22]). Two rings (with identity) R and S are said to be Morita equivalent if the two categories Mod-R and Mod-S are equivalent, that is, if there is an additive equivalence of categories Mod-R → Mod-S. It can be proved that this condition is left/right symmetric, that is, the categories Mod-R and Mod-S are equivalent if and only if the categories R -Mod and S -Mod are equivalent. Two Morita equivalent rings have isomorphic centers and isomorphic lattices of two-sided ideals. Two rings R and S are Morita equivalent if and only if there exists a bimodule R PS that is balanced (i.e., every endomorphism of PS is left multiplication by an element of R and every endomorphism of R P is right multiplication by an element of S) and with both R P and PS progenerators. (A module MR is a progenerator if it is a finitely generated projective generator, that is, if MR is a direct summand of a direct sum of finitely many copies of RR , and RR is a direct summand of a direct sum of finitely many copies of MR .) Let R, S be rings and let S PR , R QS be bimodules. If (ϑ, ϕ) is a pair of bimodule homomorphisms ϑ : P ⊗ R Q → S SS

and

ϕ : Q ⊗S P → R RR

ϕ(x ⊗ f )y = xϕ(f ⊗ y)

and

f ϑ(x ⊗ g) = ϕ(f ⊗ x)g

such that

for every x, y ∈ P and every f, g ∈ Q, then (ϑ, ϕ) is called a Morita pair for (P, Q). If S PR , R QS are bimodules and (ϑ, ϕ) is a Morita pair for (P, Q) with ϑ, ϕ epic, then R and S are Morita equivalent [Anderson and Fuller, Exercises 22.5–22.7]. Let us apply this to the study of the endomorphism rings in a preadditive category. Let A, B be nonzero objects of a preadditive category C. We can consider the bimodules EndC (B) PEndC (A)

:= HomC (A, B)

and

EndC (A) QEndC (B)

:= HomC (B, A)

and the bimodule homomorphisms ϑ : P ⊗ Q → EndC (B), defined by ϑ(f ⊗ g) = f g, and ϕ : Q ⊗ P → EndC (A), defined by ϕ(g ⊗ f ) = gf for all f ∈ P and g ∈ Q. One has that ϑ(f ⊗ g)f  = f ϕ(g ⊗ f  ) and gϑ(f ⊗ g  ) = ϕ(g ⊗ f )g  for allf, f  ∈ P, g, g  ∈ Q, and so the couple (θ, φ) defines a Morita pair for (P, Q). Moreover, when θ and φ are both epic, the rings EndC (A) and EndC (B) turn out to be Morita equivalent.

233

7.6. Notes on Chapter 7

Theorem 7.17. Let A, B be nonzero objects of a preadditive category C. The following conditions are equivalent: (a) The mappings α : L(EndC (A)) → L(EndC (B)), I ∈ L(EndC (A)) → AI (B, B) ∈ L(EndC (B)), and β : L(EndC (B)) → L(EndC (A)), K ∈ L(EndC (B)) → AK (A, A) ∈ L(EndC (A)), are mutually inverse bijections. (b)

HomC (A, B) HomC (B, A) = EndC (B) and

HomC (B, A) HomC (A, B) = EndC (A). (c) add(A) = add(B). (d) In the Morita pair (θ, φ) for the bimodules EndC (B) PEndC (A)

:= HomC (A, B) and

EndC (A) QEndC (B)

:= HomC (B, A),

both mappings θ and φ are epic. Moreover, if the category C is additive and has splitting idempotents, the previous four conditions (a)–(d) are also equivalent to: (e) A is isomorphic to a direct summand of B n and B is isomorphic to a direct summand of Am for suitable nonnegative integers n and m. Proof. The equivalence of (a), (b), (c), and (e) follows from Lemma 7.16. The equivalence of (b) and (d) is trivial.  Notice that the condition (d) is strictly stronger than the condition “The rings EndC (A) and EndC (B) are Morita equivalent.” For instance, it is possible to construct examples of abelian groups G that are not free, but whose endomorphism rings are isomorphic to Z. The simplest of such examples is probably the subgroup G of Q generated by all p−1 , where p ranges over the set of all prime numbers. The group G contains Z as a subgroup. Since G is torsion-free of rank 1, its endomorphism ring End(GZ ) is isomorphic to the subring of Q whose elements are all rational numbers q with qG  ⊆ G. Now qG ⊆ G if and only if qp−1 ∈ G for every prime p, if and only  if q ∈ p pG }. Since G/Z is the socle of the torsion group Q/Z, we have that p pG ⊆ Z. Thus End(GZ ) ∼ = Z. In particular, End(GZ ) and Z are Morita equivalent, but clearly add(GZ ) = add(ZZ ).

7.6 Notes on Chapter 7 Most of the content of this chapter is taken from [Facchini and Perone 2011]. Definition 4.61 was suggested to us by P. Pˇr´ıhoda.

Chapter 8

Modules of Type ≤ 2. Uniserial Modules 8.1 Rings and Modules of Type n A completely prime ideal P of a ring S is a proper ideal P of S with the property that for every x, y ∈ S, xy ∈ P implies that either x ∈ P or y ∈ P . Notice that if ϕ : S → D is a ring morphism of S into an integral domain D (e.g., a division ring D), then the kernel of ϕ is a two-sided completely prime ideal of S. Lemma 8.1. Let S be a ring, P1 , P2 , . . . , Pn completely prime two-sided ideals of S, n and L a right ideal of S contained in i=1 Pi . Then L ⊆ Pi for some i = 1, 2, . . . , n. Proof. The statement

n is trivial for n = 1. Assume n ≥ 2. Let L be a right ideal of S contained in i=1 Pi , but not contained in Pi for every i = 1, 2, . . . , n. Let xi be an element of L not in Pi . Then x1 · · · xi xi+1 · · · xn is in Pj for every j = i, n n but not in Pi . Thus x := i=1 x1 · · · xi xi+1 · · · xn ∈ / i=1 Pi . But x ∈ L because n ≥ 2. This is a contradiction.  Let n ≥ 1 be an integer. We say that a ring S has type n if the ring S/J(S) is a direct product of n division rings, and we say that S is a ring of finite type if it has type n for some integer n ≥ 1. Thus the rings of finite type form a subclass of the class of semilocal rings, because if a ring S has finite type, then the ring S/J(S) is a direct product of n division rings, and hence S/J(S) is semisimple Artinian. More precisely, if S is of finite type n, then S is a semilocal ring of dual Goldie dimension n, because S/J(S)S is the direct sum of n simple S-modules, hence a semisimple right module of Goldie dimension n, so codim(SS ) = n (Proposition 3.12). We leave to the reader to show that a commutative ring has finite type if and only if it is semilocal. Clearly, a ring S has type 1 if and only if S/J(S) is a division ring, that is, if and only if S is a local ring. Our first proposition generalizes this fact. © Springer Nature Switzerland AG 2019 A. Facchini, Semilocal Categories and Modules with Semilocal Endomorphism Rings, Progress in Mathematics 331, https://doi.org/10.1007/978-3-030-23284-9_8

235

236

Chapter 8. Modules of Type ≤ 2. Uniserial Modules

Proposition 8.2. Let S be a ring and n ≥ 1 an integer. The following conditions are equivalent: (a) The ring S has type n. (b) n is the smallest of the positive integers m for which there is a local morphism of S into a direct product of m division rings. (c) The ring S has exactly n distinct maximal right ideals, and they are all twosided ideals in S. (d) The ring S has exactly n distinct maximal left ideals, and they are all twosided ideals in S. Proof. (a) ⇒ (b) If S has type n, then S/J(S) is the direct product of n division rings, so that the canonical projection S → S/J(S) is a local morphism into the direct product of n division rings (Example 3.21(1)). If ϕ : S → D1 × · · · × Dm is a local morphism of S into the direct product of m division rings Di , then n = codim(S) ≤ codim(D1 × · · · × Dm ) = m (Theorem 3.27). (b) ⇒ (c) Assume that (b) holds, so that there exists a local morphism ϕ : S → D1 × · · · × Dn with D1 , . . . , Dn division rings. Let πi be the canonical projection of D1 × · · · × Dn onto Di and let Pi be the kernel of the composite morphism

n πi ϕ, so that Pi is a completely prime two-sided ideal of S. Then U (S) = S \ ( i=1 Pi ), because ϕ is a local morphism.

n It follows that all proper right ideals of S are contained in i=1 Pi , hence in one of the ideals Pi by Lemma 8.1. Thus the unique maximal right ideals of S are at most the n ideals P1 , . . . , Pn , and they are all two-sided ideals. Suppose that the Pi ’s are not all distinct, or that one of them is not a maximal right ideal. In both cases, there exist two distinct indices i, j = 1, . . . , n with Pi ⊆ Pj . It is easily seen that π1 ϕ × · · · × πi−1 ϕ × πi+1 ϕ · · · × πn ϕ : S → D1 × · · · × Di−1 × Di+1 × · · · × Dn is a local morphism as well, and this contradicts the minimality of n. (c) ⇒ (a) Assume that Q1 , . . . , Qn are all the maximal right ideals of S, that they are all distinct, and that they are all two-sided ideals. Then n they are pairwise comaximal ideals, so that the canonical projection π : S → i=1 S/Qi is a surjective mapping by the Chinese remainder theorem. If the idealsQi are all n the maximal ideals of S, it follows that ker π = J(S). Thus S/J(S) ∼ = i=1 S/Qi , and the rings S/Qi do not have nontrivial right ideals. Hence the rings S/Qi are division rings. (a) ⇔ (d) If S has type n, its opposite ring S op has type n, that is, “having type n” is a left/right symmetric condition. Thus it suffices to apply the equiva lence (a) ⇔ (c) to the ring S op . Recall that a ring R is called right duo if every right ideal of R is two-sided. A ring R is right quasi-duo if every maximal right ideal of R is two-sided. Similarly for left duo rings and left quasi-duo rings. The following is an immediate corollary of Proposition 8.2.

8.1. Rings and Modules of Type n

237

Corollary 8.3. The following conditions are equivalent: (a) The ring S has finite type. (b) The ring S is right quasi-duo and semilocal. (c) The ring S is left quasi-duo and semilocal. Thus, in a ring of finite type, the set of all maximal right ideals (= maximal among all proper right ideals), the set of all maximal left ideals (= maximal among all proper left ideals), and the set of all maximal two-sided ideals (= maximal among all proper two-sided ideals) coincide. Therefore we will talk simply of maximal ideals, without mentioning the side. As we have already remarked, if S is a ring of finite type, the canonical projection S → S/J(S) is a local morphism onto a direct product of finitely many division rings, and every ring morphism of an arbitrary ring S into a direct product D1 × · · · × Dm of division rings D1 , . . . , Dm is of the form ϕ1 × · · · × ϕm : S → D1 × · · · × Dm , where the ϕi ’s are morphisms of S into Di . Our next result is a corollary of Proposition 8.2. Corollary 8.4. Let ϕ1 × · · · × ϕm : S → D1 × · · · × Dm be a local morphism of a D1 , . . . , Dm . Then S ring S into a direct product of finitely many division rings m is a ring of finite type n ≤ m, S is a directly finite ring, i=1 ker ϕi is the set of all noninvertible elements of S, and the maximal ideals of S belong to the set { ker ϕi | i = 1, 2, . . . , m }. In particular, if the type of S is m, then  { ker ϕi | i = 1, 2, . . . , m } is the set of all the m maximal ideals of S, J(S) = m i=1 ker ϕi , and ∼ m S/ ker ϕi . S/J(S) = i=1 Proof. The ring S has type n ≤ m by Proposition 8.2. In particular, S is semilocal, hence directly finite (Corollary 3.32), so that the set of all invertible elements coincides with the set of all right invertible elements and with the set of all left invertible elements. Since ϕ1 × · · · × ϕm is a local morphism, an element s ∈ S is invertible in S if and only if (ϕ1 ×· · ·×ϕm )(s) is invertible in D1 ×· · ·×Dm , that is, if and only if ϕi (s) = 0 for every i, that is, if and only if s ∈ / ker ϕi for every i, i.e., if . Let ϕ ker Q . . , Q , . distinct maximal be all the / m and only if s ∈ i 1 n i=1

n right ideals of S, so that they are all two-sided ideals by Proposition 8.2. Then j=1 Qj is the

n

m set of all elements of R that are not right invertible. Thus j=1 Qj = i=1 ker ϕi .

m ker ϕi , it follows that for every j = 1, . . . , n there exists i = From Qj ⊆ i=1 1, . . . , m with Qj ⊆ ker ϕi (Lemma 8.1). But Qj is maximal and ker ϕi is a proper ideal, so that Qj = ker ϕi . Now the case n = m follows immediately (Chinese  remainder theorem). m m Corollary 8.5. If i=1 ϕi : S → i=1 Di is a local morphism of a ring S of type n ≤ m into a direct product of finitely many division rings D1 , . . . , Dm , then there X of {1, 2, . . . , m} of cardinality n such that the ring morphism exists a subset   : S → ϕ i i∈X i∈X Di is a local morphism.

238

Chapter 8. Modules of Type ≤ 2. Uniserial Modules

Proof. Let M1 , . . . , Mn be the maximal ideals of S, so that $m $n Mi = ker ϕj i=1

j=1

is the set of all noninvertible elements of S (Corollary 8.4). By Lemma 8.1, for every i = 1, . . . , n there exists j(i) = 1, . . . , m such that Mi ⊆ ker ϕj(i) . Since Mi is maximal, it follows that Mi = ker ϕj(i) . Thus 1, . . . , n } is the set  { ker ϕj(i) | i = of all maximal ideals of S. The morphism ni=1 ϕj(i) : S → j(i) Dj(i) is a local  n  is invertible in j(i) Dj(i) , then morphism, because if s ∈ S and i=1 ϕj(i) (s)

n

n ϕj(i) (s) = 0 for every i = 1, . . . , n, so that s ∈ / i=1 ker ϕj(i) = i=1 Mi ; hence s  is invertible in S. If a ring S has exactly one maximal right ideal M , then M = J(S), so that M is necessarily two-sided. A similar property holds when there are two maximal right ideals, as the next proposition shows. Proposition 8.6. If a ring S has exactly two maximal right ideals M1 , M2 , then M1 and M2 are two-sided ideals. Proof. Let S be a ring with exactly two maximal right ideals M1 , M2 . There is a canonical right S-module embedding S/J(S) → S/M1 ⊕ S/M2 , so that the right S-module S/J(S) is Artinian, that is, S/J(S) is a right Artinian ring. By Proposition 3.12, S is semilocal, hence a directly finite ring (Corollary 3.32). Thus M1 ∪ M2 is exactly the set of all noninvertible elements of S. Therefore every proper left ideal L of S is contained in M1 ∪ M2 . If there exist elements x ∈ L \ M1 / M2 , so that x + y ∈ / M1 and x + y ∈ / M1 ∪ M2 . This and y ∈ L \ M2 , then x + y ∈ is a contradiction, because x + y ∈ L. It follows that either L ⊆ M1 or L ⊆ M2 . Let x be an element of M1 , and consider the left ideal Sx. If x ∈ M1 ∩ M2 = J(S), then Sx ⊆ J(S) ⊆ M1 . If x ∈ M1 but x ∈ / M2 , then, as we have just seen, either / M2 , we cannot have Sx ⊆ M2 , so that Sx ⊆ M1 . Sx ⊆ M1 or Sx ⊆ M2 . Since x ∈ This proves that M1 is a two-sided ideal. Similarly for M2 .  Thus if a ring has at most two maximal right ideals, then all maximal right ideals are two-sided, but there are rings with exactly three maximal right ideals that are not two-sided. We leave to the reader to show that the ring R of all 2 × 2 matrices with entries in the field with 2 elements has exactly three maximal right ideals, which are not two-sided because R is a simple ring. Here is another example of a ring of type n. If R is a local ring with maximal ideal I, then the Jacobson radical of the ring S of all n×n upper triangular matrices with entries in R consists of all elements of S whose entries on the diagonal are all in I. It follows that S/J(S) is isomorphic to the direct product of n copies of the division ring R/I. Hence S is a ring of type n. We say that a right module MR over a ring R has type nif its endomorphism ring End(MR ) is a ring of type n. We will consider the zero module to be the unique module of type 0. We will say that a module MR has finite type if it has

8.2. The Canonical Functor A → A/I1 × · · · × A/In

239

type n for some n ≥ 0. In the next sections of this chapter, we will give several examples of modules of type ≤ 2. Modules of type 1 are exactly the modules with a local endomorphism ring. We leave to the reader to check that an injective module has type n if and only if it is the direct sum of n pairwise nonisomorphic indecomposable modules. The modules considered in the Krull–Schmidt–Azumaya theorem are direct sums of modules with local endomorphism ring, that is, direct sums of modules of type 1. Proposition 8.7. Let M and N be R-modules with M isomorphic to a direct summand of N k for some integer k ≥ 1 and M of finite type. Then M is isomorphic to a direct summand of N . Proof. Let M be of type n and let P1 , . . . , Pn be the maximal ideals of EndR (M ). Since M is isomorphic to a direct summand of N k , there exist f : M → N k and k g : N k → M such that gf = 1M . Thus 1M = i=1 gιi πi f , where ιi : N → N k is the embedding in the ith component and πi : N k → N is the canonical projection on the ith factor. Since EndR (M )/J(EndR (M )) is canonically isomorphic to  n j=1 EndR (M )/Pj , for every j = 1, . . . , n there exists an endomorphism qj of M  k such that qj ∈ / Pj and qj ∈ l=j Pl . Then i=1 (qj gιi )(πi f qj ) = qj qj ∈ / Pj . Thus, for every j = 1, . . . , n, there exists an index i(j) with (qj gιi(j) )(πi(j) f qj ) ∈ / Pj . that fj : M → N and gj : N → M have Set fj := πi(j) f qj and gj := qj gιi(j) , so  the property that gj fj ∈ / Pj and gj fj ∈ l=j Pl . Now define the homomorphisms f  : M → N and g  : N → M by n n f  := fj qj and g  := qj gj . j=1

j=1

 

Clearly, g f is not contained in any of the n maximal ideals of EndR (M ); hence g  f  is invertible. Then ((g  f  )−1 g  )f  = 1M , so that M is isomorphic to a direct  summand of N .

8.2 The Canonical Functor A → A/I1 × · · · × A/In We will later consider full subcategories A of Mod-R in which the endomorphism rings of all nonzero objects are rings of type n. This leads us to study when, for a preadditive category A and ideals I1 , . . . , In of A, the canonical functor A → A/I1 × · · · × A/In is local. In order to study the canonical functor A → A/I1 × · · · × A/In , it is convenient to introduce some noncommutative polynomials pn = pn (x, y1 , . . . , yn ) with coefficients in the ring Z of integers. More generally, let x, y1 , y2 , y3 , . . . be infinitely many noncommuting indeterminates over the ring Z. There is a strictly ascending chain Zx, y1  ⊂ Zx, y1 , y2  ⊂ Zx, y1 , y2 , y3  ⊂ · · ·

240

Chapter 8. Modules of Type ≤ 2. Uniserial Modules

of noncommutative integral domains, where Zx, y1 , . . . , yn  indicates the ring of polynomials in the noncommuting indeterminates x, y1 , . . . , yn with coefficients in Z. Proposition 8.8. Let Z be the ring of integers and x, y1 , y2 , y3 , . . . noncommuting indeterminates over Z. Let Zx, y1 , . . . , yn  be the ring of noncommutative polynomials in the indeterminates x, y1 , . . . , yn with coefficients in Z for every n ≥ 1. Then there exists, for each n ≥ 1, a unique polynomial pn = pn (x, y1 , . . . , yn ) ∈ Zx, y1 , . . . , yn  such that 1 − pn x = (1 − y1 x)(1 − y2 x) · · · (1 − yn x).

(8.1)

Moreover, the polynomials pn , n ≥ 1, have the following properties: (a) 1 − xpn = (1 − xy1 )(1 − xy2 ) · · · (1 − xyn ) for every n ≥ 1. (b) p1 = y1 , and pn+1 = yn+1 + pn (1 − xyn+1 ) for every n ≥ 1. (c) For every n ≥ 1, pn = 1≤i≤n yi − 1≤i1 c. Here is a second example. Example 10.15. Let G = (V, E) be a graph that consists of exactly two cycles c1 = 1 , . . . , n and c2 = e1 , . . . , em , of lengths n ≥ 3 and m ≥ 3 respectively, which intersect each other in a unique vertex. Thus the two cycles are disjoint, but not completely disjoint. We will show that the weak Krull–Schmidt theorem holds for G. Without loss of generality, we can assume that the unique vertex in which the two cycles intersect is a vertex in common of 1 , n , e1 , em . If n and m are both even, then the graph G is bipartite, so that the weak Krull–Schmidt theorem holds for G by Corollary 10.12. (Notice that in this case, the cycles of G are all of the form c1n1 c2m1 c1n2 c2m2 . . . c1nt c2mt , where n1 , n2 , . . . , nt , m1 , m2 , . . . , mt are integers, and all these cycles have even length. Every relation in V (G) is a sum of trivial relations as in Proposition 10.7(a) and the two relations δ1 + δ3 + · · · + δn−1 = δ2 + δ4 + · · · + δn and δe1 + δe3 + · · · + δem−1 = δe2 + δe4 + · · · + δem .) Assume that n and m are both odd. We will show that the weak Krull– Schmidt theorem holds for G using Proposition 10.9. Let G be the graph that consists of a unique cycle c = 1 , . . . , n , e1 , . . . , em of even length n+m. This is a bipar˙ ), where both tite graph; hence it is contained in a complete bipartite graph B(X ∪Y n+m X and Y have cardinality 2 . Consider the injective mapping ϕ : E → EX×Y defined by ϕ( i ) = i and ϕ(ej ) = ej . Let us show that (i) and (ii) of Proposint mt 1 n2 m2 tion 10.9(c) are satisfied. The cycles of G are of the form cn1 1 cm 2 c1 c2 . . . c1 c2 with n1 , n2 , . . . , nt , m1 , m2 , . . . , mt integers. The cycles of even length are those with n1 + · · · + nt + m1 + · · · + mt even. Hence the cycles of even length of G are concatenations of the cycles c21 , c22 , c1 c2 , c−1 1 c2 and their inverses. In order to prove that condition (i) of Proposition 10.9(c) holds, we can consider only the cycles in which there is no edge that appears both in an even position and in an odd position. They are the cycles that are concatenations of the cycles c1 c2 , c−1 1 c2 and their inverses. Consider the cycle c1 c2 = 1 , . . . , n , e1 , . . . , em . Applying ϕ, we get the cycle ˙ ). For the other cycle, we have c−1

1 , . . . , n , e1 , . . . , em in the graph B(X ∪Y 1 c2 =

340

Chapter 10. The Krull–Schmidt Theorem in the Case Two

n , . . . , 1 , e1 , . . . , em . Applying ϕ, we get the sequence of edges 

n , . . . , 1 , e1 , . . . , em .

The edges in odd position are n , n−2 , . . . , 1 , e2 , e4 , . . . , em−1 , and they can be   permuted and become 1 , . . . , n−2 . The edges in even posi, n , e2 , e4 , . . . , em−1       tion are n−1 , n−3 , . . . , 2 , e1 , e3 , . . . , em , and they can be permuted and become  , n−1 , e1 , e3 , . . . , em . Inserting the edges in even position in those in

2 , . . . , n−3  ˙ ). This odd position, we get the cycle 1 , . . . , n , e1 , . . . , em in the graph B(X ∪Y shows that condition (i) holds. ˙ ) constructed with edges For (ii), notice that the cycles in the graph B(X ∪Y in ϕ(E) are the cycles cn with n a positive integer and their inverses. Their inverse images via ϕ are the cycles c1 c2 c1 c2 . . . c1 c2 (n factors c1 c2 ) and their inverses, which are cycles of even length in the graph G. Hence (ii) holds as well. Now assume n even and m odd (the case n odd and m even is similar). The cycles of G are of the form cn1 1 c2m1 cn1 2 c2m2 . . . c1nt c2mt , where n1 , n2 , . . . , nt , m1 , m2 , . . . , mt are integers. The cycles of even length are those with m1 +m2 +· · ·+mt even. Thus every cycle of even length is a concatenation of the cycle c1 , the cycles c2 ct1 c2 (with t any integer), c2 ct1 c−1 2 , and their inverses. We will prove that the weak Krull–Schmidt theorem holds for G using Proposition 10.9. Let G be the graph that consists of exactly two cycles c1 = 1 , . . . , n and c2 = e1 , . . . , e2m , of lengths n and 2m respectively, which intersect each other in a unique vertex, which is the  . This is a bipartite graph; hence it is contained vertex in common of 1 , n , e1 , e2m ˙ ), where X has cardinality n2 + m and Y in a complete bipartite graph B(X ∪Y has cardinality n2 + m − 1. Let ϕ : E → EX×Y be the injective mapping defined by ϕ( i ) = i and ϕ(ej ) = ej . We will now show that (i) and (ii) of Proposint mt 1 n2 m2 tion 10.9(c) are satisfied. The cycles of G are of the form cn1 1 cm 2 c1 c2 . . . c1 c2 , where n1 , n2 , . . . , nt , m1 , m2 , . . . , mt are integers. The cycles of even length are those for which m1 + · · · + mt is even. Hence the cycles of even length of G are concatenations of the cycles c1 , c22 , c2 ct1 c2 (with t an integer), c2 ct1 c−1 2 , and their inverses. In order to prove condition (i) of Proposition 10.9(c), we can consider only the cycles in which there is no edge that appears both in an even position and in an odd position. They are the cycles that are concatenations of the cycle  c1 and its inverse c−1 1 . Applying ϕ to the edges of c1 , we get a cycle of G . Thus condition (i) holds. As far as (ii) is concerned, note that the cycles in the graph ˙ ) constructed with edges in ϕ(E) are the cycles (c1 )n with n an integer. B(X ∪Y Their inverse images via ϕ are the cycles cn1 , which are cycles of even length in G. This proves that (ii) holds, and we can conclude by Proposition 10.9. Example 10.16. Let B be a graph with at most one cycle of odd length (except for repetitions of the cycle an odd number of times, in either direction) in each connected component. Let us prove that the weak Krull–Schmidt theorem holds for G. To this end, we can assume G := (V, E) connected by Corollary 10.10. By Corollary 10.12, we can suppose that the connected graph G has exactly one cycle 1 , . . . , n of odd length n ≥ 3. Notice that the edge 1 is not on any

10.5. Another Realization Theorem

341

cycle of even length (except for the repetitions of the cycle 1 , . . . , n an even number of times), because if 1 , e1 . . . , em were any cycle of even length m + 1, then e1 . . . , em , n , n−1 , . . . , 2 would be another cycle of odd length m + n − 1, which would contradict our hypothesis. Thus the graph G and the graph G = (V, E \ { 1 }) have the same cycles of even length, so that V (G) ∼ = V (G ) ⊕ N0 . Now it suffices to apply Proposition 10.9 and Corollary 10.12.

10.5 Another Realization Theorem In this section, for any ring R, the class of all finitely generated projective right R-modules PR with End(PR ) semilocal will be denoted by SR . We introduced this notation in Section 4.15, before the statement of Theorem 4.68. Let us begin with a remark relating that theorem with the notion of module of finite type. Remark 10.17. (a) Notice that the ring R in Theorem 4.68 is not a semilocal ring in general. Thus the factor ring R/J(R) is not necessarily a semisimple Artinian ring. Nevertheless, the indecomposable modules in SR/J(R) , that is, the finitely generated indecomposable projective R/J(R)-modules with a semilocal endomorphism ring, are simple modules. To prove this, fix an indecomposable finitely generated projective R/J(R)-module P with a semilocal endomorphism ring. From Corollary 2.25, we know that J(EndR/J(R) (P )) = HomR/J(R) (P, rad(P )) = 0. Therefore EndR/J(R) (P ) is a semisimple ring. But P is indecomposable, so that EndR/J(R) (P ) has only the trivial idempotents. It follows that EndR/J(R) (P ) must be a division ring; hence in particular, it is a local ring. By Lemma 3.9, the R/J(R)module P is the projective cover of a simple module. But rad(P ) = 0, so that the module P itself is a simple module. (b) What we have seen in (a) can be partially generalized. In the notation (I) of Theorem 4.68, let P ∈ Ob(SR ) be such that h−1 τ (P ) is an element of N0 whose coordinates are n ones and all the other coordinates are zero. Then the Rmodule P is of type n. To prove this, notice that by Corollary 2.25, the canonical homomorphism EndR (P ) → EndR/J(R) (P/P J(R)) is surjective and has kernel J(EndR (P )). From the commutativity of diagram (10.2), we have that P/P J(R) is the direct sum of n nonisomorphic finitely generated indecomposable projective R/J(R)-modules with semilocal endomorphism rings. By (a), P/P J(R) = P1 ⊕ · · · ⊕ Pn , where P1 , . . . , Pn are pairwise nonisomorphic simple modules. The ring Di := EndR (Pi ) is a division ring (i = 1, . . . , n), and EndR (P )/J(EndR (P )) ∼ = EndR/J(R) (P/P J(R)) ∼ = D1 × · · · × Dn , so that P is a module of type n. In the next theorem, we consider only “small” graphs, that is, graphs having sets of vertices and edges (not classes). Theorem 10.18. If k is a field and G is a graph, then there are a k-algebra R and a set C of finitely generated indecomposable projective right R-modules of type 2 ∼ G. such that G(C) =

342

Chapter 10. The Krull–Schmidt Theorem in the Case Two (V )

Proof. Let k be a field, G = (V, E) a graph, and N0 the free commutative (V ) monoid with free set of generators the set V of vertices of G. The elements of N0 will be denoted in the form (nv )v∈V , where the nonnegative integers nv are zero (V ) for almost all v ∈ V . Let M denote the Krull submonoid of N0 consisting of (V ) (V ) all the elements (nv )v∈V ∈ N0 such that v∈V nv is even. Let T : M → N0 denote the inclusion. By Theorem 4.68, there are a k-algebra R and two monoid (V ) isomorphisms M → V (SR ) and h : N0 → V (SR/J(R) ) such that if τ : V (SR ) → V (SR/J(R) ) denotes the monoid morphism induced by the canonical projection π : R → R/J(R), then the diagram T

N0 (V ) −→ M ∼ ∼ ↓= h↓ = τ V (SR ) −→ V (SR/J(R) )

(10.2)

commutes. Let { δv | v ∈ V } be the free set of generators of the commutative (V ) monoid N0 . The atoms of M are the elements 2δv (v ∈ V ) and the elements δv + δw (v, w ∈ V, v = w). Let C denote a set of representatives of the finitely generated indecomposable projective right R-modules P{v,w} ∈ SR with {v, w} ∈ E. The modules P = P{v,w} with v = w are modules of type 2 by Remark 10.17(b). The commutativity of diagram (10.2) implies that P/P J(R) is the direct sum of two nonisomorphic finitely generated indecomposable projective R/J(R)-modules with a semilocal endomorphism ring. By Remark 10.17(a), P/P J(R) = P1 ⊕ P2 , where P1 , P2 are pairwise nonisomorphic simple modules. Thus End(Pi ) is a division ring Di , and from the canonical local morphism ∼ D1 × D2 , End(P ) → End(P/P J(R)) = End(P1 ⊕ P2 ) = we get two canonical ring homomorphisms πi : End(P ) → Di (i = 1, 2). By Corollary 8.4, ker(π1 ) and ker(π1 ) are the two maximal right ideals of End(P ). To conclude, it suffices to show that there is a one-to-one correspondence between the ideals in the category C associated to a maximal ideal of the endomorphism ring of a module in C and the set V of vertices of the graph G. That is, if PR , QR ∈ SR , P/P J(R) = P1 ⊕ P2 , Q/QJ(R) = Q1 ⊕ Q2 , and π1 : End(PR ) → End(P1 ), π1 : End(QR ) → End(Q1 ) are the two canonical onto ring morphisms, then the ideals in the category C associated to ker(π1 ) and ker(π1 ) are equal if and only if P1 ∼ = Q1 . By Lemma 9.3, the ideals associated  to the kernels ker(π1 ) and ker(π1 ) coincide if and only if there are two morphisms ϕ : PR → QR , ψ : QR → PR such that / ker(π1 ), ψ ker(π1 )ϕ ⊆ ker(π1 ), ϕ ker(π1 )ψ ⊆ ker(π1 ). (10.3) ψϕ ∈ / ker(π1 ), ϕψ ∈

10.6. Complete Graphs

343

If such morphisms ϕ and ψ exist, they induce morphisms ϕ : P/P J(R) = P1 ⊕ P2 → Q/QJ(R) = Q1 ⊕ Q2 and

ψ : Q/QJ(R) = Q1 ⊕ Q2 → P/P J(R) = P1 ⊕ P2 .

Writing them in matrix form as 2 × 2 matrices, we can write     ϕ11 ϕ12 ψ 11 ψ 12 ϕ= ψ= , . ϕ21 ϕ22 ψ 21 ψ 22 Notice that EndR/J(R) (P/P J(R)) ∼ = D1 × D2 implies that the 2 × 2 matrices representing endomorphisms of P/P J(R) and Q/QJ(R) are diagonal. Moreover, an endomorphism f of PR is in ker(π1 ) if and only if the matrix of f has the   0 0 form . Since the mapping (π1 × π2 ) : End(PR ) → End(P1 ) × End(P2 ) ∼ = 0 ∗   0 0 D1 × D2 is onto, there exists an endomorphism f of PR with f = . Thus 0 1 ψϕ ∈ / ker(π1 ) is equivalent to ψ 11 ϕ11 + ψ 12 ϕ21 = 0. Also, ψ ker(π1 )ϕ ⊆ ker(π1 ) implies that       ψ 11 ψ 12 ϕ11 ϕ12 0 0 0 0 = , ϕ21 ϕ22 0 1 0 ∗ ψ 21 ψ 22 so that ψ 12 ϕ21 = 0. Thus ψ 11 ϕ11 is a nonzero element in the division ring End(P1 ), that is, it is an automorphism of P1 . Thus ϕ11 : P1 → Q1 is a splitting monomorphism. But the modules P1 , Q1 are indecomposable projective, so that ϕ11 is an isomorphism of P1 onto Q1 . Conversely, suppose P1 isomorphic to Q1 . Since PR and QR are projective modules, there exist two homomorphisms ϕ : PR → QR and ψ : QR → PR with     ϕ11 0 ψ 11 0 ϕ= ψ= , 0 0 0 0 and such that ϕ11 , ψ 11 are mutually inverse isomorphisms. It is now easy to check that conditions (10.3) hold for these two morphisms ϕ and ψ. 

10.6 Complete Graphs If α is a nonempty class, the complete graph on α is the (large) graph in which α is the class of vertices, and any two vertices in α are adjacent. It will be denoted ˙ by Kα . If α and β are nonempty classes, the complete bipartite graph B(α∪β) will be denoted by Kα,β .

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Chapter 10. The Krull–Schmidt Theorem in the Case Two

Proposition 10.19. If a graph G contains a subgraph isomorphic to the complete graph K4 , then the weak Krull–Schmidt theorem does not hold for the graph G. Proof. It suffices to show that the weak Krull–Schmidt theorem does not hold for the complete graph G = K4 (Corollary 10.11). Suppose the contrary, that is, that the weak Krull–Schmidt theorem holds for the complete graph K4 . Now, K4 has six edges 1 , 2 , . . . , 6 , suitably labeled, such that δ1 +δ2 = δ3 +δ4 = δ5 +δ6 in ˙ ) with an V (K4 ). By Proposition 10.9, there is a complete bipartite graph B(X ∪Y ˙ injective monoid homomorphism ψ : V (K4 ) → V (B(X ∪Y )) that sends atoms to ˙ ) such that δ = ψ(δi ). Then the edges atoms. Let i denote the edge of B(X ∪Y i    ˙ ) such that δ + δ = δ + δ =

1 , 2 , . . . , 6 are six distinct edges of B(X ∪Y 2 1 4 3 δ5 + δ6 . It follows that any vertex of 1 is incident both to 3 or 4 and to 5 or ˙ )

6 . Thus any vertex of 1 has degree at least three in the subgraph G of B(X ∪Y  with the six edges i , i = 1, . . . , 6. Since the sum of the degrees of the vertices in a graph is equal to double the number of edges, that is, is equal to 12 in this case, it ˙ ) has four vertices of degree three. Hence follows that the subgraph G of B(X ∪Y  G is isomorphic to K4 . In particular, K4 is a bipartite graph, which is not the case.  We have seen in Theorem 10.18 that if k is a field and G is a graph having a set of vertices (not a class), then there exist a k-algebra R and a set C of finitely generated indecomposable projective right R-modules of type 2 such that the graph G(C) of Section 10.2 is isomorphic to G. Applying this result to the graph K4 , we obtain that there exist a ring R and a set C of finitely generated indecomposable projective right R-modules of type 2 with G(C) ∼ = K4 . By Proposition 10.19, we get that the weak Krull–Schmidt theorem does not hold for the class T2 of all indecomposable right R-modules of type 2. Hence a “global” weak Krull–Schmidt theorem for the category T2 of all indecomposable modules of type 2 does not hold in general.

10.7 Condition (DSP) Let R be a ring. A class C of right R-modules is said to satisfy condition (DSP) if ∼ C ⊕ D implies D ∈ C [Bican, Definition 1.13]. for every A, B, C ∈ C, A ⊕ B = Clearly, every class C of right R-modules has a (DSP)-closure, that is, there exists a smallest class C  of right R-modules that satisfies condition (DSP), contains C, and is closed under isomorphism. To this end, define a sequence of classes by  ∼ := Cn ∪ { D | there exist A, B, C ∈ Cn with setting C0 := C and Cn+1

A⊕B = C ⊕ D } for every n ≥ 0. Then the class of right R-modules C  := n≥0 Cn turns out to be the (DSP)-closure of C. Recall that for every right module MR , δ(MR ) denotes the dual Goldie dimension codim(End(MR )) of the endomorphism ring of MR (Section 3.5).

10.7. Condition (DSP)

345

Lemma 10.20. If C is a class of indecomposable modules of type ≤ 2, then the (DSP)-closure C  of C is a class of indecomposable modules MR with δ(MR ) ≤ 2. Proof. We will show that every module MR ∈ Cn is such that δ(MR ) ≤ 2 by  induction on n. If A, B, C ∈ Cn−1 , then δ(A), δ(B), and δ(C) are ≤ 2 by the ∼ C ⊕ D implies δ(D) < δ(C) + δ(D) because inductive hypothesis. Thus A ⊕ B = the indecomposable module C is nonzero, so that δ(D) < δ(C ⊕ D) = δ(A ⊕ C ≤ 4 (Proposition 3.38). Thus δ(D) ≤ 3. If δ(D) = 3, then δ(A) = 2, δ(B) = 2, and ∼C⊕D δ(C) = 1; hence C has a local endomorphism ring, and therefore A ⊕ B = implies C ∼ = A or C ∼ = B, which contradicts the fact that δ(A) = 2, δ(B) = 2 and δ(C) = 1. Hence δ(D) ≤ 2, which concludes the proof by induction. Finally, if D ∈ Cn is not indecomposable, then δ(D) ≤ 2 implies that D is a direct sum of two modules with a local endomorphism ring; hence they are  isomorphic to either A or B, and we get a contradiction again. With a similar proof, it is possible to show that if C is a class of indecomposable modules of type 2, then its (DSP)-closure C  is a class of indecomposable modules MR with δ(MR ) = 2. Notice that if C is a class of right R-modules with semilocal endomorphism rings, A, B, C ∈ C, and A ⊕ B ∼ = C ⊕ D, then D is uniquely determined up to isomorphism by Proposition 3.36(a). Moreover, in the monoid corresponding to the graph G(C), we have that δV (A) + δV (B) = δV (C) + δV (D) . The following lemma is a continuation of Lemma 10.20. Lemma 10.21. If C is a class of indecomposable modules of type 2, G(C) is its graph, C  is the (DSP)-closure of C, and A ⊕ B ∼ = C ⊕ D with A, B, C ∈ C and D ∈ C  , then one of the following conditions holds: (a) V (D) is already an edge of the graph G(C). (b) D is an indecomposable module that is not of type 2. (c) D is an indecomposable module of type 2 and V (A), V (C), V (B), V (D) is a cycle of length 4 in G(C  ). Proof. There are three possibilities for the edges V (A) and V (B) of G(C). First possibility: V (A) = V (B). In this case, let CA be the smallest full subcategory of Mod-R for which Ob(CA ) contains A and is closed under isomorphisms, finite direct sums, and direct summands. Then the monoid V (CA ) is a Krull monoid (Theorem 7.14). More precisely, if M and N are the two maximal ideals of CA , that is, the ideals of CA associated to the two maximal ideals of the endomorphism ring of the R-module A, then the direct-summand reflecting functor F : CA → CA /M × CA /N of Theorem 7.13(c) induces a divisor homomorphism ∼ N2 (Corollary 7.10). Thus all modules V (F ) : V (CA ) → V (CA /M) ⊕ V (CA /N ) = 0 of type 2 in CA are isomorphic to A. Therefore A, B, C ∈ C, A ⊕ B ∼ = C ⊕ D, and ∼ B imply that A ∼ A= =B∼ =C∼ = D. Thus V (D) = V (A) is already an edge of the graph G(C).

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Second possibility: V (A) = V (B), but V (A) and V (B) are incident. In this case, let CA,B be the smallest full subcategory of Mod-R for which Ob(CA,B ) contains A and B and is closed under isomorphisms, finite direct sums, and direct summands. Then there is a divisor homomorphism V (CA,B ) → N03 , with A → ∼ C⊕D and A, B, C ∈ C (1, 1, 0) and B → (1, 0, 1), say. Thus the hypotheses A⊕B = imply that either C → (1, 1, 0) (that is, C ∼ = B) = A) or C → (1, 0, 1) (that is, C ∼ or C → (0, 1, 1) (that is, V (C) is incident to both V (A) and V (B)). In the first two cases, V (D) is already an edge of the graph G(C). In the third case, D → (2, 0, 0), that is, D is an indecomposable module that is not of type 2. Third possibility: V (A) and V (B) are not incident. If CA,B is the smallest full subcategory of Mod-R for which Ob(CA,B ) contains A and B and is closed under isomorphisms, finite direct sums, and direct summands, then there is a divisor homomorphism V (CA,B ) → N04 , with A → (1, 1, 0, 0) and B → (0, 0, 1, 1), ∼ C ⊕ D and A, B, C ∈ C imply that either say. Thus the hypotheses A ⊕ B = ∼ ∼ B) or C → C → (1, 1, 0, 0) (that is, C = A) or C → (0, 0, 1, 1) (that is, C = ∼ C⊕D (0, 1, 0, 1), (0, 1, 1, 0), (1, 0, 0, 1) or (1, 0, 1, 0). In these last four cases, A⊕B = and A, B, C ∈ C imply that V (C) is incident to both V (A) and V (B), so that V (A), V (C), V (B) is a path of length 3, V (A), V (C), V (B), V (D) is a cycle of  length 4, and D is of type 2. From Lemma 10.21, we get the following corollary: Corollary 10.22. I f C is a class of indecomposable modules of type 2, G(C) is its graph, and C  is the (DSP)-closure of C, then one of the following conditions holds: (a) C  contains an indecomposable module that is not of type 2. (b) C  consists only of indecomposable modules of type 2 and G(C  ) is a graph in which any two vertices in the same connected component always have distance ≤ 2. Lemma 10.23. Let U, U  , W be right modules of type 2 such that the edges V (U ) ∼ W ⊕ X for some module X, then X is a and V (U  ) are not incident. If U ⊕ U  = module of type 2. This has been proved as the “third possibility” in the proof of Lemma 10.21. Lemma 10.23 motivates the following definition. A class C of indecomposable modules of type 2 is said to satisfy weak (DSP) if for every U, V, W ∈ C such that the edges V (U ) and V (U  ) are not incident and for every module X such that U ⊕ U ∼ = W ⊕ X, one has that X ∈ C. For instance, Lemma 10.23 says that the class of all indecomposable modules of type 2 always satisfies weak (DSP). Moreover, as we saw in the proof of Lemma 10.21 (third possibility), if a class C of indecomposable modules of type 2 satisfies weak (DSP), then in the graph G(C), any two distinct vertices connected by a path of length 3 are adjacent. These graphs are very special, as the next lemma shows.

10.7. Condition (DSP)

347

Lemma 10.24. Let G be a connected graph in which any two distinct vertices connected by a path of length 3 are adjacent. Then G is either a complete graph or a complete bipartite graph. Proof. Let G = (V, E) be a connected graph in which any two distinct vertices connected by a path of length 3 are adjacent. The lemma is trivial when |V | ≤ 2, so that we can assume |V | ≥ 3. Fix a vertex v0 ∈ V . Define the two subsets X := { v ∈ V | v is adjacent to v0 } and Y := V \ X of V , so that in particular v0 ∈ Y and X = ∅. Let G be the subgraph of G with the same set V of vertices as G and set of edges E  := { {v, w} ∈ E | {v, w} ∩ X = ∅ and {v, w} ∩ Y = ∅ }. The graph G is clearly a bipartite graph. Let us show that it is a complete bipartite graph. If x ∈ X and y ∈ Y , then x is a vertex adjacent to v0 , and y is a vertex not adjacent to v0 . If y = v0 , there is an edge in G between x and y. If y = v0 , then there is a path in G between y and v0 , which can be shortened to a path of length at most 2. Since y = v0 is not adjacent to v0 , there exists a path of length 2 in G that connects y to v0 . Thus there is a path of length 3 in G between y and x, so that x and y are adjacent in G by hypothesis. This proves that G is a complete bipartite graph. Now if E  = E, then G is a complete bipartite graph and we are done. Suppose E  properly contained in E. There is an edge in E with both vertices either in X or in Y . Suppose that x0 , x0 are two vertices in X with {x0 , x0 } ∈ E. Fix any two distinct vertices x, x ∈ X. Then {x, v0 }, {v0 , x0 }, {x0 , x0 }, {x0 , v0 }, {v0 , x } is a path in E of length 5, possibly with equal consecutive edges. In any case, this path can be shortened to a path of length 1 between x and x . Thus all vertices in X are adjacent. Now fix any two distinct vertices y, y  in Y . Then {y, x0 }, {x0 , x0 }, {x0 , y  } ∈ E, so that there is a path of length 3 in G between the two distinct vertices y, y  of G. Thus y and y  are adjacent in G by hypothesis. This proves that G is a complete graph. Similarly if there is an edge in E with  both vertices in Y . Theorem 10.25. Let C be a nonempty class of indecomposable modules of type 2 and let C  be its (DSP)-closure. Then exactly one of the following two conditions holds: (a) The class C  consists only of indecomposable modules of type 2, the graph G(C) is bipartite, the connected components of G(C  ) are complete bipartite graphs, and the weak Krull–Schmidt theorem holds for both C and C  . (b) The class C  contains an indecomposable module that is not of type 2, and the graph G(C) is not bipartite. Proof. The cases (a) and (b) correspond to the two cases of G(C) bipartite or not. Assume the graph G(C) bipartite. Let us prove by induction on n that Cn consists of indecomposable modules of type 2 and that the graph G(Cn ) is bipartite. The case n = 0 holds by hypothesis. Suppose that Cn−1 consists only of indecomposable modules of type 2 and that the graph G(Cn−1 ) is bipartite. Assume that A ⊕ B ∼ =

348

Chapter 10. The Krull–Schmidt Theorem in the Case Two

C ⊕ D, where A, B, C ∈ Cn−1 . We have seen in Lemma 10.21 that we have three possibilities: either D ∈ Cn−1 , or V (A), V (B) are distinct adjacent edges (so that V (A), V (B), V (C) form a cycle of length 3, because A, B, C ∈ Cn−1 are of type 2), or V (A), V (B) are distinct nonadjacent edges (in which case V (A), V (C), V (B) form a path of length 3 that is not a cycle). The second possibility cannot take place, because the bipartite graph G(Cn−1 ) cannot contain cycles of length 3. In the third possibility, if the set of vertices of Cn−1 is bipartite by X and Y say, then V (A) connects a vertex in X to one in Y , V (C) connects a vertex in Y to one in X, and V (B) connects a vertex in X to one in Y . Thus V (D) also connects a vertex in X to one in Y . In other words, the graph G(Cn ) has the same set of vertices as G(Cn−1 ) and is bipartite by the same subsets X and Y as G(Cn−1 ). Moreover, in this case D is also of type 2, so that Cn consists only of indecomposable modules of type 2. This concludes the proof of the inductive step. Since C  is the union of all the classes Cn , we obtain that C  consists only of indecomposable modules of type 2 and G(C  ) is bipartite by the subsets X and Y . Now the connected components of G(C  ) are either complete graphs or complete bipartite graphs (Lemma 10.24). Since G(C  ) is bipartite, any connected component of G(C  ) that is a complete graph has necessarily only two vertices; hence it is also a complete bipartite graph. This shows that all the connected components of G(C  ) are complete bipartite graphs. In particular, the weak Krull– Schmidt theorem holds for both C and C  , and so (a) holds. Now suppose that the graph G(C) is not bipartite. We want to show that (b) holds. Assume the contrary, so that C  is a class of indecomposable modules MR with δ(MR ) = 2 (remark in the paragraph immediately after the proof of Lemma 10.20), and C  contains only modules of type 2 because we are assuming the negation of (b). Then G(C  ) must have a cycle of odd length n. Since paths of length 3 can be shortened to paths of length 1 in G(C  ), it follows that G(C  ) contains a cycle of length 3. Let t be the smallest positive integer such that G(Ct ) has a cycle of length 3. Then, by the (DSP) property, Ct+1 contains an indecomposable  module that is not of type 2. This contradiction shows that (b) holds. Clearly, if C is a class of indecomposable modules of type 2, then the graph K1 ∼ K1,1 cannot appear as a connected component of the graph G(C). Notice that K2 = is the unique graph that is both a complete graph and a complete bipartite graph. Proposition 10.26. Let α be a class. Then the monoid V (Kα ) is a free commutative monoid if and only if the class α has at most three elements. Proof. The monoid V (K3 ) is a free commutative monoid of rank 3, that is, is a graph isomorphic to the monoid N30 . In fact, the monoid V (K3 ) is the submonoid of N03 (because K3 has three vertices) generated by the three elements (1, 1, 0), (1, 0, 1), and (0, 1, 1) (because K3 has three vertices), and the three elements form a free set of generators for V (K3 ), because they are linearly independent elements

10.7. Condition (DSP)

349

⎞ 1 1 0 of the Q-vector space Q3 (the determinant of the matrix ⎝ 1 0 1 ⎠ with entries 0 1 1 in Z is nonzero). The monoids V (K1 ) and V (K2 ) are free commutative monoids of rank 0 and 1 respectively. Conversely, in any free commutative monoid, the set of atoms is the unique free set of generators. If α has at least four distinct elements, then there are four atoms δi , i = 1, 2, 3, 4, in the monoid V (Kα ), which are subject to the relation δ1 + δ2 = δ3 + δ4 . Hence V (Kα ) is not free in this case.  ⎛

Lemma 10.27. Let α, β, α , β  be nonempty classes with |α| ≥ |β| and |α | ≥ |β  |. Then: (a) The commutative monoid V (Kα,β ) is a free commutative monoid if and only if the class β has exactly one element. In this case, V (Kα,β ) is a free commutative monoid whose class of free generators is equipotent to the class α. (b) The commutative monoids V (Kα,β ) and V (Kα ,β  ) are isomorphic if and only if α is equipotent to α and β is equipotent to β  . Proof. (a) If both α and β have at least two distinct elements a1 , a2 and b1 , b2 respectively, then there are four atoms δ{ai ,bj } , i, j = 1, 2, in the monoid V (Kα,β ), which are subject to the relation δ{a1 ,b1 } + δ{a2 ,b2 } = δ{a1 ,b2 } + δ{a2 ,b1 } . Hence V (Kα,β ) is not free in this case. Conversely, if β := {b} has one element, there is a monoid isomorphism N(α) → V (Kα,β ) that maps each atom a ∈ α of N(α) to the atom δ{a,b} of V (Kα,β ). (b) Any monoid isomorphism V (Kα,β ) → V (Kα ,β  ) must send the class X of all atoms of V (Kα,β ) onto the class Y of all atoms of V (Kα ,β  ). Now the atoms of V (Kα,β ) are the elements of the form δ{a,b} , with a ∈ α and b ∈ β. That is, the elements of X correspond to edges of the graph Kα,β . Let F denote the family of all the subclasses X  of X such that the divisor-closed submonoid [[X  ]] of V (Kα,β ) generated by X  is a free monoid with free class of generators X ∩ [[X  ]]. (It is convenient to recall here that the divisor-closed submonoid [[X  ]] of V (Kα,β ) generated by X  is the class of all elements s ∈ V (Kα,β ) such that there exist n ≥ 1, x1 , . . . , xn ∈ X  , and t ∈ V (Kα,β ) with s + t = x1 + · · · + xn . Graphically, this implies that if there are in X  an edge from a ∈ α to b ∈ β and another edge from a ∈ α to b ∈ β, then in X  there are also the edge from a ∈ α to b ∈ β and the edge from a ∈ α to b ∈ β.) Thus, if δ{a,b} and δ{c,d} are two atoms in a subclass X  belonging to F , then the two corresponding edges {a, b} and {c, d} must be adjacent, because otherwise there would be a nontrivial relation δ{a,b} + δ{c,d} = δ{a,d} + δ{c,b} between elements of X ∩ [[X  ]]. It follows that the elements X  ∈ F correspond to subgraphs of Kα,β that have the form of a star. If we partially order the class F by inclusion, then any maximal element of F corresponds to the class of all edges of Kα,β incident in a single vertex. Thus there is a one-to-one correspondence between maximal elements of F and vertices of Kα,β . Similarly for the graph Kα ,β  and its monoid V (Kα ,β  ). Hence any

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monoid isomorphism between V (Kα,β ) and V (Kα ,β  ) induces an order-preserving bijection between the class F and the corresponding class F  for the graph Kα ,β  . Thus the size of the maximal elements of F and F  is preserved by the monoid isomorphism, that is, α = α and β = β  .  Proposition 10.28. The following conditions are equivalent for a class C of indecomposable right modules of type 2 satisfying the weak (DSP) property. (a) The weak Krull–Schmidt theorem holds for the class C. (b) The graph G(C) does not contain subgraphs isomorphic to the complete graph K4 . (c) Every connected component of the graph G(C) is either a complete bipartite graph or a graph isomorphic to K3 . Proof. (a) ⇒ (b) is Proposition 10.19. (b) ⇒ (c) From the weak (DSP) property, we know that in the graph G(C) any two distinct vertices connected by a path of length 3 are adjacent. Thus the graph G(C) is a disjoint union of complete bipartite graphs and complete graphs by Lemma 10.24. The only complete graphs that do not contain subgraphs isomorphic to K4 are K1 , K2 and K3 . Now K1 cannot appear as a connected component of G(C) and K2 ∼ = K1,1 . It follows that (b) ⇒ (c). (c) ⇒ (a) follows from Corollary 10.10, Corollary 10.12, and Example 10.16.  By Lemma 10.23, the class T2 of all indecomposable modules of type 2 satisfies weak (DSP). From Proposition 10.28, we obtain the following: Theorem 10.29. Let T2 be the class of all right R-modules of type 2. Then the weak Krull–Schmidt theorem holds for T2 if and only if G(T2 ) does not contain subgraphs isomorphic to K4 , if and only if the connected components of G(T2 ) are either isomorphic to K3 or complete bipartite graphs. Theorem 10.30 (Dichotomy). For any ring R, exactly one of the following two conditions holds: (a) There exist two right R-modules U1 , U2 of type 2 such that U1 ⊕U2 has exactly three nonisomorphic direct-sum decompositions. (b) There exist two ideals I, K of the full subcategory Mat(T2 ) of Mod-R whose objects are all finite direct sums of right R-modules of type 2 with Mat(T2 )/I

and

Mat(T2 )/K

amenable semisimple categories and the canonical functor F : Mat(T2 ) → Mat(T2 )/I × Mat(T2 )/K isomorphism reflecting.

10.7. Condition (DSP)

351

Proof. Let T2 be the class of all indecomposable right R-modules of type 2. The dichotomy corresponds to whether (a) the graph G(T2 ) contains a subgraph isomorphic to K4 , or (b) G(T2 ) does not contain any subgraph isomorphic to K4 . For (a), it suffices to prove that a graph G = (V, E) contains a copy of the graph K4 if and only if there exist six distinct edges 1 , 2 , . . . , 6 ∈ E such that δ1 + δ2 = δ3 + δ4 = δ5 + δ6 in V (G). Now, if v, w, x, y ∈ V are four distinct vertices pairwise incident in G, then 1 = {v, w}, 2 = {x, y}, 3 = {v, x}, 4 = {w, y}, 5 = {v, y}, 6 = {w, e} have the required properties. Conversely, if there exist six distinct edges 1 , 2 , . . . , 6 ∈ E such that δ1 + δ2 = δ3 + δ4 = δ5 + δ6 , then 1 ∪ 2 = 3 ∪ 4 = 5 ∪ 6 is a subset of V consisting of four distinct pairwise incident vertices of G, so that G(T2 ) contains a subgraph isomorphic to K4 . For (b), suppose that G(T2 ) does not contain subgraphs isomorphic to K4 . Equivalently (Theorem 10.29), assume that the connected components of G(T2 ) are either isomorphic to K3 or complete bipartite graphs. For every connected component of G(T2 ) isomorphic to K3 , fix a module M with M  in the connected component (we are using the axiom of choice for classes, Section 1.7). Let C  be the full subcategory of T2 whose objects are all the modules in T2 that are not isomorphic to any of the fixed modules M . Now the graph G(C  ) is bipartite, because we have eliminated an edge from all the triangles in G(T2 ). Notice that the graphs G(C  ) and G(T2 ) have the same class V of vertices. Let Mat(C  ) be the full subcategory of Mat(T2 ) whose objects are all direct sums of finitely many objects of C  . Let Mat(FT-R) be the full subcategory of Mod-R whose objects are all right R-modules that are direct sums of finitely many right R-modules of finite type. Now we have full subcategories Mat(C  ) ⊆ Mat(T2 ) ⊆ Mat(FT-R) ⊆ Mod-R. Correspondingly, we have a commutative diagram of canonical functors Mat(C  ) ↓ F 

→

Mat(T2 ) ↓ F

→

Mat(FT-R) ↓F

E

⊕M∈V Mat(C  )/M −→ ⊕M∈V Mat(T2 )/M −→ ⊕M∈V Mat(FT-R)/M, (10.4) where the canonical functor F is full and isomorphism reflecting (Proposition 9.13, (a) and (c)). Here we have denoted by V the class Max(FT-R) of all maximal ideals of the category FT-R. Since F is isomorphism reflecting, F  must be also ˙ 2 be a bipartition of V corresponding to isomorphism reflecting. Let V = X1 ∪X the bipartite graph G(C  ). The square on the left in (10.4) becomes Mat(C  ) ↓ F1 ×F2 ⊕M∈X1 Mat(C  )/M × ⊕M∈X2 Mat(C  )/M

→ E1 ×E2

−→

Mat(T2 ) ↓ F1 ×F2 ⊕M∈X1 Mat(T2 )/M × ⊕M∈X2 Mat(T2 )/M.

Now consider the ideals Ker Fi of the category Mat(T2 ), i = 1, 2, kernels of the functors Fi . The functor Fi : Mat(T2 ) → ⊕M∈Xi Mat(T2 )/M induces a faithful functor Gi : Mat(T2 )/Ker Fi → ⊕M∈Xi Mat(T2 )/M. In order to conclude, it suffices to show that the faithful functor Gi is an equivalence. Now, F full implies

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F  full, hence Fi full, so that Gi is full. Similarly for Mat(C  ). Hence we find a faithful functor Gi : Mat(C  )/Ker Fi → ⊕M∈Xi Mat(C  )/M, which now is not only full, but clearly also essentially surjective, hence an equivalence. We claim that Mat(C  )/M ∼ = Mat(T2 )/M for every M ∈ V . From the claim, it will follow that Ei is an equivalence. Now Fi and Ei essentially surjective implies Fi essentially surjective. This will prove that Gi is an equivalence. Hence, now it suffices to prove the claim. The inclusion Mat(C  ) → Mat(T2 ) clearly induces a full and faithful functor Mat(C  )/M → Mat(T2 )/M. We must prove that this functor is essentially surjective, and to this end, it suffices to show that for any of the originally fixed modules M there exists an object M  ∈ Mat(C  ) isomorphic to M in the category Mat(T2 )/M. If M is a vertex of G(T2 ) that is not one of the two vertices of the edge M , we can take M  = 0. Suppose that M is one of the two vertices of M . We can assume that the connected component of G(T2 ) is the triangle with three edges M1 , M2 , M3 , with three vertices I1 , I2 , I3 , that each edge is Mi  = {Ik , Il } ({i, k, l} = {1, 2, 3}), that M = M1 , and that M = I2 . Then M1 ∼ = M3 in the category Mat(T2 )/M, so that M  = M3  has the required property. This concludes the proof of the claim. Let R be a ring, let G(T2 ) be the graph of the category T2 of all indecomposable right R-modules of type 2, and let { Cλ | λ ∈ Λ } be the set of all connected components of G(T2 ). Then V (T2 ) = ⊕λ∈Λ V (Cλ ) by Lemma 10.5. Equivalently, every element of V (T2 ) is a sum of elements in the monoids V (Cλ ) in a unique way. We therefore have the following result. Proposition 10.31. Every module in Mat(T2 ) has a direct-sum decomposition, unique up to isomorphism, whose direct summands are indexed in the connected components of G(T2 ). Now, the connected components of G(T2 ) are either complete graphs or complete bipartite graphs. Thus we have two cases, which are treated in Proposition 10.32 and Proposition 10.33, respectively. Proposition 10.32. Let M1 , . . . , Mm , N1 , . . . , Nn be right R-modules of type 2. Suppose that M1 , . . . , Mm are all in the same connected component of G(T2 ), and that this connected component is a complete graph. Let P1 , P2 be the two maximal ideals of End(M1 ), . . . , P2m−1 , P2m the two maximal ideals of End(Mm ), Q1 , Q2 the two maximal ideals of End(N1 ), . . . , Q2n−1 , Q2n the two maximal ideals of End(Nn ). Then M1 ⊕ · · · ⊕ Mm ∼ = N1 ⊕ · · · ⊕ Nn if and only if m = n and there exists a permutation σ of {1, 2, . . . , 2m} such that APi = AQσ(i) for every i = 1, 2, . . . , 2m. Proof. By Corollary 9.12, the two modules M := M1 ⊕ · · ·⊕ Mm and N1 ⊕ · · ·⊕ Nn are isomorphic if and only if dimM (M ) = dimM (N ) for every maximal ideal of any additive category to which all the modules M1 , . . . , Mm , N1 , . . . , Nn belong, that is, counting multiplicity, if and only if the maximal ideals associated to the maximal ideals of the endomorphism rings of M1 , . . . , Mm coincide, up to their order, with the maximal ideals associated to the maximal ideals of the endomorphism rings of

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N1 , . . . , Nn , that is, if and only if there is a permutation σ of {1, 2, . . . , 2m} such  that APi = AQσ(i) for every i = 1, 2, . . . , 2m. Thus the direct-sum decompositions of the module M1 ⊕ · · · ⊕ Mm up to isomorphism correspond to partitions of the set {P1 , P2 , . . . , P2m } into m subsets each of cardinality 2 (but some care is needed here, because if APi = APj , then the subset {Pi , Pj } of cardinality 2 does not correspond to a direct summand of M1 ⊕ · · · ⊕ Mm , and the subsets {Pi , Pk } and {Pj , Pk } correspond to the same direct summand of M1 ⊕ · · · ⊕ Mm up to isomorphism). We leave to the reader to check that the set {1, 2, . . . , 2m} has 2(2m)! m ·m! partitions into m subsets of cardinality 2. Thus, under the hypotheses of Proposition 10.32, the module M1 ⊕ · · · ⊕ Mm has at most 2(2m)! m ·m! nonisomorphic direct-sum decompositions into direct sums of modules of type 2. (It has exactly 2(2m)! m ·m! nonisomorphic direct-sum decompositions into direct sums of modules of type 2 if the edges M1 , . . . , Mm  in the graph G(T2 ) are pairwise nonincident.) Let us move on to consider the case in which all the modules are in a connected component that is a complete bipartite graph. Proposition 10.33. Let C = (VC , EC ) be a connected component of G(T2 ). Assume ˙ C be a corresponding biparthat C is a bipartite complete graph and let VC = XC ∪Y tition. Let M1 , . . . , Mm , N1 , . . . , Nn be right R-modules of type 2 in the connected component C, so that it is possible to label the maximal ideals of EndR (Mi ) and EndR (Nj ) in such a way that the associated ideals AP1 , . . . , APm , AP1 , . . . , APn are in X and are orderly associated to maximal ideals P1 , . . . , Pm , P1 , . . . , Pn of End(M1 ), . . . , End(Mm ), End(N1 ), . . . , End(Nn ) respectively, and the associated ideals AQ1 , . . . , AQm , AQ1 , . . . , AQn are in Y and are orderly associated to the other maximal ideals Q1 , . . . , Qm , Q1 , . . . , Qn of End(M1 ), . . . , End(Mm ), End(N1 ), . . . , End(Nn ), respectively. Then M1 ⊕ · · · ⊕ Mm ∼ = N1 ⊕ · · · ⊕ Nn if and only if m = n and there exist two permutations σ, τ of {1, . . . , n} such that  and AQi = AQτ (i) for every i = 1, . . . , n. APi = APσ(i) As before, it is possible to see that under the hypotheses of Proposition 10.33, the module M1 ⊕ · · · ⊕ Mm has at most m! direct-sum decompositions into direct sums of modules of type 2 (and exactly m! direct-sum decompositions into direct sums of modules of type 2 if the edges M1 , . . . , Mm  in the graph G(T2 ) are pairwise nonincident).

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It is interesting to remark that any two nonincident indecomposable modules of type 2 in the same connected component determine whether the connected component is a complete graph or a complete bipartite graph. In fact, let MR , NR be any two indecomposable right R-modules of type 2. Then there are exactly three possibilities: (1) The edges MR  and NR  are incident or in distinct connected components. In this case, the module MR ⊕ NR has exactly one direct-sum decomposition into indecomposable modules of type two. (2) MR ⊕ NR has exactly two direct-sum decompositions into indecomposable modules of type two. This happens if and only if the edges MR  and NR  are not incident but belong to the same connected component which is a bipartite complete graph. (3) MR ⊕ NR has exactly three direct-sum decompositions into indecomposable modules of type two. This happens if and only if the edges MR  and NR  are not incident but belong to the same connected component, which is a complete graph. It is not known what the graph G(T2 ) for the ring Z of integers is. By [Calugareanu], Z-modules of type 2 are reduced torsion-free modules. By [Corner], every countable reduced torsion-free ring is the endomorphism ring of a countable reduced torsion-free abelian group. This allows us to construct several examples of indecomposable abelian groups of type 2. For instance, if A is any finite algebraic extension of Z, p, q are two distinct primes of Z, and Ap , Aq are the corresponding localizations, then the intersection Ap ∩ Aq is a countable reduced torsion-free ring of type 2.

10.8 Completely Prime Ideals: The Functorial Point of View Let us go back to some notions we have already seen. We have introduced and studied completely prime ideals of a category in Section 8.11. We have already considered pairs of functors Fi (i = 1, 2) into amenable semisimple categories Ai in Sections 8.4 and 8.7. Let us see the connections between these notions and the graph G(T2 ). In the rest of this chapter, when we restrict a functor F : A → B to a subcategory C of A, the restriction C → B of the functor F : A → B is still denoted  Thus, by F . For any preadditive category A, add(A) denotes the category Mat(A). if A is a full subcategory of Mod-R, add(C) is equivalent to the full subcategory of Mod-R whose objects are all right R-modules that are isomorphic to direct summands of direct sums of finitely many modules in C. Proposition 10.34. Let C be a class of indecomposable right R-modules and let C2 denote its subclass consisting of all the modules in C of type 2. Then the following conditions are equivalent: (a) There are two completely prime ideals P, Q of the category C such that for every U ∈ C, the set of all automorphisms of U is EndR (U ) \ (P(U, U ) ∪ Q(U, U )).

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(b) All the modules in C have type ≤ 2, and G(C2 ) is a bipartite graph. (c) There exist two additive functors Fi : add(C) → Ai , i = 1, 2, of the category add(C) into two amenable semisimple categories Ai such that Fi (U ) is a simple object of Ai for every U ∈ Ob(C), i = 1, 2, and F1 × F2 : add(C) → A1 × A2 is a local functor. (d) There are two amenable semisimple categories Ai , i = 1, 2, and two additive functors Fi : C → Ai of the full subcategory C of Mod-R into the categories Ai such that for every U ∈ C, Fi (U ) is a simple object of Ai and for every morphism f ∈ EndR (U ), f is an automorphism of U if and only if F1 (f ) is an automorphism of F1 (U ) and F2 (f ) is an automorphism of F2 (U ). Proof. (a) ⇒ (b) Let P and Q be two ideals satisfying condition (a). Let U be a module in C. Clearly, one of the following two conditions holds: (1) Either the ideals P(U, U ) and Q(U, U ) of EndR (U ) are comparable, and in this case EndR (U ) is a local ring with maximal ideal P(U, U ) ∪ Q(U, U ), or (2) the ideals P(U, U ) and Q(U, U ) of the ring EndR (U ) are not comparable, in which case they are the two distinct maximal right (left, two-sided) ideals of EndR (U ), the ring EndR (U ) has type exactly 2, the Jacobson radical is J(EndR (U )) = P(U, U ) ∩ Q(U, U ), and End(UR )/J(End(UR )) is canonically isomorphic to the direct product of the two division rings EndR (U )/P(U, U ) × EndR (U )/Q(U, U ). In both cases, the ring EndR (U ) has type ≤ 2 for every U ∈ C. Now let I be a vertex of G(C2 ), so that I is the ideal associated to a maximal ideal I of EndR (U ) for some object U of C2 . Since the set of all automorphisms of U is EndR (U ) \ (P(U, U ) ∪ Q(U, U )), and the ring EndR (U ) has type 2, we have that either I = P(U, U ) or I = Q(U, U ), but not both. Thus either P ⊆ I or Q ⊆ I, but not both. Let X denote the set of all the vertices I with P ⊆ I and let Y be the set of the vertices I with Q ⊆ I. Then the set of vertices of G(C2 ) is ˙ . For every U ∈ Ob(C), the edge V (U ) connects the ideal of C associated to X ∪Y P(U, U ) and the ideal of C associated to Q(U, U ). That is, it connects a vertex of X and one of Y . Hence G(C2 ) is a bipartite graph. (b) ⇒ (c) All the modules in C have type ≤ 2, and G(C2 ) is a bipartite ˙ 2 be a corresponding bipartition of the class of vertices graph. Let V = X1 ∪X V of the bipartite graph G(C2 ). Let X be the collection of all the ideals in the category C associated to the maximal ideals of the rings EndR (U ) with U any object of C. Fix an ideal I ∈ X and an object U in C. By Lemma 7.1, either I(U, U ) = End(U ) or I  := I(U, U ) is a maximal ideal of End(U ) and I is the ideal of C associated to I  . If I(U, U ) = End(U ), then U = 0 in add(C)/I. If I  := I(U, U ) is a maximal ideal of End(U ) and I is the ideal of C associated to I  , then I is a maximal ideal by Proposition 7.9, and I can be extended to a maximal ideal of add(C) by Lemma 4.29. Now each category add(C)/I can be embedded into the full subcategory fgss-S of Mod-S whose objects are all finitely generated semisimple right modules over a simple Artinian ring S (Corollary 7.10).

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Here S := EndC (A)/M(A, A), where A is an object of C such that I is the ideal associated to a maximal ideal of the ring EndR (U ). But A is an indecomposable module of type 1 or 2, so that S = EndC (A)/M(A, A) is a division ring. Moreover, each category add(C)/I is equivalent to the category of all finite-dimensional right vector spaces over the division ring S, so that add(C)/I is amenable semisimple, and for every object U of C, either U = 0 in add(C)/I (when I(U, U ) = End(U )) or U is a simple object of add(C)/I (when I(U, U ) is a maximal ideal of End(U ), so that U corresponds to a one-dimensional vector space over the division ring S). Thus the canonical projection add(C) → add(C)/I is a functor that sends an object U of C either to 0 (when I(U, U ) = End(U )) or to a simple object of add(C)/I (when I(U, U ) is a maximal ideal of End(U )). Recall that in Theorem 7.13, we saw that the canonical functor F : add(C) → ⊕I∈X add(C)/I, induced by the collection of all canonical projections add(C) → add(C)/I, I ∈ X, is isomorphism reflecting. For i = 1, 2, set Ai := ⊕I∈Xi add(C)/I, so that Ai is an amenable semisimple category. On applying the canonical functor Fi : add(C) → ⊕I∈Xi add(C)/I, any object U of C becomes a simple object Fi (U ) in Ai , because every object U of C goes to zero in the canonical projection add(C) → add(C)/I for all I except for one (the maximal ideal I associated to the maximal ideal of End(U ) when U has type 1, and the maximal ideal I associated to the maximal ideal of End(U ) not in Xi when U has type 2). It remains to show that F1 × F2 : add(C) → A1 × A2 is a local functor. To this end, it clearly suffices to prove that the canonical functor F : add(C) → of all canonical projections, is local ⊕I∈X add(C)/I, induced by the collection  (Example 4.44). The kernel of F is I∈X I, which is equal to the Jacobson radical of add(C) (Theorem 7.13(a)). Hence, in order to show that F is local, it suffices to prove that F is full (Example 4.47). From Proposition 9.14(c), we know that the canonical functor G : add(FT-R) → ⊕M∈Max(FT-R) add(FT-R)/M, induced by the collection of canonical functors GM : add(FT-R) → add(FT-R)/M, M ∈ Max(add(FT-R)) = Max(FT-R), is full. Now, C is a full subcategory of FT-R, add(C) is a full subcategory of add(FT-R), the ideals I of X are associated to the maximal ideals of the endomorphism rings of objects of C, and the ideals M of Max(FT-R) are associated to the maximal ideals of the endomorphism rings of objects of FT-R. Thus the maximal ideals of C can be extended to maximal ideals of FT-R in a unique way, and we can view X as a subset of Max(add(FT-R)). Thus, let A and B be two objects of add(C) and suppose that fI : A → B is a homomorphism in add(C)/I for every I ∈ X. Set fM := 0 : A → B (homomorphism in add(FT-R)/M for every M ∈ Max(FT-R) \ X). Then there exists f : A → B in add(FT-R) congruent to fM modulo M for every M ∈ Max(FT-R).

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Thus f : A → B is a morphism in add(C) congruent to fI modulo I for every I ∈ X. Thus F is full, hence local. (c) ⇒ (d) It suffices to restrict the functors Fi from add(C) to C.  For (d) ⇒ (a), it suffices to set P := KerF1 and Q := KerF2 . Example 10.35. We will now give an example of a ring R, a full subcategory C of R -Mod whose objects are all indecomposable left modules of type 2, two functors Fi : C → Ai , i = 1, 2, of C into two amenable semisimple categories Ai such that for every object U of C, Fi (U ) is a simple object of Ai and for every f ∈ EndR (U ), f is an automorphism of U if and only if F1 (f ) and F2 (f ) are automorphisms of F1 (U ) and F2 (U ) respectively, but F1 × F2 : C → A1 × A2 is not a local functor. Hence these functors F1 , F2 satisfy condition (d) in Proposition 10.34, but do not satisfy condition (c). This occurs because if a pair of functors add(C) → Ai satisfies (c), then their restrictions to C satisfy (d), but if (d) holds for a pair of functors Fi : C → Ai , then there is another, possibly different, pair of functors Fi : add(C) → Ai satisfying (c). Fix two distinct primes p and q and let Zp,q be the localization of the ring Z of integers at the multiplicatively closed subset of Z consisting of all integers not divisible by p and q. Thus Zp,q is a semilocal PID with two maximal ideals. Consider the additive group G := Q ⊕ Q ⊕ (Q/Zp,q ). The factor group Q/Zp,q is ufer groups Z(p∞ ) and ⎞ isomorphic to the direct Z(q ∞ ). Con⎛ sum of the two Pr¨ Q 0 0 Q Q 0 ⎠ of End(G). sider the subring R := ⎝ Hom(Q, Q/Zp,q ) Hom(Q, Q/Zp,q ) Zp,q Then R/J(R) ∼ = Q × Q × Z/pZ × Z/qZ, R acts on G on the left, and the lattice of all submodules of the R-module G is the disjoint union of (1) the direct product (ω + 1) × (ω + 1) with the componentwise order, and (2), above, a linearly ordered set of cardinality two. Here ω = {0, 1, 2, . . . } and ω + 1 = ω ∪ {ω}. In particular, G is an Artinian left R-module, its socle is isomorphic to Z/pZ ⊕ Z/qZ, and End(R G) ∼ = Zp,q . The subgroup H := 0 ⊕ Q ⊕ (Q/Zp,q ) of G is an essential submodule of R G and a maximal submodule of R G, and End(R H) ∼ = Zp,q . Thus R H ⊆ R G are indecomposable R-modules of type 2. Now let C be the full subcategory of R -Mod whose objects are all indecomposable Artinian left R-modules of type 2 with socle isomorphic to Z/pZ ⊕ Z/qZ. Let A be the category of all semisimple Z-modules of finite length. Let F1 : C → A be the functor that associates to each module R A in C its p-socle, that is, the submodule consisting of the elements annihilated by p, and similarly for F2 with the q-socle. Notice that for every object R A in C and every endomorphism f of R A, if F1 (f ) and F2 (f ) are automorphisms of F1 (A) and F2 (A), then the restriction of f to the socle of R A is an automorphism, so that f is injective because the socle is essential in Artinian modules. Since injective endomorphisms of Artinian modules are automorphisms, f is an automorphism of R A. Nevertheless, F1 × F2 : C → A × A is not a local functor, because the inclusion f : R H → R G is not an isomorphism, whereas F1 (f ) and F2 (f ) are isomorphisms.

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10.9 Direct Summands of Serial Modules of Finite Goldie Dimension In this section, we will prove that direct summands of serial modules of finite Goldie dimension are serial, a result due to Pavel Pˇr´ıhoda [Pˇr´ıhoda 2004]. We begin with some elementary lemmas. Lemma 10.36. Let Xn be the set {1, 2, . . . , n}. Let σ : Xn → Xn be a mapping. Let G = (Xn , σ) be the directed graph with set of vertices Xn and an edge from i to j if and only if σ(i) = j, i, j ∈ Xn . Dually, let G∗ = (Xn , σ ∗ ) be the directed graph with set of vertices Xn and an edge from i to j if and only if σ(j) = i, i, j ∈ Xn . Then both G and G∗ contain an oriented cycle. Proof. Consider the sequence σ m (1), m ≥ 0. Since the set Xn is finite, there exist p < q with σ p (1) = σ q (1). Then σ p (1) → σ p+1 (1) → · · · → σ q−1 (1) → σ q (1) is an oriented cycle in G and σ q (1) → σ q−1 (1) → · · · → σ p+1 → σ p (1) is an oriented cycle in G∗ .  Lemma 10.37. Let W ⊆ U be modules with W uniform. Let ε : W → U be the embedding, and let α : W → U be any morphism. Then at least one of the two morphisms α and ε − α is injective. Proof. Clearly, ker(α) ∩ ker(ε − α) = 0. Since W is uniform, either α is injective  or ε − α is injective. Lemma 10.38. Let W be a proper submodule of a couniform module U . Let α : W → U be an epimorphism and let ε : W → U be the embedding. Then ε − α is an epimorphism. Proof. Suppose the contrary, so that we have W ⊂ U , with U couniform, α : W → U is an epimorphism, and ε − α is not an epimorphism. Then α(W ) ⊆ ε(W ) + (ε − α)(W ) = W + (ε − α)(W ) ⊂ U because U is couniform. Thus α : W → U is not surjective, a contradiction.



Proposition 10.39. Let U1 , . . . , Un be n ≥ 1 nonzero uniserial modules, and suppose S := U1 ⊕ · · · ⊕ Un = M ⊕ N . Then either M or N contains a nonzero uniserial direct summand of S. Proof. The case n = 1 is trivial, so that we can suppose n ≥ 2. Let πi : U1 ⊕ · · · ⊕ Un → Ui , πM : U1 ⊕ · · · ⊕ Un → M , and πN : U1 ⊕ · · · ⊕ Un → N be the canonical projections relative to the direct-sum decompositions S = U1 ⊕ · · · ⊕ Un = M ⊕ N . Fix an index i = 1, 2, . . . , n. Consider the 2n submodules πi (πM (Uj )), πi (πN (Uj )), j = 1, 2, . . . , n,

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of Ui . As far as their sum is concerned, we have that n 

(πi (πM (Uj )) + πi (πN (Uj ))) ⊇

n 

πi (πM (Uj ) + πN (Uj )) ⊇

j=1

j=1

n 

πi (Uj ) = Ui .

j=1

Hence the sum of these 2n submodules is Ui . Since Ui is couniform, one of the submodules is equal to Ui . Hence there is an index j = 1, 2, . . . , n with πi πM (Uj ) = Ui or πi πN (Uj ) = Ui . Set, correspondingly, Ai := πM (Uj ) or Ai := πN (Uj ). Then Ai is a uniserial module contained either in M or in N , and πi (Ai ) = Ui . In particular, Ai is nonzero and the restriction πi |Ai : Ai → Ui of the canonical projection πi is an epimorphism (*). Now consider the restrictions πk |Ai : Ai → Uk of πk to Ai , k = 1, 2, . . . , n. Then n n * * %k ⊕ · · · ⊕ Un ) ∩ Ai = 0. (U1 ⊕ · · · ⊕ U ker(πk |Ai ) = k=1

k=1

Since Ai is uniform, there exists an index si = 1, 2, . . . , n such that πsi |Ai is in∼ Ai . jective (**). Let Vsi be the image of the monomorphism πsi |Ai , so that Vsi =  More precisely, the corestriction πi : Ai → Vsi of πsi |Ai : Ai → Ui is an isomorphism. Let ϕi : Vsi → Ai be the inverse of πi . Set fk,i := πk |Ai ϕi : Vsi → Uk for every index k = 1, 2, . . . , n, and ⎞ ⎞ ⎛ ⎛ U1 f1,i ⎟ ⎟ ⎜ ⎜ fi := ⎝ ... ⎠ : Vsi → ⎝ ... ⎠ = S. fn,i

Un

The image of fi is Ai . We now distinguish two cases. First case: there exist indices i, j = 1, 2, . . . , n such that πj |Ai : Ai → Uj is an isomorphism. In this case, let (πj |Ai )−1 : Uj → Ai be its inverse. If ε : Ai → S is the embedding, then (πj |Ai )−1 πj ε : Ai → Ai is the identity of Ai , so that S = %j ⊕ · · · ⊕ Un , and this concludes the ε(Ai ) ⊕ ker((πj |Ai )−1 πj ) = Ai ⊕ U1 ⊕ · · · ⊕ U proof of the proposition, because Ai is contained either in M or in N . Second case: πj |Ai : Ai → Uj is not an isomorphism for every i, j = 1, 2, . . . , n. Let us see some consequences of this assumption: (a) si = i for every i = 1, 2, . . . , n. In fact, suppose that there exists i with si = i. We have seen that πi |Ai : Ai → Ui is an epimorphism in (*) and that πsi |Ai : Ai → Ui is injective in (**). Thus, for si = i, we have that πi |Ai : Ai → Ui is an isomorphism, which is false in this second case. (b) Vsi = Usi for every i = 1, 2, . . . , n. In fact, if Vsi = Usi for some i, then πsi |Ai : Ai → Usi is an isomorphism. (c) fsi ,i : Vsi → Usi is not surjective for every i = 1, 2, . . . , n. In fact, suppose fsi ,i : Vsi → Usi surjective for some i. From fsi ,i = πsi |Ai ϕi , it follows that πsi |Ai : Ai → Usi is surjective. But then πsi |Ai must be an isomorphism by (**), which is false in this second case.

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(d) fi,i : Vsi → Ui is not a monomorphism for every i. In fact, suppose the contrary, so that fi,i : Vsi → Ui is a monomorphism for some index i. Since fi,i = πi |Ai ϕi is a monomorphism and ϕi is an isomorphism, it follows that πi |Ai is a monomorphism. Moreover, πi |Ai : Ai → Ui is an epimorphism by (*). Thus πi |Ai is an isomorphism, contradiction. Therefore, in this second case, we have, for every i = 1, 2, . . . , n, that si = i, Vsi = Usi , and fi,i : Vsi → Ui is not a monomorphism. Notice that πi |Ai : Ai → Ui is an epimorphism by (*) and that ϕi is an isomorphism, so that fi,i = πi |Ai ϕi is an epimorphism (but not a monomorphism in this second case). Since Vsi = Usi , we have that Vsi ⊂ Usi for every i. Thus there exists a mapping σ : Xn → Xn , where Xn = {1, 2, . . . , n}, such that σ(i) = si for every i. In the notation of Lemma 10.36, there is an oriented cycle in the graph G∗ . Notice that for every i, j ∈ Xn , if there is an edge in G∗ from j to i, then there are a submodule A of Uj and an epimorphism g : A → Ui . In fact, if there is an edge j → i in G∗ , then j = σ(i) = si , and fi,i : Vsi = Vj → Ui is the required epimorphism. From this, it follows that the existence of a path in G∗ from j to i implies that there is an epimorphism g : A → Ui for some submodule A of Uj (it suffices to take the composition of suitable restrictions of the epimorphisms corresponding to the edges along the path). By Lemma 10.36, there is a cycle in G∗ . Relabeling the indices of the uniserial modules Ui , we can assume that the edge 1 → 2 belongs to an oriented cycle. Thus there is a path in G∗ from 2 to 1, hence an epimorphism g : W2 → U1 for a suitable submodule W2 of U2 . Since 1 → 2 is an edge, we have the noninjective epimorphism f2,2 : V1 → U2 . Now A2 is the image of f2 : V1 → S, and f1,2 : V1 → U1 is the inclusion of the submodule V1 in U1 , because for every v1 ∈ V1 , f1,2 (v1 ) = π1 |A2 ϕ2 (v1 ) = π1 (ϕ2 (v1 )) = π1 ((π2 )−1 (v1 )) = π1 ((π1 )−1 (v1 )) = v1 . Let ε2 : W2 → U2 be the inclusion, ιi : Ui → Ui the identity morphism, and X the image of the morphism ⎞ ⎞ ⎛ ⎛ ⎛ ⎞ 0 ... 0 g U1 W 2 ⎜ ε2 0 . . . 0 ⎟ ⎜ U2 ⎟ ⎟ ⎜ U3 ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ 0 ι3 ⎜ U3 ⎟ ⎜ ⎟ 0 ⎟: ⎜ . ⎟→⎜ = S. (10.5) G := ⎜ ⎟ ⎟ ⎝ .. ⎠ ⎜ ⎜ . ⎟ .. ⎠ ⎝ ⎝ .. ⎠ . Un Un 0 ... 0 ιn 

   g U1 (For n = 2, one has G := = S.) : W2 → ε2 U2 We will now prove that S = X ⊕ A2 . Set W1 := (f2,2 )−1 (W2 ) ⊆ U1 . Let ε1 : W1 → U1 be the embedding. Notice that U1 = 0 implies W2 = 0, so W1 = 0 because f2,2 is an epimorphism. But f2,2 is not a monomorphism, so that f2,2 |W1 is not a monomorphism either (because W1 = 0, ker(f2,2 ) = 0, and U1 is uniform). Thus gf2,2 |W1 : W1 → U1 is not a monomorphism either, so that ε1 − gf2,2 |W1 is a monomorphism by Lemma 10.37. Let us show that this implies X ∩ A2 = 0. If s = (u1 , . . . , un ) ∈ X ∩ A2 , then there exist v1 ∈ V1 and w2 ∈ W2 such that u1 =

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361

f1,2 (v1 ) = g(w2 ), u2 = f2,2 (v1 ) = w2 , and s = f1 (v1 ). Now f2,2 (v1 ) = w2 implies that v1 ∈ W1 , so that (ε1 −gf2,2 |W1 )(v1 ) = v1 −g(w2 ) = v1 −f1,2 (v1 ) = v1 −v1 = 0. Hence v1 = 0, from which s = f1 (v1 ) = 0. This proves that X ∩ A2 = 0. We must now show that X + A2 = S. To this end, it suffices to prove that U1 and U2 are contained in X + A2 , because clearly Ui ⊆ X for every i ≥ 3. Now gf2,2 |W1 (W1 ) = gf2,2 (W1 ) = gf2,2 (f2,2 )−1 (W2 ) = g(W2 ) because f2,2 is an epimorphism, and g(W2 ) = U1 because g is an epimorphism. Moreover, W1 = (f2,2 )−1 (W2 ) ⊆ (f2,2 )−1 (U2 ) = V1 ⊂ U1 . Thus gf2,2 |W1 : W1 → U1 is surjective and the embedding ε1 : W1 → U1 is not surjective. It follows that ε1 − gf2,2 |W1 is surjective by Lemma 10.38. We are ready to prove that U1 ⊆ X + A2 . Suppose u1 ∈ U1 . Since X is the image of the morphism G (10.5) and A2 is the image of f2 : V1 → S, in order to show that u1 ∈ X + A2 , we must prove that there exist w2 ∈ W2 , ui ∈ Ui for i = 3, . . . , n and v1 ∈ V1 such that ⎧ ⎪ u1 = g(w2 ) + f1,2 (v1 ), ⎨ 0 = w2 + f2,2 (v1 ), (10.6) ⎪ ⎩ 0 = ui + fi,2 (v1 ) for i = 3, . . . , n. Since ε1 − gf2,2 |W1 : W1 → U1 is surjective, there exists w1 ∈ W1 such that (ε1 − gf2,2 |W1 )(w1 ) = u1 , that is, w1 − gf2,2 (w1 ) = u1 . The solution of the system (10.6) is then w2 := −f2,2 (w1 ) and v1 := w1 (hence the second equation is trivially satisfied) and ui := −fi,2 (v1 ) for i = 3, . . . , n (so that the third equation is satisfied). As far as the first equation is concerned, we have that g(w2 ) + f1,2 (v1 ) = −g(f2,2 (w1 )) + v1 = −g(f2,2 (w1 )) + w1 = u1 . This shows that U1 ⊆ X + A2 . In order to prove that U2 ⊆ X + A2 , set W2 := g −1 (V1 ). Notice that g : W2 → U1 is an epimorphism between uniserial modules and V1 is a proper submodule of U1 , so that W2 is a proper submodule of U2 . Moreover, f2,2 g|W2 (W2 ) = f2,2 g(W2 ) = f2,2 g(g −1 (V1 )) = f2,2 (V1 ) = U2 , because g and f2,2 are surjective mappings. Hence we can apply Lemma 10.38 to the proper submodule W2 of U2 and the epimorphism f2,2 g|W2 : W2 → U2 , so if ε : W2 → U2 denotes the embedding, we have that ε − f2,2 g|W2 : W2 → U2 turns out to be an epimorphism. We can now show that U2 ⊆ X + A2 . Let u2 be an element of U2 . Exactly as in the case of U1 , to show that u1 ∈ X + A2 , it suffices to prove that there exist w2 ∈ W2 , ui ∈ Ui for i = 3, . . . , n, and v1 ∈ V1 such that ⎧ ⎪ 0 = g(w2 ) + f1,2 (v1 ), ⎨ u2 = w2 + f2,2 (v1 ), (10.7) ⎪ ⎩ 0 = ui + fi,2 (v1 ) for i = 3, . . . , n.

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Since ε − f2,2 g|W2 : W2 → U2 is onto, there exists w2 ∈ W2 ⊆ W2 with (ε − f2,2 g|W2 (w2 ) = u2 . That is, w2 − f2,2 g(w2 ) = u2 . Set v1 := −g(w2 ). Then the second equation of the system (10.7) is satisfied. Since f1,2 (v1 ) = v1 , the first equation is also satisfied. For the third, it suffices to take ui = −fi,2 (v1 ) for every i ≥ 3. This concludes the proof that S = X ⊕ A2 . Hence A2 is a direct summand  of S contained either in M or in N . Theorem 10.40. Direct summands of serial modules of finite Goldie dimension are serial. Proof. Let S be a serial module of finite Goldie dimension, and suppose that S = U1 ⊕ · · · ⊕ Un = M ⊕ N , with U1 , . . . , Un nonzero uniserial submodules of S. We will prove that M and N are serial modules by induction on n. If n = 1, S is uniserial, hence S has only the two trivial direct summands, and the theorem is true. Suppose n > 1. By Proposition 10.39, either M or N contains a nonzero uniserial direct summand of S. Without loss of generality, we can suppose that M contains a nonzero uniserial direct summand U , S = U ⊕ S  say. Then M = U ⊕ (S  ∩ M ). By Proposition 8.36, there exist two distinct indices i, j = 1, . . . , n and a direct-sum decomposition U  ⊕U  = Ui ⊕Uj of Ui ⊕Uj such that U ∼ = U  and  ∼   S = U ⊕(⊕k=i,j Uk ). The module U is uniserial (Lemma 8.39). Thus S = M ⊕N can be written as U ⊕ S  = U ⊕ (S  ∩M )⊕ N , with S  a direct sum of n− 1 nonzero uniserial modules. From the cancellation property, we get that S  ∼ = (S  ∩ M ) ⊕ N .  By the inductive hypothesis, S ∩ M and N are serial, so that M = U ⊕ (S  ∩ M )  is also serial. Recall that a graph G = (V  , E  ) is a subgraph of a graph G = (V, E) if V  ⊆ V and E  ⊆ E. A full subgraph of G = (V, E) is a subgraph G = (V  , E  ) of G such that for any pair of vertices v, w ∈ V  of G , v is adjacent to w in G if and only if v is adjacent to w in G . If G = (V, E) is a graph and L is a set of edges of G, the full subgraph of G generated by L is the graph whose set of vertices V  is the set of all the vertices of the edges in L, and whose edges are all the edges in E whose vertices belong to V  . To sum up what we saw in this chapter, we have associated to each ring R and each class C of indecomposable right R-modules of type 2 a graph G(C) = (V, E). The class V of vertices of G(C) is the class of all maximal ideals of C (= the ideals P of the category C associated to P , where P is one of the two maximal ideals of End(MR ) and M ranges over C). The class E of edges of G(C) is the class V (C). For every object M ∈ V (C), the endomorphism ring End(MR ) has two maximal ideals P and Q. If AP and AQ are their associated ideals in C, then the edge M ∈ V (C) of G(G) joins the two vertices AP and AQ . The graph G(C) has no multiple edges and no loops. In the special case in which C = T2 , the class of all indecomposable right Rmodules of type 2, the connected components of G(T2 ) are either complete graphs Kα or complete bipartite graphs Kβ,γ . Now consider the subgraph GU of G(T2 ), where U is the class of all uniserial right R-modules of type 2.

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Proposition 10.41. The graph GU is a full subgraph of G(T2 ). Proof. If v, w are two vertices of GU , that is, if v, w are the maximal ideals associated to two maximal ideals M, N of the endomorphism rings EndR (V ), EndR (W ) of two uniserial modules V, W respectively, then v is adjacent to w in G if and only there exists an indecomposable module AR of type 2 such that v and w are the maximal ideals associated to the two maximal ideals of the endomorphism ring End(AR ). It suffices to prove that AR is a direct summand of V ⊕ W , because if AR is a direct summand of V ⊕ W , then AR is serial by Theorem 10.40, and hence AR is uniserial because it is indecomposable, so that v is adjacent to w in GU . Let us show that AR is a direct summand of V ⊕ W . Let Pv , Pw be the two maximal right ideals of End(AR ) with associated ideals v, w respectively, and let Pv , Pw be the maximal right ideals of EndR (V ), EndR (W ) with associated ideals v, w, respectively. Apply Lemma 9.3 to the category FT-R and to the ideal v associated to both Pv and Pv , getting that there exist two right R-module morphisms ϕv : A → V and ψv : V → A such that ψv ϕv ∈ / Pv , ϕv ψv ∈ / Pv ,   / Pv ∪ Pw , so that / Pw , then ψv ϕv ∈ ψv Pv ϕv ⊆ Pv , and ϕv Pv ψv ⊆ Pv . If ψv ϕv ∈ ψv ϕv is an automorphism of AR that factors through V ; hence AR is isomorphic to a direct summand of V , and we are done. Similarly, we can apply Lemma 9.3 to the ideal w associated to both Pw and Pw , getting two right R-module morphisms ϕw : A → W and ψw : W → A such that ψw ϕw ∈ / Pw . If ψw ϕw ∈ / Pv , then ψw ϕw is an automorphism of AR ; hence AR is isomorphic to a direct summand of W , and we conclude. Thus we can suppose that ψv ϕv ∈ Pw \ Pv and ψw ϕw ∈ Pv \ Pw . In this case, ψv ϕv + ψw ϕw ∈ / Pv ∪ Pw , so ψv ϕv + ψw ϕw = (ψv , ψw ) ϕϕwv is an automorphism of A that factors through V ⊕ W . Thus the composite mapping of  ϕv  ⊕ : A V W and (ψv ϕv + ψw ϕw )−1 (ψv , ψw ) : V ⊕ W → A is the identity → ϕw mapping of A, whence A is isomorphic to a direct summand of V ⊕ W , which  concludes our proof.

10.10 Direct Summands of Direct Sums of Modules Whose Endomorphism Rings Have at Most Two Maximal Right Ideals The classes of modules we have studied in this chapter, for instance the class of all uniserial right modules over a fixed ring R, consist of modules whose endomorphism rings have at most two maximal right ideals. Thus it is convenient to extend the techniques of the previous sections to this case, for instance the graph G(C), moving on from indecomposable modules of type 2 to indecomposable modules of type ≤ 2. Thus let C be a full subcategory of Mod-R whose objects are indecomposable right R-modules of type ≤ 2. It is possible to associate to the category C the (large) graph G(C) defined as follows. The vertices of G(C) are the ideals

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in the category C associated to maximal ideals of the endomorphism rings of the modules in Ob(C). The edges of G(C) are the isomorphism classes MR  of the indecomposable right R-modules MR of type 2 belonging to Ob(C). If MR is an indecomposable module of type 2, its endomorphism ring End(MR ) has two maximal ideals P and Q. Their associated ideals P and Q are distinct vertices of the graph, and the edge MR  connects the vertices P and Q of G(C). Therefore the extension from the case of type 2 to the case of type ≤ 2 is really trivial. We have just put a new isolated point in the graph for each isomorphism class of modules in C of type 1. Clearly, the graph G(C) is bipartite if and only if G(C2 ) is bipartite, where C2 denotes the full subcategory of C whose objects are all modules in C of type 2. The graph G(C2 ) is the graph G(C) in which the isolated points have been deleted. In the special case in which Ob(C) is the class of all indecomposable right R-modules of type ≤ 2, we will denote the graph G(C) by G(R). Thus, in the graph G(R), any two distinct vertices of G(R) connected by a path of length three are adjacent, and hence by Lemma 10.24, the connected components of G(R) are either complete graphs Kα (α a nonempty class) or complete bipartite graphs Kα,β (α, β nonempty classes). In Section 10.2, we saw that it is possible to associate a commutative monoid V (G) to any graph G = (V, E). Given a graph G = (V, E), where the elements of E are subsets of V of cardinality 2, we can construct the free commutative monoid (V ) N0 having the set of all δv : V → N0 , v ∈ V , with δv (v) = 1 and δv (w) = 0 for every w ∈ V , w = v, as free set of generators. If = {v, w} ∈ E is any edge of G, (V ) (V ) set δ := δv + δw ∈ N0 . Then V (G) is the submonoid of N0 generated by the (V ) elements δ ∈ N0 , where ranges over E. (X)

Now let D be a divisor closed submonoid of a free commutative monoid N0 . It is possible to construct a graph G(D) associated to D. For every x ∈ X, let (X) δx ∈ N0 be defined, for every y ∈ X, by δx (y) = 1 if y = x and δx (y) = 0 if (X) y = x. Then { δx | x ∈ X } is the free set of generators of N0 . The set of vertices of G(D) is { x ∈ X | δx ∈ D } ∪ { x ∈ X | there exists y ∈ X with y = x and δx + δy ∈ D }. The set of edges of G(D) is { {x, y} | x, y ∈ X, x = y and δx + δy is an atom of D }. The proof of Theorem 10.18 now shows that if R is the ring whose existence is guaranteed by Theorem 4.68, then the graph G(R) associated to the ring R contains a subgraph isomorphic to G(D). We know n that if M1 , . . . , Mn are modules of type 1, then every direct summand of i∈X Mi for some subset X of {1, . . . , n} i=1 Mi is isomorphic to (Krull–Schmidt–Azumaya theorem). We have seen in Theorem 10.40 that every direct summand of a direct sum of finitely many uniserial modules is a direct sum of finitely many uniserial modules. We now prove with an example that this cannot be extended to arbitrary modules of type 2, that is, there exists a ring R

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365

with three modules of type 2 whose direct sum has a direct summand that is not a direct sum of modules of type 2. Example 10.42. Let Z6 be the direct sum of six copies of the abelian group Z, Z/3Z the cyclic group of order 3, and ϕ : Z6 → Z/3Z the homomorphism defined by ϕ(x1 , x2 , x3 , x4 , x5 , x6 ) = x1 + x2 + x3 − x4 − x5 − x6 + 3Z for all (x1 , x2 , x3 , x4 , x5 , x6 ) ∈ Z6 . Set I := ker ϕ∩N60 . First of all, we will prove that (1, 0, 0, 1, 0, 0), (0, 1, 0, 0, 1, 0), (0, 0, 1, 0, 0, 1), (1, 1, 1, 0, 0, 0) and, (0, 0, 0, 1, 1, 1) are atoms of the commutative monoid I. Consider the augmentation map γ : Z6 → Z defined by γ(x1 , x2 , x3 , x4 , x5 , x6 ) = 6i=1 xi . This map γ is a group homomorphism. No element of N60 whose augmentation is 1 belongs to I. Thus all the elements of I of augmentation 2 are necessarily atoms. In particular, (1, 0, 0, 1, 0, 0, (0, 1, 0, 0, 1, 0),

and (0, 0, 1, 0, 0, 1)

are atoms. Since I does not have elements of augmentation 1, all the elements of I of augmentation 3 are also atoms. In particular, (1, 1, 1, 0, 0, 0) and (0, 0, 0, 1, 1, 1) are atoms of I. There is the relation (1, 0, 0, 1, 0, 0) + (0, 1, 0, 0, 1, 0) + (0, 0, 1, 0, 0, 1) = (1, 1, 1, 0, 0, 0) + (0, 0, 0, 1, 1, 1) between atoms of I. By Theorem 4.68, there exists a ring R with three indecomposable finitely generated projective modules P1 , P2 , P3 and two indecomposable finitely generated projective modules Q1 , Q2 such that P1 ⊕ P2 ⊕ P3 ∼ = Q1 ⊕ Q2 . By Remark 10.17(b), the modules P1 , P2 , P3 are of type 2 and the modules Q1 , Q2 are of type 3. The projective modules P1 , P2 , P3 of type 2 belong to the same connected component of the graph G(R), and this connected component is a complete bipartite graph. More precisely, in the monoid I there are exactly nine elements whose coordinates are two ones and four zeros. Hence there are exactly nine finitely generated projective right R-modules of type 2 up to isomorphism (Remark 10.17(b)). They are the nine edges of a complete bipartite graph K3,3 . Moreover, the elements (3, 0, 0, 0, 0, 0) and (0, 0, 0, 3, 0, 0) of I have augmentation 3, and so are also atoms of I. From the relation (1, 0, 0, 1, 0, 0)+(1, 0, 0, 1, 0, 0)+ (1, 0, 0, 1, 0, 0) = (3, 0, 0, 0, 0, 0) + (0, 0, 0, 3, 0, 0) between atoms of I, we see that there exist two indecomposable finitely generated projective modules U1 , U2 such ∼ U1 ⊕ U2 . It is interesting to observe here that the modules that P1 ⊕ P1 ⊕ P1 = ∼ M3 (Di ) for U1 and U2 are not of finite type, because EndR (Ui )/J(EndR (Ui )) = some division rings Di , so that EndR (Ui )/J(EndR (Ui )) is not a finite direct product of division rings.

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This Example 10.42 shows that it is not true that if A1 , . . . , An are modules of type ≤ 2, then every direct summand of A1 ⊕· · ·⊕An is a direct sum of modules of type ≤ 2. In the next proposition, we consider this situation in detail. Proposition 10.43. Let R be a ring and let M1 , . . . , Mt be indecomposable right R-modules of type 2. Suppose that the full subgraph of G(R) generated by the edges M1 , . . . , Mt  is not a bipartite graph. Then there are two distinct indices i, j = 1, 2, . . . , t such that Mi ⊕Mi ⊕Mj has an indecomposable direct summand that is not a module of finite type. In particular, (M1 ⊕· · ·⊕Mt )2 has an indecomposable direct summand that is not a module of finite type. Proof. Let G denote the full subgraph of G(R) generated by the edges M1 , . . . , Mt . Suppose that G is not a bipartite graph, so that t ≥ 2. The connected components of G(R) are either complete graphs or complete bipartite graphs, so that the connected components of G are also either complete graphs or complete bipartite graphs. But G is not bipartite, so that G must have a connected component that is not a complete bipartite graph. Thus there exist two distinct edges Mi , Mj , i, j = 1, 2, . . . , t, in this connected component. Since G is a finite graph, this connected component is a complete graph Kn for some n ≥ 3. First case: the two edges Mi  and Mj  are incident. In this case, there is a module N of type 2 such that Mi , Mj , N  is a cycle of length 3. If the edges Mi  and Mj  are not incident, then G contains a complete graph Kn with ∼ n ≥ 4 and there are three nonisomorphic direct-sum decompositions Mi ⊕ Mj = N1 ⊕N2 ∼ = N3 ⊕N4 with N1 , N2 , N3 , N4 pairwise nonisomorphic modules of type 2: Mi N2 N4

N3 N1 Mj

Notice that we have chosen the notation in such a way that Mi , N1 , N3  is a cycle of length 3. Now if A, B, C is any cycle of length 3, then Hom(A ⊕ B ⊕ C, A), Hom(A ⊕ B ⊕ C, B), and Hom(A ⊕ B ⊕ C, C) turn out to be three projective right modules over the ring End(A⊕B ⊕C), and they are the projective covers of the modules S1 ⊕ S2 , S1 ⊕ S3 , and S2 ⊕ S3 respectively, where S1 , S2 , S3 are three pairwise nonisomorphic simple End(A⊕B ⊕C)-modules (Proposition 9.1 and Lemma 9.4): S1 A

S2

B

C

S3

Thus Hom(A ⊕ B ⊕ C, C) is a direct summand of Hom(A ⊕ B ⊕ C, A) ⊕ Hom(A ⊕ B ⊕ C, B) (Theorem 2.27). Moving on to the direct complement, we see

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that there exists a finitely generated projective right End(A ⊕ B ⊕ C)-module that is a projective cover of S12 . Now apply the equivalence between the category of direct summands of finite direct sums of A ⊕ B ⊕ C and the category of finitely generated projective End(A ⊕ B ⊕ C)-modules we saw in Theorem 2.35. We can see that there exists an indecomposable right R-module D not of finite type such ∼ C ⊕ D. Thus when the edges Mi  and Mj  are incident, then that A ⊕ B = Mi ⊕ Mj has an indecomposable direct summand that is not of finite type. Second case: the edges Mi  and Mj  are not incident. In this case, the module Mi ⊕ N1 has an indecomposable direct summand that is not of finite type, where N1 is a direct summand of Mi ⊕ Mj . Therefore Mi ⊕ Mi ⊕ Mj has an indecomposable direct summand that is not of finite type.  By Proposition 10.43, when the graph G(R) is not bipartite, there always exist modules M1 , . . . , Mn of type ≤ 2 such that M1 ⊕ · · · ⊕ Mn has a direct summand that is not a direct sum of modules of type ≤ 2 (and this can also occur when the graph G(R) is bipartite). In the rest of this chapter, we will show that under suitable further conditions, it is true that all direct summands of a finite direct sum of modules of type ≤ 2 are direct sums of modules of type ≤ 2. To this end, we move on from graph theory to the functorial point of view.

10.11 Under Further Hypotheses, All Direct Summands of a Finite Direct Sum of Modules of Type ≤ 2 Are Direct Sums of Modules of Type ≤ 2 Let R be a ring and let M := M1 ⊕ M2 ⊕ · · · ⊕ Mt be a direct sum of finitely many indecomposable right R-modules Mi of type ≤ 2. We want to determine conditions under which the direct summands of M are direct sums of (necessarily finitely many) right R-modules of type ≤ 2. As we have already shown after the proof of Proposition 10.2, modules of type 1 have good behavior, because if M1 ⊕ M2 ⊕ · · · ⊕ Mt = A ⊕ B and M1 is of type 1, then M1 is isomorphic to a direct summand of either A or B, so that we can cancel M1 on both sides, and we can study the direct summands of M2 ⊕ M3 ⊕ · · · ⊕ Mt . Therefore we can consider direct sums M1 ⊕ M2 ⊕ · · · ⊕ Mt of finitely many indecomposable right modules Mi all of type 2. Indecomposable modules of type 2 are described by the graph G(R), which is a disjoint union of complete graphs Kα and complete bipartite graphs Kα,β . By Proposition 10.43, if G(R) has a connected component that is a complete graph Kα that is not bipartite, that is, with |α| ≥ 3, then there are direct summands of a direct sum of finitely many modules of type 2 that are not direct sums of modules of finite type. Thus we can consider modules Mi that generate a full subgraph of G(R) that is a bipartite graph. This section is devoted to studying

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this case, that is, the case in which the modules Mi are all in the same connected component C of G(R) and C is a complete bipartite graph. Let us be more precise. Let R be a ring, C = (V  , E  ) a connected component of G(R) that is a complete bipartite graph, and C the full subcategory of Mod-R whose objects are the right R-modules M of type 2 with M  an edge in C. By Proposition 10.34, there exist two additive functors Fi : C → Ai , i = 1, 2, of the full subcategory C into two amenable semisimple categories Ai such that for every U ∈ C, Fi (U ) is a simple object of Ai , and for every f ∈ EndR (U ), f is an automorphism of U if and only if F1 (f ) and F2 (f ) are automorphisms of F1 (U ) and F2 (U ) respectively. In Example 10.42, we saw that a direct summand of a direct sum of finitely many modules in C is not necessarily a direct sum of modules of type 2. Hence we must impose further conditions, and this is what we will do in this section. Our conditions may appear extremely technical, but we will see in Section 10.12 that they apply to all our previous examples. In the rest of this section, C will be a full subcategory of Mod-R and  will be the full additive subcategory of Mod-R with splitting add(C) := Mat(C) idempotents generated by the class Ob(C). Thus add(C) is the full subcategory of Mod-R whose objects are the right R-modules that are isomorphic to direct summands of direct sums of finitely many modules belonging to Ob(C). Lemma 10.44. Let A be an amenable semisimple category and F : add(C) → A an additive functor. Assume that for every object C in add(C), F (C) = 0 implies C = 0. Then, for every nonzero object C of add(C), there exist objects U, U  in C and morphisms f : U → C, f  : C → U  in add(C) such that F (f  f ) = 0. Proof. Let C be a nonzero object of add(C), so that there exist objects U1 , . . . , Un of C and morphisms α : U1 ⊕ · · · ⊕ Un → C and β : C → U1 ⊕ · · · ⊕ Un with αβ the identity morphism of C and βα an idempotent endomorphism of U1 ⊕ · · · ⊕ Un . Then F (α)F (β) is the identity morphism of F (C) and F (β)F (α) is an idempotent endomorphism of F (U1 ⊕ · · · ⊕ Un ). Since C = 0, we have that F (C) = 0, so that F (α)F (β) = F (1C ) = 0. Thus F (α)F (β)F (α)F (β) = F (1C ) = 0. From this we obtain that F (β)F (α) = 0. Thus there exist two indices k, t = 1, . . . , n such that the composite map F (πt β)F (αεk ) : F (Uk ) → F (Ut ) is nonzero, where for every i = 1, . . . , n, εi : Ui → U1 ⊕· · ·⊕Un denotes the embedding and πi : U1 ⊕· · ·⊕Un → Ui the canonical projection. It follows easily that f := αεk and f  := πt β have the  properties required in the statement. We fix some further hypotheses for the rest of this section. For each i = 1, 2, let Fi : add(C) → Ai be an additive functor of add(C) into an amenable semisimple category Ai . Suppose that Fi (U ) is a simple object of Ai for every U ∈ Ob(C) and both indices i = 1 and i = 2. Assume that the product functor F1 ×F2 : C → A1 ×A2 is local. Also, suppose that if C is an object of add(C) and F2 (C) = 0, then C = 0.

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Finally, assume that the ring morphism EndR (U1 ⊕ · · · ⊕ Un ) → EndA1 (F1 (U1 ) ⊕ · · · ⊕ F1 (Un )) × EndA2 (F2 (U1 ) ⊕ · · · ⊕ F2 (Un )), induced by the functor F1 × F2 , is a local morphism for every n ≥ 1 and every U1 , . . . , Un ∈ C. Remark 10.45. Under these hypotheses, all objects of C are indecomposable modules, as we have seen in Remark 8.29. The fact that the functor F1 × F2 : C → A1 × A2 is a local functor does not imply that its extensions Mat(F1 × F2 ) : Mat(C) → A1 × A2 to the additive closure Mat(C) of C and add(F1 × F2 ) : add(C) → A1 × A2 to add(C) are local functors (Remark 4.62). Now, for every pair of objects A, B of add(C), set [A]i = [B]i if there are morphisms f : A → B and g : B → A such that Fi (f ) and Fi (g) are both isomorphisms. Thus [A]i = [B]i if and only if A and B belong to the same Fi -class. Cf. Section 8.6. Clearly, [A]i = [B]i implies Fi (A) ∼ = Fi (B). Lemma 10.46. Let U and U  be objects of C such that [U ]1 = [U  ]1 and [U ]2 = [U  ]2 . Then U ∼ = U . Proof. Assume U, U  ∈ Ob(C), [U ]1 = [U  ]1 , and [U ]2 = [U  ]2 . Thus there exist two morphisms f : U → U  and g : U → U  such that F1 (f ) = 0 and F2 (g) = 0. If F2 (f ) is also nonzero, then f is an isomorphism between U and U  , because F1 × F2 : C → A1 × A2 is a local functor. Hence, if F2 (f ) = 0, we are done. Similarly, if F1 (g) = 0, then g : U → U  is an isomorphism. It remains to consider the case F2 (f ) = 0 and F1 (g) = 0. In this case, we have that F1 (f + g) = 0 and F2 (f + g) = 0, so f + g is the required isomorphism.  Lemma 10.47. If U, V, X, Y are objects of C with [U ]1 = [X]1 , [V ]1 = [Y ]1 , [U ]2 = ∼ X ⊕Y. [Y ]2 , and [V ]2 = [X]2 , then U ⊕ V = Proof. From the hypotheses, it follows that we have eight morphisms f1 : U → X, g1 : X → U, h1 : V → Y, 1 : Y → V, f2 : U → Y, g2 : Y → U, h2 : V → X, 2 : X → V,

(10.8)

where the functor F1 applied to the four morphisms in the first line yields four isomorphisms, and the functor F2 applied to the four homomorphisms in the second line produces four isomorphisms. If one of the eight homomorphisms in (10.8) ∼ X, is an isomorphism, for instance if f1 : U → X is an isomorphism, so that U = ∼ then [V ]2 = [X]2 = [U ]2 = [Y ]2 . This and [V ]1 = [Y ]1 imply that V = Y by ∼ X ⊕ Y , and we have proved the lemma. Lemma 10.46, so U ⊕ V = Thus we can suppose that all eight morphisms in (10.8) are not isomorphisms, so that the functor F2 applied to the four homomorphisms in the first line and the

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functor F1 applied to the four homomorphisms in the second line produce eight zero morphisms. The eight morphisms in (10.8) define two morphisms     g1 g2 f 1 h2 : U ⊕ V → X ⊕ Y and : X ⊕ Y → U ⊕ V.

2 1 f 2 h1 

  g1 g2 f 1 h2 The composite mapping α := is an endomorphism of f 2 h1

2 1 U ⊕ V , and applying F1 to it, we get that    f 1 h2 g1 g2 F1 (α) = F1 f 2 h1

2 1     0 F1 (g1 f1 ) F1 (g1 f1 + g2 f2 ) F1 (g1 h2 + g2 h1 ) = . = F1 ( 2 f1 + 1 f2 ) F1 ( 2 h2 + 1 h1 ) 0 F1 ( 1 h1 ) Similarly, applying F2 to the endomorphism α of U ⊕ V , we obtain that   F2 (g2 f2 ) 0 . F2 (α) = 0 F2 ( 2 h2 ) But the functor F1 × F2 induces a local morphism of rings EndR (U ⊕ V ) → EndA1 (F1 (U ) ⊕ F1 (V )) × EndA2 (F2 (U ) ⊕ F2 (V )), so that α turns out to be an automorphism of U ⊕ V . Similarly for the composite mapping    f 1 h2 g1 g2 β := .

2 1 f 2 h1   f 1 h2 : U ⊕ V → X ⊕ Y is an isomorphism.  Thus f 2 h1 Define the associated bipartite graph B(F1 , F2 ) = (V, E) as follows. Set Xi := { [U ]i | U ∈ Ob(C) }. The set of vertices V of B(F1 , F2 ) is the disjoint union ˙ 2 . The set E of edges of B(F1 , F2 ) is E := { U  | U ∈ C }. The edge U  X1 ∪X connects the vertex [U ]1 in X1 to the vertex [U ]2 in X2 . By Lemma 10.46, the graph B(F1 , F2 ) has no multiple edges. Remark 10.48. Let us briefly analyze the difference between the graph G(C) we have defined at the beginning of Section 10.10 and the graph B(F1 , F2 ) = (V, E) we have just introduced. In both cases, we have fixed a category C whose objects are indecomposable modules of type 1 or 2. In G(C) the vertices are the ideals in the category C associated to maximal ideals of the endomorphism rings of the objects of C, and the edges are the isomorphism classes of the objects of C that are indecomposable modules of type 2. In B(F1 , F2 ) the vertices are the classes [U ]1 and [U ]2 with U any object of C (both of type 1 and of type 2). The edges of B(F1 , F2 ) are the isomorphism classes of all objects of C (also those that are modules of type 1). The graph G(C) is not necessarily a bipartite graph (in

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Proposition 10.34, we saw that the graph G(C) is bipartite if and only if there exist two functors F1 , F2 with suitable properties). In contrast, the graph B(F1 , F2 ) is always bipartite (its construction depends on the two functors F1 and F2 ). Essentially, the new bipartite graph B(F1 , F2 ) can be regarded as a “refinement” of the graph G(C), because G(C) is constructed only from the category C, whereas B(F1 , F2 ) is constructed from more data, that is, also from the two functors F1 and F2 . In the transition from the graph G(C) to the graph B(F1 , F2 ), we essentially do not modify the structure relative to the indecomposable modules of type 2, but in B(F1 , F2 ) we replace the isolated points of G(C), which correspond to modules U of type 1, with an edge from [U ]1 to [U ]2 . The motivation of the difference between the constructions of the graphs G(C) and B(F1 , F2 ) as far as the modules of type 1 in C are concerned and the similarity between the two graphs as far as the modules of type 2 in C are concerned depends on the fact that the graph B(F1 , F2 ) is essentially constructed from the completely prime ideals P1,U and P2,U of EndR (U ) for every U ∈ Ob(C), where Pi,U = { f ∈ EndR (U ) | Fi (f ) = 0 }. On the one hand, if U is an indecomposable module of type 2, then P1,U and P2,U are the two maximal ideals of EndR (U ) and the ideal Pi,U associated to Pi,U corresponds to the Fi -class [U ]i of U . On the other hand, if U is a module of type 1, then one of the ideals Pi,U is the maximal ideal of EndR (U ), but the other is not necessarily the maximal ideal, though U still has the two Fi -classes [U ]1 and [U ]2 . Proposition 10.49. If U and V are objects of C and i = 1, 2, then [U ]i = [V ]i if and only if the ideals Pi,U and Pi,V of C coincide. Proof. Suppose [U ]i = [V ]i . Then there exist two morphisms f : U → V and g : V → U with both Fi (f ) and Fi (g) isomorphisms. It follows that gf ∈ Pi,U , f g ∈ Pi,V , gPi,V f ⊆ Pi,U , and f Pi,U g ⊆ Pi,V . Therefore Pi,U = Pi,V by Lemma 9.3. Conversely, if Pi,U = Pi,V , then there are two morphisms f : U → V and g : V → U such that gf ∈ Pi,U and f g ∈ Pi,V . Hence Fi (f g) = 0 and Fi (gf ) = 0. Therefore Fi (f ) and Fi (g) are isomorphisms.  Corollary 10.50. If U and V are objects of C and the edges U  and V  of the graph B(F1 , F2 ) are incident in B(F1 , F2 ), then U and V have the same type. Thus all modules in the same connected component of B(F1 , F2 ) have the same type. Proof. Assume, to the contrary, that U has type 1 and V has type 2. By symmetry, we can suppose [U ]1 = [V ]1 , so that the associated ideals Pi,U and Pi,V of C are equal (Proposition 10.49). Now P1,V is a maximal ideal of EndR (V ), so that P1,U = P1,V is a maximal ideal of C. Thus the proper ideal P1,U = P1,U (U, U ) = P1,V (U, U ) is a maximal ideal of EndR (U ) (Lemma 9.8). Moreover, there exist two morphisms f : U → V and g : V → U such that gf ∈ P1,U . It follows that gf is an automorphism of U , so U is isomorphic to a direct summand of V . The objects ∼ V , which is a contradiction, of C are indecomposable (Remark 10.45), so U = because U has type 1 and V has type 2. 

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The connected component of B(F1 , F2 ) corresponding to any module of type 1 has the shape of a star, that is, it is isomorphic to the complete bipartite graph K1,β for some nonempty class β. Thus the graph B(F1 , F2 ) is the disjoint union of its connected components, some of which consist entirely of modules of type 1 and the others of modules of type 2. In the transition from the graph B(F1 , F2 ) to the graph G(C), the components consisting of modules of type 2 are not changed, and the components consisting of modules of type 1, which are of the form K1,β , are replaced with |β| isolated vertices. In particular, we have the following: Corollary 10.51. If C2 is the full subcategory of C whose objects are the objects of C that are modules of type 2, then the graph G(C2 ) is isomorphic to the full subgraph of B(F1 , F2 ) whose vertices are the classes [U ]i with i = 1, 2 and U ∈ Ob(C2 ). If we transfer the terminology introduced in Section 10.7 for the graph G(C) to the graph B(F1 , F2 ), we can say that a full subcategory C of Mod-R satisfies weak (DSP) if for every three objects U, V, W of the category C such that the edges U  and V  are not incident in the graph B(F1 , F2 ) and for every module X, U ⊕ V ∼ = W ⊕ X implies X ∈ C. Lemma 10.52. Let U, V, W ∈ Ob(C) be three pairwise nonisomorphic modules of type 2. The module W is isomorphic to a direct summand of U ⊕ V if and only if U , W , V  is a path of length 3 in B(F1 , F2 ), that is, if U  and W  are incident and W  and V  are incident. Proof. Assume that U , W , V  form a path of length 3 in the bipartite graph B(F1 , F2 ). Without loss of generality, we can suppose [U ]1 = [W ]1 and [W ]2 = [V ]2 . Thus there are morphisms f : U → W , g : W → U , h : V → W , and k : W → V such that F1 (f g) = 0 and F2 (hk) = 0. Now the modules U , V , and W are pairwise nonisomorphic, [U ]1 = [W ]1 and [W ]2 = [V ]2 , so that [U ]2 = [W ]2 and [W ]1 = [V ]1 . Thus F2 (f ), F2 (g), F1 (h), and F1 (k) are all zero morphisms. It follows that F1 (f g + hk) = 0 and F2 (f g + hk) = 0. Therefore f g + hk is an automorphism of W , hence W is isomorphic to a direct summand of U ⊕ V . For the converse, suppose W isomorphic to a direct summand of U ⊕V . Then mapping there are homomorphisms W → U ⊕V and U ⊕V → W whose composite   g :W → is the identity of W . In matrix notation, there are homomorphisms h   g U ⊕ V and (f, h) : U ⊕ V → W with (f, h) = 1W . Equivalently, f g + hk = h 1W . Hence F1 (f g + hk) = 0 and F2 (f g + hk) = 0. Since U and W are not isomorphic, either [U ]1 = [W ]1 or [U ]2 = [W ]2 . Suppose for instance [U ]1 = [W ]1 . Then F1 (f g) = 0, so that F1 (hk) = 0. Thus [V ]1 = [W ]1 . But V and W are not isomorphic, hence [V ]2 = [W ]2 . Therefore U , W , V  is a path of length 3 in  the graph B(F1 , F2 ). If the subcategory C2 of Mod-R satisfies weak (DSP), then any two distinct vertices of B(F1 , F2 ) connected by a path of length 3 are adjacent in B(F1 , F2 ).

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From Lemma 10.24, we get the following result: Proposition 10.53. All the connected components of B(F1 , F2 ) are stars or complete bipartite graphs if and only if the subcategory C of Mod-R satisfies the weak (DSP) property. Notice that in Theorem 8.22, we proved a weak form of the Krull–Schmidt theorem under the hypotheses of this section.. Theorem 10.54. If the bipartite graph B(F1 , F2 ) is a complete bipartite graph, then every object of add(C) is a direct sum of finitely many objects of C. That is, every right R-module that is isomorphic to a direct summand of a direct sum of finitely many modules in C is isomorphic to a direct sum of finitely many modules in C. Proof. Suppose that M = U1 ⊕ · · · ⊕ Un = A ⊕ B, where U1 , . . . , Un are objects of C and A, B are nonzero right R-modules. Let πi : M → Ui , πA : M → A, and πB : M → B denote the canonical projections relative to the direct-sum decompositions M = U1 ⊕ · · · ⊕ Un and M = A ⊕ B, respectively. Let εi : Ui → M , εA : A → M and εB : B → M denote the embeddings. First of all, we show that either the module A or the module B contains a direct summand isomorphic to an object of C. To this end, notice that 1U1 = π1 εA πA ε1 + π1 εB πB ε1 . If one of the two endomorphisms π1 εA πA ε1 and π1 εB πB ε1 of U1 is an isomorphism, then U1 is isomorphic to a direct summand of either A or B, and we are done. Hence, in the following, we can assume that π1 εA πA ε1 and π1 εB πB ε1 are both nonisomorphisms. Since the product functor F1 ×F2 : C → A1 × A2 is a local functor, we can suppose without loss of generality, possibly exchanging the notation for the modules A and B, that F1 (π1 εA πA ε1 ) = 0, F2 (π1 εA πA ε1 ) = 0, F1 (π1 εB πB ε1 ) = 0, and F2 (π1 εB πB ε1 ) = 0. We claim that there are a module U in C and morphisms α : U → A and β : A → U such that F2 (βα) = 0. By Lemma 10.44, there exist modules U, U  ∈ Ob(C) and morphisms α : U → A, β  : A → U  such that F2 (β  α) = 0. Thus F2 (β  α) is an isomorphism. But B(F1 , F2 ) is a complete bipartite graph, so that there exists V ∈ Ob(C) with [U  ]1 = [V ]1 and [U ]2 = [V ]2 . Hence there are morphisms f : V → U  and g : V → U for which F1 (f ) = 0 is an isomorphism and F2 (g) = 0. Then h := β  αg : V → U  is such that F2 (h) = 0. Hence we have that either f or h or f + h is an isomorphism. In all these three cases, we obtain that ∼ U  , so [U ]2 = [U  ]2 . Now let : U  → U be a morphism with F2 ( ) = 0. Then V = α and β = β  : A → U are morphisms with F2 (βα) = 0. This concludes the proof of our claim. Since B(F1 , F2 ) is a complete bipartite graph, there exists W ∈ Ob(C) with [U1 ]1 = [W ]1 and [U ]2 = [W ]2 . But F1 (π1 εA πA ε1 ) = 0 and F2 (βα) = 0, so that there are morphisms f1 , g1 ∈ Hom(W, A) and f2 , g2 ∈ Hom(A, W ) with F1 (f2 f1 ) = 0 and F2 (g2 g1 ) = 0. If the composite mapping f2 f1 is an automorphism of W , then the module A has a direct summand isomorphic to W , as we wanted to prove. Similarly, when g2 g1 is an automorphism of W , the module A has a direct

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summand isomorphic to W , and we are done again. If the two composite morphisms f2 f1 and g2 g1 not automorphisms, then f2 f1 + g2 g1 is an automorphism of W , so W is isomorphic to a direct summand of A ⊕ A. By Proposition 8.7, we have that W is isomorphic to a direct summand of A, and we are done in this case as well. The rest of the proof of the theorem is now by induction on n. The case n = 1 is trivial, because in this case M is indecomposable, so that either A = 0 or B = 0. If M = U1 ⊕ · · · ⊕ Un = A ⊕ B, then either A or B has a direct summand isomorphic to a module U in C. By symmetry, we can suppose without loss of generality A = U ⊕ A . Then U1 ⊕ · · · ⊕ Un = U ⊕ A ⊕ B, whence there exist morphisms fi : U → Ui and gi : Ui → U (i = 1, . . . , n) such that ni=1 gi fi = 1U . Therefore there exist indices i and j for which F1 (gi fi ) = 0 and F2 (gj fj ) = 0. Then [U ]1 = [Ui ]1 and [U ]2 = [Uj ]2 . ∼ Ui by Lemma 10.46. We can suppose, for simplicIf i = j, we have that U = ity of notation, that i = j = 1. By the cancellation property of modules with a ∼ A ⊕ B. By the inducsemilocal endomorphism ring, it follows that U2 ⊕ · · ·⊕ Un =  tive hypothesis, both A and B are finite direct sums of modules in C. Therefore A and B are also finite direct sums of modules in C. Thus we can assume that i = j. For simplicity of notation, we will suppose i = 1 and j = 2. Since B(F1 , F2 ) is a complete bipartite graph, there exists V ∈ Ob(C) with [V ]1 = [U2 ]1 and [V ]2 = [U1 ]2 . By Lemma 10.46, [U ]1 = [U1 ]1 and [U ]2 = [U2 ]2 imply that U1 ⊕ U2 ∼ = U ⊕ V . Now ∼ U1 ⊕ U2 ⊕ · · · ⊕ Un ∼ U ⊕ A ⊕ B = = U ⊕ V ⊕ U3 ⊕ · · · ⊕ Un . By the cancellation property, we get that V ⊕ U3 ⊕ · · · ⊕ Un ∼ = A ⊕ B. Hence, by the inductive hypothesis, both A and B are finite direct sums of modules  in C. Therefore A and B are also finite direct sums of modules in C.

10.12 Examples In this section, we will give some examples of functors F of the category add(C) into an amenable semisimple category A that satisfy (or do not satisfy) the hypotheses of Lemma 10.44 and Theorem 10.54. Example 10.55. Let P : Mod-R → Spec(Mod-R) be the canonical functor of Mod-R into its spectral category Spec(Mod-R) (Section 5.1). Let U denote the full additive subcategory of Mod-R whose objects are all right R-modules of finite Goldie dimension. Let A be the full subcategory of Spec(Mod-R) whose objects are all semisimple objects of finite length, so that A is an amenable semisimple category. The functor P restricts to a functor U → A. If U is a uniform right

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375

R-module, that is, a module of Goldie dimension 1, then P (U ) is a simple object ∼ P (E(B)) is a of A. If B is an object of U of Goldie dimension m, then P (B) = direct sum of m simple objects of Spec(Mod-R). In particular, P (B) = 0 implies B = 0. It follows that P : U → A satisfies the hypotheses of Lemma 10.44. Example 10.56. Let F : Mod-R → (Mod-R) be the canonical functor in the construction dual to the construction of the spectral category (Section 5.6). Consider the product functor P × F : Mod-R → Spec(Mod-R) × (Mod-R) . If f : M → N is a morphism and P (f ), F (f ) are two isomorphisms, then f is both an essential monomorphism and a superfluous epimorphism (Propositions 5.3 and 5.34(c)). Thus f is an isomorphism. This proves that the product functor P × F : Mod-R → Spec(Mod-R) × (Mod-R) is local. Now let C be the full additive subcategory of Mod-R whose objects are all right R-modules of finite dual Goldie dimension. Let A be the full subcategory of (Mod-R) whose objects are all semisimple objects of finite length, so that A turns out to be an amenable semisimple category. The restriction of F to C is a functor C → A with the property that for every couniform R-module U , the object F (U ) of A is simple. If C is a nonzero object of C of dual Goldie dimension m, then there is a superfluous epimorphism of C onto the direct sum of m couniform modules U1 , . . . , Um , so that F (C) ∼ = F (U1 ⊕ · · · ⊕ Um ) is the direct sum of m simple objects of A . In particular, for every object C of C, F (C) = 0 implies C = 0. Thus F : C → A satisfies the hypothesis of Lemma 10.44. Let B be the full subcategory of Mod-R whose objects are all biuniform right R-modules. The objects of add(B) belong both to the category U of Example 10.55 and to the category C. In particular, the restriction of the product functor P × F : add(B) → A × A is a local functor. Also, if A ⊕ B ∼ = C ⊕ D and A, B, C ∈ Ob(B), then D ∈ Ob(B). Thus the connected components of the graph B(P, F ) corresponding to the category B are stars or complete bipartite graphs. Hence every direct summand of a finite direct sum of biuniform modules is a direct sum of biuniform modules. Example 10.57. Let R be a ring and let P : Mod-R → Spec(Mod-R) be the left exact covariant functor we have already considered in Example 10.55. Let P (1) : Mod-R → Spec(Mod-R) be the first right derived functor of P (Section 5.3). We will now show that the product functor P × P (1) : Mod-R → Spec(Mod-R) × Spec(Mod-R) is a local functor. Let A, A be two R-modules and let f : A → A be a morphism with both P (f ) and P (1) (f ) isomorphisms. Then f can be lifted to the minimal injective resolutions of A and A , i.e., it is possible to construct a commutative diagram with exact rows 0 → 0 →

A ↓f A

→ →

E ↓ f0 E

→ →

F ↓ f1 F .

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Since P (f ) = P (f0 ) and P (1) (f ) = P (f1 ) are both isomorphisms, the two morphisms f0 and f1 are essential monomorphisms. Since essential monomorphisms between injective modules are isomorphisms, the two morphisms f0 and f1 are isomorphisms. From the commutativity of the diagram, we get that f is also an isomorphism. Therefore the functor P × P (1) : Mod-R → Spec(Mod-R) × Spec(Mod-R) is local. Similarly, let D denote the full subcategory of Mod-R whose objects are all kernels of all nonzero morphisms ϕ : L0 → L1 , where L0 and L1 are injective right R-modules, L1 is indecomposable and ker ϕ is essential in L0 . Equivalently, Ob(D) is the class of all right R-modules K whose minimal injective resolution 0 → K → L0 → L1 → L2 → · · · is such that L1 is an indecomposable module. Let add(D) denote the full additive subcategory of Mod-R with splitting idempotents generated by Ob(D) and let A be the amenable semisimple full subcategory of Spec(Mod-R) whose objects are all semisimple objects of finite length. Here the notation is the same as in Example 10.55. Then the derived functor P (1) restricts to a functor add(D) → A. For any module K ∈ Ob(D) with a minimal injective ∼ P (L1 ) is a resolution 0 → K → L0 → L1 → L2 → · · · , we get that P (1) (K) = simple object of A. Note that for every K ∈ Ob(D) and every injective R-module Q, one has that K ⊕ Q ∈ Ob(D). Thus Q ∈ add(D), but P (1) (Q) = 0. Now let K be the full subcategory of Mod-R whose objects are all nonzero modules that are kernels of nonzero morphisms between indecomposable injective modules. That is, Ob(K) is the class of all right R-modules K whose minimal injective resolution 0 → K → L0 → L1 → L2 → · · · has both L0 and L1 indecomposable modules. Equivalently, it is the class of all right R-modules K with both P (K) = P (L0 ) and P (1) (K) = P (L1 ) simple objects in the spectral category Spec(Mod-R). The modules in add(K) are contained in the class U of Example 10.55 and in the class add(D). Since P and P (1) are additive functors, they preserve finite direct sums, so that for every A ∈ Ob(add(K)), both P (A) and P (1) (A) are semisimple objects of finite length in the category Spec(Mod-R). More precisely, this follows from the fact that if 0 → A → E → F is the minimal injective resolution of A, then E and F are direct sums of finitely many indecomposable injective modules. Thus the product functor P × P (1) restricts to a local functor add(K) → A × A. It is easily checked that if A ⊕ B ∼ = C ⊕ D and A, B, C ∈ Ob(K), then D ∈ Ob(K). Thus the connected components of the graph B(P, P (1) ) corresponding to the category K are either stars or complete bipartite graphs. It follows that Theorem 10.54 can be applied to the class of kernels of nonzero morphisms between two indecomposable injective modules. Example 10.58. Let R be any ring. The category (Mod-R) of Example 10.56 is not abelian in general, because it does not have kernels in general (Example 5.37). Hence it is not possible to compute the left derived functors F(n) : Mod-R → (Mod-R) .

10.12. Examples

377

Nevertheless, in Section 5.44, we defined a sort of derived functor F(1) : C → (Mod-R) , where C is the full subcategory of Mod-R whose objects are all right R-modules with a projective cover, and we showed that the functor F × F(1) : C → (Mod-R) × (Mod-R) is a local functor (Theorem 5.45). In particular, let S be the full subcategory of Mod-R whose objects are all couniformly presented right R-modules UR (Example 8.44(5)). Let A be the full subcategory of (Mod-R) whose objects are semisimple objects of finite length, so that A is amenable semisimple (Example 10.56). We will continue to denote by F(1) : S → A the restriction of the functor F(1) : C → (Mod-R) . For every module UR in S, the object F(1) (UR ) = F (ker πU ) is a simple object of A . By Proposition 4.20 and the universal property of the embedding functor Mat(S) → add(S) of the free additive category Mat(S) into  (page 120), the functor F(1) : S → A its idempotent completion add(S) = Mat(C) extends to a functor, which we still denote by F(1) , of the full additive subcategory add(S) of Mod-R with splitting idempotents generated by S into the category A . It is not difficult to prove that the functor F × F(1) : P → A × A is a local functor (for the details, see [Amini, Amini, and Facchini 2011b, Example 6.5]). Clearly, if M ⊕ N and M have projective covers, then so does N . It follows that ∼ C ⊕ D and A, B, C ∈ Ob(S), then D ∈ Ob(S). So the connected if A ⊕ B = components of the graph B(F, F(1) ), corresponding to the category S, are either stars or complete bipartite graphs. This shows that Theorem 10.54 can be applied to the class of all couniformly presented R-modules. Example 10.59. Let R be a ring and let S1 , S2 be two simple nonisomorphic Rmodules. In Example 10.35, we have considered the case of the category C of all indecomposable Artinian R-modules with socle isomorphic to S1 ⊕ S2 . There are two functors Fi , i = 1, 2, that associate to every object A of C the trace of Si in A. We gave that example because for the ring R in that example, the product functor F1 × F2 : C → A × A was not a local functor. For a different example, consider the ring R of Example 10.42. In this second case, the category C consists of nine finitely generated indecomposable projective R-modules Pi,j (i = 1, 2, 3, j = 4, 5, 6) say, R has six nonisomorphic simple modules Si (i = 1, 2, 3, 4, 5, 6), Pi,j /Pi,j J(R) ∼ = Si ⊕ Sj , the corresponding graph G(C) is the complete bipartite graph K3,3 , and there are functors F1 , F2 that associate to every module P in C the factor module of P modulo the reject of S1 ⊕ S2 ⊕ S3 , S4 ⊕ S5 ⊕ S6 respectively. Nevertheless, P1,4 ⊕P2,5 ⊕P3,6 has a direct-sum decomposition into two indecomposable modules of type 3.

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10.13 Notes on Chapter 10 Sections 10.1–10.8 are taken from [Facchini and Pˇr´ıhoda 2011b]. Lemma 10.27(b) appears in [Amini, Amini, and Facchini 2009, Lemma 4.1]. The material in Section 10.9 is due to Pavel Pˇr´ıhoda [Pˇr´ıhoda 2004]. Pˇr´ıhoda’s original proof is written rather densely. We have preferred a simpler presentation. Our presentation is based in part on the master’s thesis of Nicola Girardi (2007). The last part of the chapter, from Section 10.10 to the end of the chapter, is taken from [Amini, Amini, and Facchini 2011b].

Chapter 11

Serial Modules of Infinite Goldie Dimension 11.1 Small, ℵ0 -Small, Quasismall Modules Definition 11.1. We say that a module MR over a ring R is: • small if for every family { Mi | i ∈ I } of right R-modules and every homomorphism ϕ : MR → ⊕i∈I Mi , there is a finite subset F ⊆ I such that ϕ(M ) ⊆ ⊕i∈F Mi . Equivalently, MR is small if and only if Hom(MR , ⊕i∈I Mi ) is canonically isomorphic to ⊕i∈I Hom(MR , Mi ) for every family { Mi | i ∈ I } of right R-modules. • ℵ0 -small if for every family { Mi | i ∈ I } of right R-modules and every homomorphism ϕ : MR → ⊕i∈I Mi , there is a subset F ⊆ I with |F | ≤ ℵ0 such that ϕ(M ) ⊆ ⊕i∈F Mi . • quasismall if for every family { Mi | i ∈ I } of right R-modules such that MR is isomorphic to a direct summand of ⊕i∈I Mi , there is a finite subset F ⊆ I such that MR is isomorphic to a direct summand of ⊕i∈F Mi . For example: (1) Finitely generated modules are small. (2) Every small module is ℵ0 -small and quasismall. In particular, every finitely generated module is ℵ0 -small and quasismall. (3) Every countably generated module is ℵ0 -small. Proposition 11.2. Every module with a local endomorphism ring is a quasismall module. © Springer Nature Switzerland AG 2019 A. Facchini, Semilocal Categories and Modules with Semilocal Endomorphism Rings, Progress in Mathematics 331, https://doi.org/10.1007/978-3-030-23284-9_11

379

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Proof. Let NR be a module with a local endomorphism ring and let { Mi | i ∈ I } be a set of modules with NR isomorphic to a direct summand of ⊕i∈I Mi . We can suppose that G = N  ⊕ P = ⊕i∈I Mi with N  ∼ = N . By [Facchini 1998, Theorem 2.8], the module N has the exchange property, so that for every i ∈ I, there exists a direct-sum decomposition Mi = Bi ⊕Ci such that G = N  ⊕(⊕i∈I Bi ). It follows that N  ∼ = ⊕i∈I Ci . Now, N  ∼ = N is indecomposable, so that there exists ∼ Cj . This proves that N is isomorphic to a direct an index j ∈ I with N  = summand of Mj . Hence M is quasismall.  The Z-module Q is an example of a module with a local endomorphism ring, hence it is quasismall (Proposition 11.2) and ℵ0 -small (example (3) above), which is not small. If F is an uncountable field, F [x] is the ring of polynomials in one indeterminate x, and F (x) is the field of fractions of F [x], then F (x), as an F [x]-module with a local endomorphism ring, is quasismall (Proposition 11.2), but not ℵ0 -small. Here is the proof. For every α ∈ F , the principal ideal generated by x − α is a maximal ideal Mα of F [x]. The localization F [x]Mα of F [x] at Mα consists of all the elements of F (x) that can be written in the form f /g with f, g ∈ F [x] and g(α) = 0. πα : F (x) → F (x)/F [x]Mα for every α ∈ F , hence There is a canonical projection  a mapping α∈F πα : F (x) → α∈F F (x)/F [x]Mα . Every element of F (x) is the form f /g with f, g ∈ F [x] and g = 0. Let R be the finite subset of F consisting of the roots of g in F . Then, for any α ∈ F \ R, we have thatg(α) = 0, so that f/g ∈ F [x]Mα , i.e., πα (f /g) = 0. This proves that the image of  α∈F πα : F (x) → (x)/F F [x] , so that (x)/F F [x] is contained in ⊕ α∈F Mα Mα α∈F πα can be α∈F viewed as an F [x]-module morphism ϕ : F (x) → ⊕α∈F F (x)/F [x]Mα . But there is not a countable subset C of F such that ϕ(F (x)) ⊆ ⊕α∈C F (x)/F [x]Mα . It is possible to prove, though we will not need it here, the following result: Proposition 11.3 ([Facchini and Herbera 2004, Prop. 3.1(a) and Cor. 3.3]). (a) Injective envelopes of small modules are quasismall. (b) Every Artinian module is ℵ0 -small and quasismall. Proposition 11.4. Every uniserial non-quasismall module is countably generated. Proof. Let U be a uniserial non-quasismall module. By definition of quasismall, there exist a family of modules { Mi | i ∈ I } and a homomorphism ϕ : U → ⊕i∈I Mi such that πj ϕ = 0 for infinitely many j ∈ I. Here πj : ⊕i∈I Mi → Mj denotes the canonical projection for every j ∈ I. For every element x ∈ U , set supp(x) := { i ∈ I | πi ϕ(x) = 0 }. Thus supp(x) is a finite subset of I for every x ∈ U . Note that if x, y ∈ U and xR ⊆ yR, then supp(x) ⊆ supp(y). We will now define by induction a sequence of elements xn ∈ U , n ≥ 0, such that supp(x0 ) ⊂ supp(x1 ) ⊂ supp(x2 ) ⊂ · · · . Set x0 := 0. If xn ∈ U has been defined, then supp(xn ) is finite, but πj ϕ = 0 for infinitely many j ∈ I. Hence there exists k ∈ I with k ∈ / supp(xn ) and πk ϕ = 0. Let xn+1 ∈ U

11.1. Small, ℵ0 -Small, Quasismall Modules

381

be an element of U with πk ϕ(xn+1 ) = 0. Then supp(xn+1 ) ⊆ supp(xn ), so that xn+1 R ⊆ xn R. Thus xn R ⊆ xn+1 R. It follows that supp(xn ) ⊂ supp(xn+1 ). This defines the sequence xn . Assume by contradiction that the elements xn do not generate the module U . / xn R for every n ≥ 0. Thus Then there exists an element v ∈ U such that v ∈ vR ⊇ xn R for every n, so that supp(v) ⊇ supp(xn ) for every n. This yields a contradiction, because supp(v) is finite and n≥0 supp(xn ) is infinite. Therefore  the elements xn generate U . Hence U is countably generated. Thus every uniserial module is ℵ0 -small, but there exist uniserial modules that are not quasismall [Puninski 2001b]. If A and B are arbitrary modules, we say that a family { fi | i ∈ I } of module morphisms from A into B is a summable family of morphisms if for every element x of A there exists a finite subset Fx of I such that fi (x) = 0 for every i ∈ I \ Fx . Clearly, if { fi | i ∈ I } is a summable from A into B, family of module morphisms it is possible to define its sum i∈I fi : A → B by i∈I fi (x) := i∈Fx fi (x) for every x ∈ A. Lemma 11.5. Let U be a uniserial module. The following conditions are equivalent: (a) The module U is quasismall. If { fi | i ∈ I } is a summable family of endomorphisms of U whose sum (b) i∈I fi : U → U is the identity of U , then at least one of the endomorphisms fi is an epimorphism. Proof. (a) ⇒ (b) Let U be a quasismall uniserial module and let { fi | i ∈ I } be a summable family of endomorphisms of U with i∈I fi = 1U but no fi an epimorphism. For every index i ∈ I, there is a cyclic submodule Ci of U that contains the image fi (U ) of fi . Let f : U → i∈I Ci be the homomorphism defined by f (x) = (fi (x))i∈I for every x ∈ U and let g : → U be the i∈I Ci homomorphism defined by g((xi )i∈I ) = i∈I xi for every (xi )i∈I ∈ i∈I Ci . The composite morphism gf is the identity morphism 1U , so that U is isomorphic to a direct summand of i∈I Ci . Now, U is quasismall, so that there is a finite subset F of I with U isomorphic to a direct summand of i∈F Ci . It follows that U is finitely generated, hence cyclic. Thus the morphisms in the summable family { fi | i ∈ I } are almost all zero. Since U is uniserial, the equality i∈I fi (U ) = U implies that at least one of the images fi (U ) is U . Thus one of the morphisms fi is an epimorphism, and this contradiction allows us to conclude. (b) ⇒ (a) Assume that (b) holds. Suppose that M = i∈I Ai = U ⊕ C. Let εi : Ai → M , πi : M → Ai , εU : U → M and πU : M → U be the inclusions and the canonical projections corresponding to these decompositions. Then { πU εi πU εU | i ∈ I } is a summable family of endomorphisms of U whose sum is the identity of U . By (b), one of these endomorphisms, πU εj πj εU say, must be an epimorphism. By Proposition 8.37, there exists an index k such that πU εk πk εU : U → U is a monomorphism. Now if πU εj πj εU is an automorphism,

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then the composite morphism of πj εU : U → Aj and (πU εj πj εU )−1 πU εj : Aj → U is the identity of U , so that U is isomorphic to a direct summand of Aj , which concludes the proof. Similarly, if πU εk πk εU is an automorphism of U , then U must be isomorphic to a direct summand of Ak . If πU εj πj εU and πU εk πk εU are not automorphisms of U , then j = k and πU εj πj εU + πU εk πk εU is an automorphism (Lemma 8.31(a)). Then U is a direct summand of Aj ⊕ Ak . This shows that U is quasismall.  Lemma 11.6. The following conditions are equivalent for a uniserial module U . (a) The module U is not quasismall. (b) U is countably generated, and for any element x ∈ U , there exists an endomorphism f of U that is not an automorphism with f (x) = x. Proof. (a) ⇒ (b) We have already remarked that U non-quasismall implies U countably generated (Proposition 11.4). If (b) does not hold, then there exists an element x0 ∈ U such that for every endomorphism f of U , f (x0 ) = x0 implies that f is an automorphism of U . We will prove that U is quasismall by making use of 11.5. If { fi | i ∈ I } is a summable family of endomorphisms of U with sum Lemma of I such that fi (x0 ) = 0 for f i∈I i = 1U , then there exists a finite subset Fx0 every i ∈ I \ Fx0 . Then i∈Fx fi (x0 ) = x0 , so that i∈Fx fi : U → U is an auto0 0 morphism of U . Hence i∈Fx fi (U ) = U , therefore one of the submodules fi (U ) 0 is U because U is couniform. Thus one of the morphisms fi is an epimorphism, and U is quasismall by Lemma 11.5. (b) ⇒ (a) Suppose that (b) holds, so that U has a countable set of generators { xn | n ≥ 0 }. Without loss of generality we can suppose 0 ⊂ x0 R ⊂ x1 R ⊂ x2 R ⊂ · · · . From (b), we get that for every n ≥ 0 there exists an endomorphism fn of U that is not an automorphism with fn (xn ) = xn . Now, U is uniserial and fn is the identity on the nonzero submodule xn R of U . Thus all the mappings fn are monomorphisms, but not epimorphisms. Define a family of endomorphisms gn , n ≥ 0, of U , setting g0 := f0 and gn := fn − fn−1 for n ≥ 1. Then { gn | a summable family of endomorphisms of U that are not n ≥ 0 } turns out to be epimorphisms and with n≥0 gn = 1U . Thus U is non-quasismall by Lemma 11.5.  Corollary 11.7. Every nonzero homomorphic image of a uniserial non-quasismall module is a uniserial non-quasismall module. In particular, if U and V are uniserial modules and [U ]e = [V ]e , then U is quasismall if and only if V is quasismall. Proof. Let U be a uniserial non-quasismall module and let A be a proper submodule of U . We must prove that the uniserial module U/A is not quasismall. For every x ∈ U there exists fx : U → U that is not onto but with fx (x) = x. Thus for every nonzero element x + A of U/A, fx induces an endomorphism of U/A that is not onto and that sends x + A to x + A. Thus U/A is non-quasismall by Lemma 11.6.

11.1. Small, ℵ0 -Small, Quasismall Modules

383

In particular, two modules with the same epigeny class are homomorphic  images of each other. In the next proposition, we show that the existence of non-quasismall uniserial modules is the real obstruction to the existence of a bijection that preserves epigeny classes. Proposition 11.8. Let N be a uniserial R-module that is not quasismall. Then there exists a countable family { An | n ≥ 1 } of uniserial R-modules such that N ⊕ (⊕n≥1 An ) ∼ = ⊕n≥1 An and [An ]e = [N ]e for every n ≥ 1. Proof. Let N be a uniserial R-module that is not quasismall. By definition of quasismall, there is a family { Mi | i ∈ I } of modules with the property that N is isomorphic to a direct summand of ⊕i∈I Mi , but NR is not isomorphic to any direct summand of ⊕i∈F Mi for every finite subset F of I. Hence there exist homomorphisms f : N → ⊕i∈I Mi and g : ⊕i∈I Mi → N such that gf = 1N . For every index j ∈ I, let εj : Mj → ⊕i∈I Mi and πj : ⊕i∈I Mi → Mj denote the corresponding embedding and canonical projection, respectively. By Proposition 11.4, the module N has a countable set { xn | n ≥ 0 } of generators. Since finitely generated modules are quasismall, the module N is not cyclic. Hence we may assume without loss of generality that the chain of cyclic submodules 0 = x0 R ⊂ x1 R ⊂ x2 R ⊂ · · · is strictly increasing. We claim that there exist a countable chain 0 = A0 ⊆ A1 ⊆ A2 ⊆ · · · of cyclic submodules of N and endomorphisms fn , n ≥ 0, of N satisfying the following three conditions: (a) xn ∈ An for every n ≥ 0; (b) fn (x) = x for every x ∈ An−1 and every n ≥ 1; (c) fn (N ) ⊆ An for every n ≥ 0. The proof of the claim will be by induction on n. For n = 0, set A0 := 0 and f0 := 0. Suppose n ≥ 1 and that A0 , . . . , An−1 , f0 , . . . , fn−1 with the required properties have been defined. The submodule An−1 of N is cyclic, but N is not cyclic. Thus there exists y ∈ N , y ∈ / An−1 . Since f (y) ∈ ⊕i∈I Mi , there exists a finite subset F of I such that f (y) = i∈F εi πi f (y). Then y = gf (y) = i∈F gεi πi f (y). Set fn := i∈F gεi πi f . From fn (y) = y, it follows that fn ∈ End(N ) is a monomorphism and that fn satisfies property (b) (because An−1 ⊆ yR). If fn (N ) = N , then fn is an automorphism of N , so the composite mapping of ⊕i∈F πi f : N → ⊕i∈F Mi

and

⊕i∈F fn−1 gεi : ⊕i∈F Mi → N

is the identity mapping of N . Hence N is isomorphic to a direct summand of ⊕i∈F Mi , a contradiction. This shows that fn (N ) ⊂ N , whence fn (N ) + xn R ⊂ N . Therefore there exists a cyclic submodule An of N with fn (N ) + xn R ⊆ An . This concludes the proof of the claim.

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Now, N is not cyclic and the submodules An are cyclic, so that [An ]e = [N ]e for every n ≥ 1. Set gn := fn − fn−1 for every n ≥ 1. Let g : N → ⊕n≥1 An be the mapping defined by g(x) = (gn (x))n≥1 . The mapping g is well defined, because gn (N ) ⊆ An for n ≥ 1 and gn (x) = 0 for every x ∈ An−2 , n ≥ 2, so that for every element x ∈ N one has that gn (x) = 0 for almost all n. Let h : ⊕n≥1 An → N An . Then be the homomorphism defined by h((xn )n≥1 ) = n≥1 xn for every xn ∈ hg = 1N , because for every x ∈ N , there exists n with x ∈ An , since N = n≥1 An , n+1 so that x ∈ An implies hg(x) = i=1 gi (x) = fn+1 (x) − f0 (x) = x. Hence, if ∼ ⊕n≥1 An . K = ker h, we get N ⊕ K = Let ε : ⊕n≥1 An → ⊕n≥1 An be the homomorphism defined by ε(y1 , y2 , y3 , . . . ) = (y1 , y2 − y1 , y3 − y2 , y4 − y3 , . . . ). Then hε = 0, so that the image of ε is contained in ker h = K. Conversely, if z = (z1 , z2 , z3 , . . . ) ∈ ⊕n≥1 An and z ∈ K, then there exists n with zi = 0 for every n−1 i ≥ n and i=1 zi = 0. If yj = ji=1 zi for every j ≥ 1, then y = (y1 , y2 , y3 , . . . ) ∈ ⊕n≥1 An and ε(y) = z. Hence the image of ε is equal to K. Finally, it is easily ∼ ⊕n≥1 An . verified that ε is a monomorphism. Hence K =  The existence of uniserial modules that are not quasismall will be discussed in Sections 11.4–11.7.

11.2 A Further Necessary Condition for Isomorphism of Direct Sums of Families of Uniserial Modules Let { Ui | i ∈ I } and { Vj | j ∈ J } be two families of uniserial nonzero modules ∼ ⊕j∈J Vj . In this section, we will show that over a ring R and suppose that ⊕i∈I Ui = there is a one-to-one correspondence preserving the epigeny classes of quasismall uniserial modules. We begin with two technical lemmas. Lemma 11.9. If C1 , C2 , . . . , Cn are couniform modules over an arbitrary ring R, n then for every couniform direct summand A of ⊕j=1 Cj there exists an index k ∈ J with [A]e = [Ck ]e . Proof. Assume M := ⊕nj=1 Cj = A⊕B. If εj : Cj → M , εA : A → M , πj : M → Cj , πA : M → A are the embeddings and the canonical projections relative to the two direct-sum decompositions of M , then ⎞ ⎛ n n   1A = πA εA = πA ⎝ εj πj ⎠ εA = πA εj πj εA , j=1

j=1

11.2. A Necessary Condition for Families of Uniserial Modules

so that

n 

385

πA εj πj εA (A) = A.

j=1

Since the module A is couniform, there exists an index k = 1, 2, . . . , n such that πA εk πk εA (A) = A. Thus πA εk πk εA is an epimorphism, so both πA εk : Ck → A and πk εA : A → Ck are epimorphisms by Lemma 8.27(b). Therefore [A]e = [Ck ]e .  Lemma 11.10. Let A1 , . . . , An be biuniform quasismall right modules over a ring R and let { Cj | j ∈ J } be a family of biuniform R-modules. Suppose that the module A1 ⊕ · · · ⊕ An is isomorphic to a direct summand of the direct sum ⊕j∈J Cj . Then there exist n distinct indices j1 , . . . , jn ∈ J such that [Ai ]e = [Cjt ]e for every t = 1, . . . , n. ∼ ⊕j∈J Cj . The proof of the lemma is by Proof. Suppose A1 ⊕ · · · ⊕ An ⊕ B = induction on n. The case n = 0 is trivial, so suppose n ≥ 1. There exists k ∈ J with [A1 ]m = [Ck ]m (Proposition 8.37). Since the module A1 is quasismall, there is a finite subset F of J with A1 isomorphic to a direct summand of ⊕j∈F Cj . By Lemma 11.9, there is an index j1 ∈ F such that [A1 ]e = [Cj1 ]e . If k = j1 , then ∼ ⊕j∈J, j=j Cj . Thus the result follows from A1 ∼ = Cj1 , so that A2 ⊕ · · · ⊕ An ⊕ B = 1 the inductive hypothesis. If k = j1 , then A1 ⊕ D ∼ = Ck ⊕ Cj1 for some biuniform module D, where [D]m = [Cj1 ]m and [D]e = [Ck ]e . Then A1 ⊕ · · · ⊕ An ⊕ B ∼ = ⊕j∈J Cj ∼ = A1 ⊕ D ⊕ (⊕j∈J,

j=k,j1 Cj ) .

∼ D ⊕ (⊕j∈J, j=k,j Cj ). But [D]e = [Ck ]e , so that Therefore A2 ⊕ · · · ⊕ An ⊕ B = 1 by the inductive hypothesis, there are n − 1 distinct indices j2 , . . . , jn ∈ J \ {j1 } with [Ai ]e = [Cji ]e for every i = 2, . . . , n. This shows that the indices j1 , . . . , jn have the required property.  We are ready to prove that, for the class of quasismall uniserial modules, a bijection preserving epigeny classes exists. Theorem 11.11. Let R be an arbitrary ring. If { Ui | i ∈ I } and { Vj | j ∈ J } are two families of nonzero uniserial right modules over R such that ⊕i∈I Ui ∼ = ⊕j∈J Vj , I  := { i ∈ I | Ui is quasismall }, and J  := { j ∈ J | Vj is quasismall }, then there exists a bijection τ  : I  → J  with [Ui ]e = [Vτ  (i) ]e for every i ∈ I  . Proof. Suppose M := ⊕i∈I Ui = ⊕j∈J Vj . For every nonzero uniserial quasismall direct summand Z of M , set IZ := { i ∈ I  | [Ui ]e = [Z]e }

and JZ := { j ∈ J  | [Vj ]e = [Z]e }.

Since the module Z is quasismall, there exists a finite subset F of I with Z isomorphic to a direct summand of ⊕i∈F Ui . From Lemma 11.9, it follows that there

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exists k ∈ F with [Z]e = [Uk ]e . By Corollary 11.7, the module Uk is quasismall. Thus k ∈ IZ . In particular, the set IZ is nonempty. Similarly, the set JZ is nonempty. Clearly, as Z ranges over the set of all nonzero uniserial quasismall direct summands of M , the sets IZ form a partition of I  and the sets JZ form a partition of the set J  . Hence, in order to prove the theorem, it suffices to prove that |IZ | = |JZ | for every nonzero uniserial quasismall direct summand Z of M . If either IZ or JZ is a finite set and |IZ | = |JZ |, we may assume by symmetry that |IZ | < |JZ |. Set n := |IZ |. Then there are n + 1 distinct indices j1 , . . . , jn+1 in JZ with [Vjt ]e = [Z]e . From Lemma 11.10, we get that [Ui ]e = [Z]e for at least n + 1 distinct indices i ∈ I. The n + 1 modules Ui are quasismall by Corollary 11.7, so that |IZ | ≥ n + 1, which is a contradiction. This proves that |IZ | = |JZ | if either IZ or JZ is a finite set. If IZ and JZ are both infinite sets, it suffices to prove that |JZ | ≤ |IZ | by symmetry. Let εk : Uk → ⊕i∈I Ui and e : V → ⊕j∈J Vj denote the embeddings, and let πk : ⊕i∈I Ui → Uk and p : ⊕j∈J Vj → V denote the canonical projections. For every index t ∈ I, set A(t) := { j ∈ J | πt ej pj εt : Ut → Ut is an epimorphism }. Let us prove that every set A(t), t ∈ I, is countable. To see this, recall that every uniserial module Ut is either small or countably generated (Proposition 11.4). Thus there exists a countable subset C of J with Ut ⊆ ⊕j∈C Vj . Then πt e p εt (Ut ) = 0 for every ∈ J \ C. Therefore the set A(t) ⊆ C is countable.

Now we will show that J  ⊆ t∈I A(t). To prove this, suppose the contrary. Then there exists j ∈ J  such that j ∈ / A(t) for every t ∈ I. The morphism πt ej pj εt : Ut → Ut is not an epimorphism for every t ∈ I. By Lemma 8.27(b), pj εt πt ej : Vj → Vj is not an epimorphism for every t ∈ I. Hence, for every index t ∈ I there exists a cyclic proper submodule Ct of Vj such that pj εt πt ej (Vj ) ⊆ Ct . For every x ∈ Vj , there exist only finitely many indices t ∈ I such that πt ej (x) = 0. Thus it is possible to define a homomorphism ψ : Vj → ⊕t∈I Ct by setting ψ(x) = (p j εt πt ej (x))t∈I . Let ω : ⊕t∈I Ct → Vj be the morphism defined by ω((ct )t∈I ) = t∈I ct . Then ωψ = 1Vj , so that Vj turns out to be isomorphic to a direct summand of ⊕t∈I Ct . Since j ∈ J  , the module Vj is quasismall. Hence there exists a finite subset F ⊆ I with Vj isomorphic to a direct summand of ⊕t∈F Ct . In particular, Vj is a finitely generated module, and so a cyclic module. Let v be a generator of Vj . There exists a finite v ∈ ⊕t∈G Ut . subset G ⊆ I with Then t∈G pj εt πt ej (v) = v implies that v ∈ t∈G pj εt πt ej (Vj ) ⊆ t∈G Ct ⊂ Vj , a contradiction. This proves that J  ⊆ t∈I A(t), as we wanted.

Now, JZ is a subset of t∈IZ A(t), because if j ∈ JZ , then, as we have seen in the previous paragraph, there exists t ∈ I with j ∈ A(t). The mapping πt ej pj εt is an epimorphism, so that [Ut ]e = [Vj ]e = [Z]e (Lemma 8.27(b)). The uniserial module Z is quasismall, so that Ut is quasismall by Corollary 11.7, that is, t ∈ IZ . It follows that  |JZ | ≤ ℵ0 |IZ | = |IZ |.

11.3. The Submodules Um and Ue

387

11.3 The Submodules Um and Ue The content of this section is due to Pˇr´ıhoda [Pˇr´ıhoda 2006b] and [Pˇr´ıhoda 2006c]. Let U be a nonzero uniserial right module over a ring R. Let Um denote the intersection of all the submodules of U that are isomorphic to U . Recall that by Theorem 8.28, the endomorphisms of U that are not injective form a completely prime two-sided ideal I of the endomorphism ring E = End(U ) of U . Clearly, Um is the intersection of the submodules f (U ) as f ranges over the multiplicatively closed subset E \ I of E. In particular, Um = U if and only if every injective endomorphism of U is an automorphism of U . Lemma 11.12. Let U be a nonzero uniserial right module over a ring R. Then: (a) For every submodule V of U , [V ]m = [U ]m if and only if V = U or Um ⊂ V . (b) soc(U/Um ) = 0. Proof. (a) (⇒) Assume [V ]m = [U ]m and V = U . Then there exists an injective endomorphism f : U → U with f (U ) ⊆ V . Then Um ⊆ f 2 (U ) ⊂ f (U ) ⊆ V . (⇐) Assume Um ⊂ V . Then there exists an injective endomorphism f of U with f (U ) ⊆ V . Thus there is a monomorphism U → V . The inclusion is a monomorphism V → U . Thus [V ]m = [U ]m . (b) This is trivial when Um = U . If Um ⊂ U and soc(U/Um ) = 0, then soc(U/Um ) = 0 implies that there exists a submodule V of U containing Um and with V /Um simple. Also, from Um ⊂ U , we get that there exists an injective endomorphism of U that is not surjective, and Um turns out to be the intersection of the images of all the injective endomorphisms f : U → U that are not surjective. For every such f , f (U ) ⊂ U implies f 2 (U ) ⊂ f (U ). Thus the set of all the images f (U ) of all these endomorphisms f does not have a least element with respect to inclusion. Thus f (U ) ⊃ Um , so that f (U ) ⊇ V , hence Um ⊇ V , contradiction.  Lemma 11.13. Let U be a uniserial right module over a ring R. Assume that there exists an endomorphism of U that is injective but not surjective, so that Um is a proper submodule of U . Then: (a) A submodule of U has the same monogeny class as U if and only if it properly contains Um . (b) Um is the largest submodule of U with monogeny class different from that of U . (c) Let W be a uniserial right R-module and f : U → W a monomorphism. Then f (Um ) ⊆ Wm . Moreover, f (Um ) = Wm if and only if [U ]m = [W ]m . Proof. (a) This follows immediately from Lemma 11.12(a) and the fact that Um ⊂ U . (b) follows immediately from (a). (c) If f (U ) ⊆ Wm , then Um ⊂ U implies f (Um ) ⊂ Wm . If [U ]m = [W ]m , then [f (U )]m = [W ]m , so that from Lemma 11.12(a), we get that Wm ⊂ f (U ), a contradiction. This proves (c) in the case f (U ) ⊆ Wm .

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Thus we can assume that f (U ) ⊃ Wm . Lemma 11.12(a) implies that [W ]m = [f (U )]m = [U ]m . It remains to prove that f (Um ) = Wm . By (b), f (Um ) is the largest submodule of f (U ) with monogeny class different from that of f (U ). Hence it suffices to show that Wm is the largest submodule of W with monogeny class different from that of W . This is true by Lemma 11.12(a).  Corollary 11.14. For every uniserial module U , Um is a fully invariant submodule of U . Proof. If Um = U , this is trivial. If Um ⊂ U , we can apply Lemma 11.13(c), getting that f (Um ) ⊆ Um for every injective endomorphism f of U . If g is an endomorphism of U that is not injective, then g − 1U and 1U are injective endomorphisms of  U , so that g(Um ) ⊆ (g − 1U )(Um ) + 1U (Um ) ⊆ Um . Now we can introduce the dual notion. Let U be a uniserial right module over a ring R. Let Ue denote the sum of all the submodules U  of U with U/U  ∼ = U . By Theorem 8.28, the endomorphisms of U that are not surjective form a completely prime two-sided ideal K of the endomorphism ring E = End(U ) of U . Thus Ue is the sum of the kernels ker(f ), where f ranges over the multiplicatively closed subset E \ K of E. In particular, Ue = 0 if and only if every surjective endomorphism of U is an automorphism of U . The following lemma is immediate: Lemma 11.15. Let U be a uniserial right module over a ring R. Then: (a) For every element u ∈ U , u ∈ Ue if and only if f (u) = 0 for some surjective endomorphism f of U . (b) If X is a proper submodule of Ue , then there exists a surjective endomorphism f of U such that f (X) = 0. Remark 11.16. If U is a uniserial module and f is an endomorphism of U that is surjective but not injective, then ker f ⊂ ker f 2 . To see this, consider the isomorphism f : U/ ker f → U , induced by f and defined by f (u) = f (u) + ker f for every −1 −1 u ∈ U . Then ker f = 0 implies f (ker f ) = 0. But f (ker f ) = ker f 2 / ker f . Lemma 11.17. Let U be a uniserial right module over a ring R. Then: (a) For every submodule V of U , [U/V ]e = [U ]e if and only if V = 0 or V ⊂ Ue . (b) rad(Ue ) = 0. In particular, either Ue = 0 or Ue is not finitely generated. Proof. (a) (⇒) Suppose [U/V ]e = [U ]e and V = 0. Then there exists a surjective endomorphism f : U → U with V ⊆ ker f . By the previous remark, V ⊆ ker f ⊂ ker f 2 ⊆ Ue . (⇐) Assume V ⊂ Ue , so that there exists a surjective endomorphism f of U with V ⊆ ker f . There is an epimorphism U/V → U . Since the canonical projection U → U/V is an epimorphism, we get that [U/V ]e = [U ]e .

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(b) The case Ue = 0 is trivial. Suppose Ue = 0 and rad(Ue ) = 0, so that Ue has a maximal submodule, and Ue is cyclic. Let u be a generator of Ue . Then f (u) = 0 for some f ∈ End(U ) surjective but not injective. Thus ker f = uR. By the previous remark, Ue ⊇ ker f 2 ⊃ ker f = uR ⊇ Ue , contradiction.  Lemma 11.18. Let U be a uniserial right module over a ring R. Assume that there exists an endomorphism of U that is surjective but not injective, so that Ue = 0. Then: (a) For every submodule V of U , [U/V ]e = [U ]e if and only if V ⊂ Ue . (b) Ue is the smallest of the submodules V of U with [U/V ]e = [U ]e . (c) Let W be a uniserial right R-module and f : W → U an epimorphism. Then We ⊆ f −1 (Ue ) and f (We ) ⊆ Ue . Moreover, if [U ]e = [W ]e , then We = f −1 (Ue ) and f (We ) = Ue . Proof. (a) follows immediately from Lemma 11.17 and the fact that Ue = 0. (b) follows from (a). (c) Let W be a uniserial right R-module and f : W → U an epimorphism. We will first prove that We ⊆ f −1 (Ue ). Assume the contrary, so that f −1 (Ue ) ⊂ We . Then ker f ⊂ We . Lemma 11.17(a) implies that [W/ ker f ]e = [W ]e , so that [U ]e = [W ]e . We have W/f −1 (Ue ) ∼ = U/Ue . Now f −1 (Ue ) ⊂ We and Lemma 11.17(a) −1 imply that [W/f (Ue )]e = [W ]e , hence [U ]e = [U/Ue ]e , which contradicts Lemma 11.17(a). This contradiction proves that We ⊆ f −1 (Ue ). To prove that f (We ) ⊆ Ue , it suffices to notice that We ⊆ f −1 (Ue ) implies f (We ) ⊆ f (f −1 (Ue )) = Ue , because f is an onto mapping. We will now prove that [U ]e = [W ]e implies We = f −1 (Ue ). This is trivially true if f is an isomorphism, so that we can assume that f is not a monomorphism. In this case, in order to prove that [U ]e = [W ]e implies We = f −1 (Ue ), assume the contrary, that is, suppose that [U ]e = [W ]e and We ⊂ f −1 (Ue ). Now [W/ ker f ]e = [U ]e = [W ]e , so that by Lemma 11.17(a), either ker f = 0 or ker f ⊂ We . Since f is not a monomorphism, it follows that ker f ⊂ We . Thus ker f ⊂ We ⊂ f −1 (Ue ) implies 0 = f (We ) ⊂ Ue . By Lemma 11.17(a) again, we get that [U/f (We )]e = [U ]e . Now ker f ⊂ We implies W/We ∼ = U/f (We ), hence [W/We ]e = [U ]e = [W ]e and W = 0. This contradicts Lemma 11.17(a). Thus [U ]e = [W ]e implies We = f −1 (Ue ). Finally, We = f −1 (Ue ) implies f (We ) = f (f −1 (Ue )) = Ue because f is an onto mapping.  Corollary 11.19. For every uniserial module U , Ue is a fully invariant submodule of U . Proof. The statement is trivial when Ue = 0. If Ue = 0, there is a surjective endomorphism that is not a monomorphism, so that we can apply Lemma 11.18(c).

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Thus f (Ue ) ⊆ Ue for every surjective endomorphism f of U . If g is an endomorphism of U that is not surjective, then g − 1U and 1U are surjective endomorphisms of U , whence g(Ue ) ⊆ (g − 1U )(Ue ) + 1U (Ue ) ⊆ Ue . 

11.4 Existence of Non-quasismall Uniserial Modules The existence of non-quasismall uniserial modules was proved by G. Puninski in [Puninski 2001a]. His proof made use of powerful model-theoretical methods. Here we will give the proof given by Pˇr´ıhoda in [Pˇr´ıhoda 2006c]. Proposition 11.20. The following conditions are equivalent for a uniserial right module U over a ring R. (a) The module U is not quasismall. (b) The module U is countably generated and Um ⊂ U = Ue . Proof. (a) ⇒ (b) Let U be a uniserial module that is not quasismall. By Lemma 11.6, U is countably generated, and for every element x ∈ U , there exists an endomorphism f of U that is not an automorphism and with f (x) = x. Thus, x is in the kernel of the epimorphism 1U − f : U → U . It follows that U = Ue . Finally, the endomorphism ring of U is not local (Proposition 11.2), so that there exist injective endomorphisms of U that are not automorphisms (Theorem 8.28). Therefore Um ⊂ U . (b) ⇒ (a) Let U be a countably generated uniserial module with Um ⊂ U = Ue . In order to prove that U is not quasismall, we will apply Lemma 11.6. That is, we will show that for any element x ∈ U , there exists an endomorphism f of U that is not an automorphism and with f (x) = x. This is trivially true for x = 0. If x ∈ U is nonzero, then x ∈ Ue , so that there exists an epimorphism g : U → U with x ∈ ker g. In particular, g is not a monomorphism. Since Um ⊂ U , there exists a monomorphism h : U → U that is not an epimorphism. By Lemma 8.31(a), g +h is an automorphism of U and (g + h)(x) = h(x). Thus x = (g + h)−1 h(x). Therefore (g + h)−1 h : U → U is not an automorphism but maps x to x, as desired.  Lemma 11.21. Let U, V be uniserial right R-modules such that [U ]m = [V ]m and Um ⊂ Ue . Then: (a) Vm ⊂ Ve . (b) U is the union of its proper submodules that are isomorphic to V . Proof. From Um ⊂ Ue ⊆ U , it follows that [Ue ]m = [U ]m (Lemma 11.12(a)). Since [U ]m = [V ]m and Um ⊂ Ue ⊆ U , we can suppose that Um ⊂ V ⊂ Ue . (a) From V ⊂ Ue , we have that there exists a surjective endomorphism f of U with f (V ) = 0 (Lemma 11.15(b)). We claim that W := f −1 (V ) is isomorphic to V . In order to prove the claim, notice that f induces an epimorphism f  : W → V . Also, W ⊆ U and [V ]m = [U ]m , so that there is a monomorphism W → V . Thus W ∼ = V by Lemma 8.31(a). This proves the claim. Now, Um ⊂ V ⊂ W ⊂ U

11.4. Existence of Non-quasismall Uniserial Modules

391

and Um is the greatest submodule of U not in the same monogeny class as U (Lemma 11.12(a)). Thus [V ]m = [W ]m = [U ]m and Vm = Wm = Um . Let g : V → W be an isomorphism and gf  : W → W the composite morphism, so that gf  is a surjective endomorphism of W with Wm ⊂ V ⊆ ker(gf  ). Thus Wm ⊂ We . The conclusion follows from the isomorphism W ∼ =V. (b) Let X be the union of all the proper submodules of Ue that are isomorphic to V . Assume by contradiction that X = Ue . By Lemma 11.15(b), there exists a surjective endomorphism f of U with f (X) = 0, so that in the proof of (a) we can suppose that the surjective endomorphism f of U has not only the property that f (V ) = 0, but also the property that f (X) = 0. As in the proof of (a), W := f −1 (V ) is isomorphic to V . Let us prove that W ⊆ Ue . Consider the surjective endomorphism f 2 of U . Then f 2 (W ) = f (f (f −1 (V ))) ⊆ f (V ) = 0. Thus W ⊆ Ue . Let us show that W ⊂ Ue . Since V ⊂ Ue , there exists an element x ∈ Ue \ V . Now f is onto, so that there is y ∈ U with f (y) = x. From x ∈ Ue , we get that there exists an epimorphism h : U → U with h(x) = 0. Then hf is a surjective endomorphism of U and hf (y) = h(x) = 0. Hence y ∈ Ue and y ∈ / W (because y ∈ W implies f (y) ∈ V , that is, x ∈ V , a contradiction). Therefore W ⊂ Ue . We have thus shown that W is a proper submodule of Ue isomorphic to V , hence W ⊆ X. Thus f (W ) = 0, that is, f (f −1 (V )) = 0. Since f is onto, we get  that V = 0. This contradiction proves that X = Ue . Lemma 11.22. The following conditions are equivalent for a ring R. (a) There exists a uniserial right R-module that is not quasismall. (b) There exists a uniserial right R-module U with Um ⊂ Ue . (c) There exists a cyclic uniserial right R-module U with Um ⊂ Ue . Proof. (a) ⇒ (c) Let W be a uniserial right R-module that is not quasismall. From Proposition 11.20, we have that Wm ⊂ W = We . Let x be an element in W but not in Wm , and let U be the cyclic submodule of W generated by x. Then [U ]m = [W ]m . By Lemma 11.21(a), we get that Um ⊂ Ue . (c) ⇒ (b) is trivial. (b) ⇒ (a) Let U be a uniserial R-module with Um ⊂ Ue . Let u be an element in Ue but not in Um , so that there exists an endomorphism f of U that is an epimorphism with f (u) = 0. By induction, it is easy to construct a sequence u0 , u1 , u2 of elements of U with u0 = u and f (ui ) = ui−1 for every i ≥ 1. Then f i (ui−1 ) = 0 and f i (ui ) = u = 0 for every i, so that the kernels of the morphisms fi form a strictly ascending chain of submodules of U . Thus the cyclic submodules ui R form a strictly ascending chain of submodules of U . Set U  := i≥0 ui R, so that f induces a surjective endomorphism f  of U  . Now Um ⊂ U  ⊆ U and Um is the greatest submodule of U not in the same monogeny class as U (Lemma 11.12(a)).   ⊂ U  , and U  = Ue because for = Um . Thus Um Thus [U  ]m = [U ]m and Um  every element x ∈ U there exists an i ≥ 1 with f i (x) = 0. Therefore U  is not  quasismall by Proposition 11.20.

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Lemma 11.23. Let R be a right chain ring and let L be a proper right ideal of R. Then (RR /L)e = RL/L. Proof. Suppose L = 0. Then every epimorphism RR → RR splits because RR is projective. But the uniserial module RR is indecomposable. Thus every epimorphism RR → RR is an automorphism of RR . Hence Re = 0, and the lemma is proved. Thus we can assume L = 0. In order to prove that RL/L ⊆ (RR /L)e , notice that every element of R is the sum of at most two invertible elements (if a is not invertible, then a is the sum of the two invertible elements 1 and a − 1.) Thus it suffices to show that uL + L/L ⊆ (RR /L)e for every u ∈ U (R). This is trivial if uL ⊆ L. If uL ⊃ L, then L ⊃ u−1 L, so that left multiplication by u−1 defines an epimorphism RR /L → RR /L, whose kernel is uL/L. Thus uL/L ⊆ (RR /L)e . This proves that RL/L ⊆ (RR /L)e . In order to prove that (RR /L)e ⊆ RL/L, we must prove that the kernel of any surjective endomorphism of RR /L is contained in RL/L. Now every epimorphism RR /L → RR /L lifts to an epimorphism RR → RR , hence is given by left multiplication by some element v ∈ U (R) such that vL ⊆ L. The kernel of such an epimorphism RR /L → RR /L is v −1 L + L/L. Hence, in order to see that (RR /L)e ⊆ RL/L, it suffices to notice that v −1 L + L/L ⊆ RL/L.  Proposition 11.24. Let U, V be two non-quasismall uniserial right modules over a ∼ V if and only if [U ]m = [V ]m . ring R. Then U = Proof. It suffices to show that [U ]m = [V ]m implies U ∼ = V . Assume [U ]m = [V ]m . We have only to prove that there is an epimorphism f : V → U (Lemma 8.31(a)). We claim that U is the union of its submodules isomorphic to V . In order to prove this, assume the contrary, so that the union X of all submodules of U isomorphic to V is a proper submodule of U . Let g : V → U be a monomorphism, ∼ f (V ) ⊆ X. By Proposition 11.20, U = Ue ⊃ X, so that there is an so V = epimorphism h : U → U with ker h ⊇ X. Then V ⊆ X ⊆ ker h ⊂ h−1 (V ) ⊆ U and [U ]m = [V ]m . Therefore [h−1 (V )]m = [V ]m . Also, there is an epimorphism h−1 (V ) → V induced by h. By Lemma 8.31(a), h−1 (V ) ∼ = V , so that X ⊇ h−1 (V ), −1 which is a contradiction because X ⊂ h (V ). This contradiction proves the claim. Since U is countably generated, U is the union of a countable chain of submodules isomorphic to V . If one of these submodules is the improper submodule of U , then U ∼ = V , and we are done. If all these countably many submodules are proper submodules of U , we can suppose without loss of generality that the countably ascending chain is strictly ascending, so that there are submodules

∼ V for every n ≥ 1. X1 ⊂ X2 ⊂ X3 ⊂ · · · ⊂ U with U = n≥1 Xn and Xn = Thus there are monomorphisms n : V → U with n (V ) = Xn for every n ≥ 1. Let { wn | n ≥ 1 } be a set of generators of V . Let f0 : V → V be the identity automorphism of V and let v0 be the zero element of V . We now define by induction pairs (fn , vn ) such that for every n ≥ 1: (a) vn is an element of V such that vn R ⊇ wn R + vn−1 R;

11.4. Existence of Non-quasismall Uniserial Modules

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(b) fn : V → V is an epimorphism with fn (wn R + vn−1 R) = 0; and / Xn . (c) n+1 fn (vn ) ∈ The pair (f0 , v0 ) has already been defined. Assume that the pair (vn−1 , fn−1 ) has been constructed for some n ≥ 1. Since V is not finitely generated, wn R + vn−1 R is properly contained in V . But Ve = V , so that there exists an epimorphism fn : V → V such that wn R + vn−1 R ⊆ ker fn . Now, n+1 fn (V ) = Xn+1 ⊃ Xn ; hence there exists vn ∈ V with n+1 fn (vn ) ∈ / Xn . This completes the construction of the pair (fn , vn ) for every n ≥ 0. Notice that for n ≥ 1, wn R + vn−1 R ⊆ ker fn and n+1 fn (vn ) ∈ / Xn , so fn (vn ) = 0 and wn R + vn−1 R ⊆ vn R. Thus (a) holds. Properties (b) and (c) also hold for every n ≥ 1. Now { n+1 fn | n ≥ 1 } is a summable family of morphisms of V into U , because fm (vn ) = 0 for m > n ≥ 1 and the vectors vn generate V . If f : V → U is the sum of the family, then n n−1 f (vn ) = i=1 i+1 fi (vn ), and i=1 i+1 fi (vn ) ∈ Xn , but n+1 fn (vn ) ∈ / Xn . Thus  f (vn ) ⊃ Xn . Therefore f is an epimorphism. Lemma 11.25. Let R be a right chain ring. If x ∈ J(R) and the two-sided ideal RxR generated by x is idempotent, then there exist r ∈ U (R) and s ∈ I such that xr = sx. Proof. The two-sided ideal (RxR)2 is the abelian group generated by all products of the form r1 xr2 xr3 . But the ideals r1 xr2 xr3 R form a chain, so that from x ∈ RxR = (RxR)2 , it follows that there exist r1 , r2 , r3 ∈ R with x ∈ r1 xr2 xr3 R, that is, x = r1 xr2 xt for some t ∈ R. If t ∈ U (R), set s := r1 xr2 and r := t−1 , and we are done. If t ∈ / U (R), then t ∈ J(R), so that 1 − t ∈ U (R). Then r1 xr2 x(1 − t) = r1 xr2 x − x = (r1 xr2 − 1)x. Set u := r1 xr2 − 1, so that u ∈ U (R). Then u−1 r1 xr2 x = x(1 − t)−1 . Hence, if we set s := u−1 r1 xr2 and r := (1 − t)−1 , we are done.  Theorem 11.26. The following conditions are equivalent for a right chain ring R: (a) There exists a uniserial right R-module that is not quasismall. (b) There exists a nonzero element x ∈ J(R) such that the ideal RxR is idempotent. Proof. (b) ⇒ (a) Let I := RxR be an idempotent ideal for some nonzero element x ∈ J(R). By Lemma 11.25, there exist r ∈ U (R) and s ∈ I such that xr = sx. Thus xR = sxR, so that left multiplication by s induces an endomorphism λs of the uniserial module RR /xR. The right annihilator r.annR (s) is a right ideal of R; hence it is comparable with xR. If r.annR (s) ⊇ xR, then sxR = 0, which is a contradiction, since xR = sxR is nonzero. Thus r.annR (s) ⊂ xR. The endomorphism λs is injective, because if a ∈ R and sa ∈ xR, then sa ∈ sxR, so that a − t ∈ r.annR (s) for some t ∈ xR. Thus a − t ∈ r.annR (s) ⊂ xR implies s ∈ xR. This proves that λs is injective. Therefore RR /xR ∼ = sR/xR. So (RR /xR)m ⊆ sR/xR. Since I = I 2 ⊆ IJ(R) ⊆ I, the ideal I is not finitely generated as a right ideal. So sR ⊂ I. It follows that (RR /xR)m ⊂ I/xR. By Lemma 11.23, we know that (RR /xR)e = RxR/xR = I/xR. Thus the uniserial

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module RR /xR satisfies condition (b) of Lemma 11.22, so that there exists a uniserial right R-module that is not quasismall. (a) ⇒ (b) If (a) holds, by Lemma 11.22 there exists a cyclic uniserial right Rmodule U with Um ⊂ Ue . The uniserial module U is isomorphic to RR /L for some proper right ideal L of R. Thus (RR /L)m ⊂ (RR /L)e = RL/L (Lemma 11.23). In particular, RL ⊃ L. Fix an element y ∈ RL \ L. Then y + L ∈ RL/L = (RR /L)e (Lemma 11.23), so that there is a surjective endomorphism of RR /L whose kernel contains y + L. Thus this surjective endomorphism induces an epi∼ RR /yR → RR /L. If we also consider the morphism (RR /L)/(yR + L/L) = canonical projection π : RR /L → RR /yR, it follows that [RR /L]e = [RR /yR]e . Apply Lemma 11.18(c) to the epimorphism π : RR /L → RR /yR, noticing that [RR /L]e = [RR /yR]e implies that there exists an endomorphism of RR /yR that is surjective but not injective. One gets that (RR /L)e = π −1 ((RR /yR)e ), that is, RL/L = RyR/L (apply Lemma 11.23 twice). Hence RL = RyR for every y ∈ RL \ L. Since (RR /L)m ⊂ (RR /L)e = RL/L, there exists a monomorphism RR /L → RL/L. That is, there exists an element x ∈ R such that xs ∈ L if and only if s ∈ L for every s ∈ R. Notice that x ∈ RL \ L. Composing this monomorphism with itself, we see that x2 ∈ / L. Thus RxR = RL ⊆ Rx2 R ⊆ RxRxR = RLRL ⊆ RL = RxR. This proves that RxR = (RxR)2 is an idempotent ideal.  Recall that a chain ring is a ring that is both a right chain ring and a left chain ring. A chain ring R is said to be nearly simple if it has exactly three two-sided ideals and (J(R))2 = 0. Notice that the three two-sided ideals of R are necessarily 0, R, and J(R), and they are all necessarily idempotent (because (J(R))2 is a two-sided ideal; for further detailed information about nearly simple chain rings, see Section 11.11). Notice that nearly simple chain domains are necessarily noncommutative. If R is a nearly simple chain ring, then for every nonzero element x ∈ J(R) the ideal RxR = J(R) is idempotent, so that R has uniserial right modules that are not quasismall (Theorem 11.26).

11.5 Ordered Groups We now adapt to the noncommutative setting what we saw in Section 1.2. We will consider only linear orders on a group G, that is, partial orders ≤ with either a ≤ b or b ≤ a for every a, b ∈ G. We will not consider more general settings. A right order ≤r on a group G is a linear order on the set G such that, for every a, b, c ∈ G, a ≤r b implies ac ≤r bc. Similarly, one defines left orders ≤l , for which a ≤l b implies ca ≤l cb. If G is a group, a subset P of G is called a generalized positive cone if P P ⊆ P , P ∪ P −1 = G, and P ∩ P −1 = {1}. In other words, generalized positive cones are the reduced submonoids P such that for every element a ∈ G, a = 1 implies that either a ∈ P or a−1 ∈ P , but not both. Here P −1 denotes the set of all the inverses a−1 of the elements a of P .

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Similarly to Lemma 1.15, where only abelian groups were considered, for every group G, there is a one-to-one correspondence between the set of all generalized positive cones of G and the set of all right orders on G. This correspondence associates to every right order ≤ on G the positive cone G+ := { x ∈ G | 1 ≤ x }. Conversely, if P is a generalized positive cone of G, the corresponding right order ≤r,P on G is defined, for every a, b ∈ G, by a ≤r,P b if ba−1 ∈ P . Similarly, there is a one-to-one correspondence between the set of generalized positive cones of G and the set of all left orders on G, which associates to every generalized positive cone P of G the left order ≤l defined, for every a, b ∈ G, by a ≤l,P b if a−1 b ∈ P . It follows that there is a one-to-one correspondence between the set of all right orders and the set of all left orders on G. In this correspondence, the left order ≤l corresponds to the right order ≤r , where for every a, b ∈ G, a ≤l b if and only if b−1 ≤r a−1 . An ordered group is a group G equipped with an order ≤ that is both a left order and a right order. Proposition 11.27. Let G be a group, P a generalized positive cone of G, and ≤r,P the right order on G corresponding to P . Then ≤r,P is a left order if and only if P is a normal submonoid of G (that is, aP a−1 ⊆ P for every a ∈ G). Proof. Let ≤r,P be a left order. Then p · 1−1 ∈ P for every element p of P , that is, 1 ≤r,P p. But ≤r,P is a left order, so that a ≤r ap for every a ∈ G and p ∈ P . This is equivalent to apa−1 ∈ P , for every a ∈ G and p ∈ P . Thus P is a normal submonoid. Conversely, suppose P normal. If a, b ∈ G and a ≤r,P b, then ba−1 ∈ P . Since P is normal, c(ba−1 )c−1 = cb(ca)−1 ∈ P for every c ∈ G. Thus ca ≤r,P cb for every c ∈ G, that is, ≤r is a left order. 

11.6 Skew Polynomials and the Ore Condition Let σ : R → R be an endomorphism of a ring R. The skew polynomial ring R[x; σ] is the ring whose elements are all polynomials r0 + r1 x + r2 x2 + · · · + rn xn with the usual addition, so that R[x; σ] = ⊕n Rxn is the free left R-module with free set of generators { xn | n ≥ 0 }, and with multiplication defined via the relations xr = σ(r)x, extending by R-bilinearity. The ring R embeds into the ring R[x; σ], and is therefore identified with a subring of R. The Hilbert basis theorem holds for these skew polynomials also. That is, if R is a right Noetherian ring, then R[x; σ] is a right Noetherian ring ([Passman, Theorem 11.1] or [Goodearl and Warfield, Corollary 1.15]). Similarly, if σ : R → R is an automorphism of a ring R, the skew Laurent ring R[x±1 ; σ] is the ring whose elements are formal expressions rm xm + rm+1 xm+1 + · · · + r−1 x−1 + r0 + r1 x + r2 x2 + · · · + rn xn , where m ≤ 0 ≤ n are integers and

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the coefficients ri are elements of R, so that R[x±1 ; σ] = ⊕n∈Z Rxn is the free left R-module with free set of generators { xn | n ∈ Z }. Now, xn r = σ n (r)xn for every integer n. Again, the Hilbert basis theorem holds, that is: Proposition 11.28 ([Goodearl and Warfield, Corollary 1.15]). If R is a right (left) Noetherian ring, then R[x±1 ; σ] is a right (left) Noetherian ring. What follows is very standard, and can be found in most books on ring theory. Let R be a ring and S a multiplicatively closed subset of R, that is, a submonoid of the multiplicative monoid of R. A right ring of fractions of R with respect to S is a ring R[S −1 ] with a ring morphism ϕ : R → R[S −1 ] such that: (a) ϕ(s) is invertible in R[S −1 ] for every s ∈ S. (b) Every element of R[S −1 ] can be written in the form ϕ(r)ϕ(s)−1 for suitable r ∈ R, s ∈ S. (c) For every r ∈ R, ϕ(r) = 0 if and only if there exists s ∈ S such that rs = 0. Exchanging ϕ(r)ϕ(s)−1 with ϕ(s)−1 ϕ(r) in (a) and rs = 0 with sr = 0 in (b), one defines the left ring of fractions [S −1 ]R. When a right ring of fractions R[S −1 ] exists, the pair (R[S −1 ], ϕ) satisfies the following universal property: for every ring T and every ring morphism ψ : R → T such that ψ(s) is invertible in T for every s ∈ S, there exists a unique homomorphism ψ : R[S −1 ] → T such that  ψ = ψϕ. Since solutions to universal problems are essentially unique when they exist (they are initial objects in a suitable category), the right ring of fractions R[S −1 ] is unique up to isomorphism, when it exists. That is, if two pairs (R[S −1 ], ϕ) and (R[S −1 ] , ϕ ) both satisfy the universal property above, there exists a unique isomorphism f : R[S −1 ] → R[S −1 ] such that ϕ = ϕ f . Moreover, the universal property is right/left symmetric, so that the following result holds: Proposition 11.29. Let R be a ring and S a multiplicatively closed subset of R. If both R[S −1 ] and [S −1 ]R exist, then they are canonically isomorphic. If R is a ring and S is a multiplicatively closed subset of R, a right ring of fractions R[S −1 ] exists if and only if the following two conditions hold: (a) For every r ∈ R and every s ∈ S, there exist r ∈ R and s ∈ S such that rs = sr . (b) For every r ∈ R and s ∈ S such that sr = 0, there exists s ∈ S with rs = 0. A multiplicatively closed subset S of R satisfying these two equivalent conditions is called a right Ore subset of R. Similarly for [S −1 ]R and left Ore subsets. When S is a right Ore subset of a ring R, the ring R[S −1 ] can be constructed as the quotient set R × S/ ∼, where ∼ is the equivalence relation on the Cartesian product R × S defined by (r, s) ∼ (r , s ) if there exist x, y ∈ R with rx = r y and sx = s y ∈ S.

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The most important case of an Ore set is that in which the ring R is a (not necessarily commutative) integral domain and S is the set of all its nonzero elements. In this case, condition (b) is always automatically verified, and (a) is equivalent to “rR ∩ sR = 0 for all nonzero elements r, s ∈ R.” Thus a right ring of fractions R[S −1 ] exists if and only if RR is a uniform module. An integral domain R with RR uniform is called a right Ore domain. Its right field of fractions R[S −1 ] is then a division ring. Similarly for left Ore domains. An integral domain that is both right Ore and left Ore is called an Ore domain. Thus, for an Ore domain, the classical right ring of quotients of R and the classical left ring of quotients of R exist, coincide, and are a division ring. Proposition 11.30. Every right Noetherian integral domain is a right Ore domain. Proof. Let r and s be nonzero elements of a right Noetherian integral domain R. The right ideals ni=0 ri sR form an increasing chain as n ranges over N. Thus there n n n+1 rn+1 s ∈ i=0 ri sR, exists a smallest n ≥ 0 with i=0 ri sR = i=0 ri sR. Hence n n+1 elements so that there s = i=0 ri sti . Then st0 = that r   nti ∈ R such  n exist n n+1 i i−1 s− sti ∈ sR ∩ rR. Notice that this r i=1 r  i=1 r sti = r r s − n i−1 sti = 0, which contradicts element is nonzero, since otherwise, rn s − i=1 r the minimality of n.  We conclude with a further example of an Ore domain, which is elementary, but very important for us. If R is a chain domain, then RR is a uniserial module over the integral domain R, so that RR is uniform. This proves the following result: Proposition 11.31. Every right chain domain is an Ore domain.

11.7 A Nearly Simple Chain Domain The example of a nearly simple chain domain we give now is taken from [Bessenrodt, Brungs, and T¨ orner]. It is based on an idea of [Dubrovin 1980]. Let Q be the field of rational numbers and G the group of affine linear functions on Q, that is, G = { α : t → at + b | a, b ∈ Q, a > 0 } . This is a group with respect to the composition of functions. We will denote the elements of G by pairs (a, b). The multiplication in G then becomes (a, b) · (c, d) = (ac, ad + b), (11.1) for every a, b, c, d ∈ Q with a, c > 0. The identity of G is (1, 0), and the inverse of (a, b) is (a−1 , −a−1 b). Recall that if H and L are two arbitrary groups and σ : L → Aut H is a group morphism of L into the automorphism group of H, then the semidirect product

Chapter 11. Serial Modules of Infinite Goldie Dimension

398

is (any group isomorphic to) the Cartesian product H × L with multiplication defined by (11.2) (h1 , l1 )(h2 , l2 ) = (h1 σ(l1 )(h2 ), l1 l2 ). The group G turns out to be the semidirect product of its subgroups H = { (1, q) | q ∈ Q }, which is isomorphic to the additive group Q of the rationals, and L = { (k, 0) | 0 < k ∈ Q }, which is isomorphic to the multiplicative group Q+ of the positive rationals. In this case the group homomorphism σ : L → Aut H is defined by σ(k, 0)(1, q) = (1, kq). In fact, one sees easily that multiplication (11.2) corresponds to multiplication (11.1). Recall that for any ring R and group G, it is possible to define the group ringR[G]. It is the free left R-module with free set of generators G: R[G] := ⊕g∈G Rg. The multiplication in R[G] is defined by      rg g sh h = rg sh (gh). g∈G

h∈G

g∈G h∈G

Here the rg ’s and the sh ’s are elements of the ring R, almost all zero. Then R is identified with R · 1G , which is a subring of R[G], and G is identified with 1R · G, which is a subgroup of the multiplicative monoid of R[G]. With these identifications, the elements of R commute with the elements of G. The idea of group ring can be extended as follows. Let R be a ring, G a group, and σ : G → Aut R a group homomorphism of G into the group of all automorphisms of the ring R. For every r ∈ R and g ∈ G, the image σ(g)(r) of the element r via the automorphism σ(r) will be denoted by rg . The skew group ring R[G, σ] is again the free left R-module ⊕g∈G Rg with free set of generators G, but where the multiplication in R[G, σ] is now defined by      sh h = rg g rg sgh (gh). g∈G

h∈G

g∈G h∈G

Again, R is identified with R · 1G , which is a subring of R[G], and G is identified with 1R · G, which is a subgroup of the multiplicative monoid of R[G], but the elements of R do not commute with the elements of G, because one has that gs = sg g for every s ∈ R, g ∈ G. The group ring R[G] is the skew group ring R[G, σ] when σ is the identity. There is a relation between skew group rings and semidirect products of groups. Given groups H, L and a homomorphism σ : L → Aut H, σ induces a group homomorphism σ : L → Aut R[H]. The group ring R[G] of the semidirect product G of the two groups H and L turns out to be canonically isomorphic to the ring R[H][L, σ].

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Proposition 11.32. Let F be a field and G the group of affine linear functions on Q. Then the group ring F [G] is an Ore domain. Proof. We already know that F [G] = F [H][L; σ], where σ : L → Aut R[H] is induced by the homomorphism σ : L ∼ = Q+ → Aut H ∼ = Aut Q with σ(k)(q) = kq for k ∈ Q+ and k ∈ Q. Now H ∼ Q is abelian, so that F [H] ∼ = = F [Q] is a commutative ring. More precisely, F [H] = F [xq | q ∈ Q], that is, F [H] can be viewed as the ring of polynomials in the indeterminate x with rational exponents. Every nonzero element of F [Q] has a “degree” (=“exponent of the monomial xq of maximum degree”), and one easily sees that F [Q] is a commutative integral domain. Thus F [H] has a field of fractions F  = F ( xq | q ∈ Q ), and every automorphism σ(k), which multiplies all the exponents of an element of F [Q] by k, extends to an automorphism σ  (k) of F  . Notice that the automorphisms σ  (k) are elements of the Galois group of the field extension F  |F , and one has that F [G] = F [H][L; σ] ⊆ F  [L; σ  ]. Now, the group L, which is isomorphic to the multiplicative group Q+ of the positive rationals, is a free abelian group  with free set of generators the set P = {p1 , p2 , p3 , . . . } of all primes: F ∼ = Q+ = n≥1 pn , where pn  denotes the cyclic group generated by pn consisting of all powers of pn with exponent in Z. Thus Q+ is the directed union of all its finitely generated free subgroups p1 , p2 , . . . , pn , with n ≥ 0: Q+ = n≥0 p1 , p2 , . . . , pn . Correspondingly, the group L is a free abelian group with free set of generators the set of all affine linear functions on Q given by multiplication by p, p ∈ P, and F  [L] can be viewed as the ring of skew Laurent polynomials in the indeterminates yp with p ∈ P. Thus F [G] = F [ xq , yp±1 | q ∈ Q, p ∈ P ], where F is contained in the center,   xq xq = xq+q , yp yp = yp yp , and yp xq = xpq yp for every q, q  ∈ Q, p, p ∈ P. We have that $ F  [yp±1 , yp±1 , . . . , yp±1 ]. F [G] ⊆ F  [L; σ  ] = F  [ yp±1 | p ∈ P ] = 1 2 n n≥0

, yp±1 , . . . , yp±1 ] is an iterated Laurent skew polynomial ring: Now, each F  [yp±1 1 2 n  ±1 ±1 ±1 ; σp 1 ][yp±1 ; σp 2 ] · · · [yp±1 ; σp n ], where each σp i is the F [yp1 , yp2 , . . . , ypn ] = F  [yp±1 1 2 n  ±1 ±1 ±1 automorphism of F [yp1 ; σp 1 ][yp2 ; σp 2 ] · · · [ypi−1 ; σp i−1 ] that acts as σ  (pi ) on F  and is the identity on yp1 , yp2 , . . . , ypi−1 . By induction, each F  [yp±1 ; σp 1 ][yp±1 ; σp 2 ] . . . [yp±1 ; σp n ] 1 2 n is a (left and right) Noetherian ring (Hilbert basis theorem, Proposition 11.28), and is clearly an integral domain. Thus all the rings F  [yp±1 ; σp 1 ][yp±1 ; σp 2 ] · · · [yp±1 ; σp n ] 1 2 n

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are Ore domains. It follows that their directed union $ , yp±1 , . . . , yp±1 F  [yp±1 ] F  [yp±1 | p ∈ P] = 2 n 1 n≥0

is an Ore domain. Hence its subring F [G] = F [ xq , yp±1 | q ∈ Q, p ∈ P ] is an Ore domain.  Fix a real irrational number ε with 0 < ε < 1. It is easy to check that the set P := { α ∈ G | ε ≤ α(ε) } is a generalized positive cone. The corresponding right order on G is therefore defined, for every α1 = (a1 , b1 ), α2 = (a2 , b2 ) ∈ G, by α1 ≤r α2 if α2 α1−1 ∈ P . It is easy to show that for the corresponding left order ≤l , one has that for every α1 = (a1 , b1 ), α2 = (a2 , b2 ) ∈ G, α1 ≤l α2 if and only if a1 ε + b1 ≤ a2 ε + b2 in the linearly ordered set of real numbers. That is, there is an embedding of linearly ordered sets (G, ≤l ) → (R, ≤), (a, b) → aε + b, whose image is the two-dimensional Q-vector subspace Q ⊕ Qε of R generated by {1, ε}. In this embedding, the identity 1G = (1, 0) of G is mapped to the real number ε. The generalized positive cone P corresponds to the real numbers ≥ ε belonging to Q ⊕ Qε. The right order ≤r corresponding to P is defined, for every α1 = (a1 , b1 ), α2 = (a2 , b2 ) ∈ G, by α1 ≤r α2 if and only if α−1 ≤l α−1 1 , that is, if and only 2 −1 −1 −1 −1 if a2 ε − α2 b2 ≤ a1 ε − α1 b1 in the linearly ordered set R. Thus there is an embedding of linearly ordered sets (G, ≤r ) → (R, ≥), where now R has the opposite order ≥ of the usual order ≤ on the reals. The embedding is defined by (a, b) → a−1 ε − a−1 b. The image of this embedding is again the two-dimensional Q-vector space Q ⊕ Qε. In this embedding, the identity 1G = (1, 0) of G is mapped to the real number ε, but the generalized positive cone P corresponds to the real numbers ≤ ε belonging to Q ⊕ Qε. Let F [P ] be the F -subalgebra of K[G] generated by P , so that nF [P ] = Let [G]\{0} be the mapping that sends an v element → F g. : G F g∈P i=1 λi gi , with λi ∈ F , gi ∈ G, g1