Algebraic K-theory: The Homotopy Approach of Quillen and an Approach from Commutative Algebra 9789811269387, 9789811269394, 9789811269400

Table of contents :
Contents
Preface
About the Author
Acknowledgment
1. Simplicial Sets
1.1. Simplicial Sets
1.2. Geometric Realization
1.2.1. The CW Structure on |K|
1.2.2. Simplicial Spaces
1.3. Bisimplicial Sets
1.4. The Homotopy Groups of Simplicial Sets
1.4.1. Higher Homotopy Groups
1.5. Exercises
2. Classifying Spaces of Categories
2.1. The Classifying Spaces of Categories
2.1.1. Properties of the classifying spaces
2.1.2. Directed and filtering limit
2.1.3. A key lemma on quasifibrations
2.2. Exact Sequences of Homotopy Groups
2.2.1. The Theorem A
2.2.2. The Theorem B
2.2.3. Fiberd and cofiberd category version of Theorems A and B
2.3. Exercises
3. Quillen K-Theory
3.1. Quillen’s Q-Construction
3.1.1. Admissible layers
3.2. K0(E) and π1(BQE, 0) of Exact Categories E
3.3. Higher K-Groups of Exact categories
3.4. Exact Sequences and Filtrations
3.4.1. Additivity theorem
3.4.2. Some examples
3.5. The Resolution Theorem
3.5.1. Extension closed subcategories
3.6. Dévissage and Localization in Abelian Categories
3.6.1. Semisimple Objects in abelian categories
3.6.2. Quillen’s localization theorem
3.7. K-Theory Spaces and Reformulations
3.8. Exercises
4. The Agreement with Classical K-Theory
4.1. Symmetric Monoidal Categories
4.2. S−1S-Construction
4.3. The Projection Functors
4.4. The X-Coordinate Functor
4.5. Split Exact Categories
4.5.1. S−1S and QE
4.6. Cofinality
4.7. Agreement of Modern and Classical K-Theory
4.7.1. The Whitehead Group
4.7.2. The Agreement of Kc1(A) and K1(A)
4.7.3. The Agreement of Kc2(A) and K2(A)
4.8. Hc-Spaces
4.9. Exercises
5. K-Theory of Rings
5.1. K-Theory of Graded Rings
5.2. Homotopy Invariance
5.3. Filtered Rings
5.4. Exercises
6. G-Theory of Schemes
6.1. Preliminary Results
6.1.1. Closed subschemes and the localization sequence
6.2. Pullback and Pushforward
6.2.1. Pullback maps
6.2.2. Pushforward maps
6.2.3. A projection formula
6.3. G-Theory of Affine and Projective Bundles
6.4. Filtration by Support
6.4.1. Gersten conjecture
6.4.2. The Chow groups
6.5. Čech Cohomology Tools
6.5.1. Application of Čech cohomology
6.5.2. A spectral sequence
6.6. Exercises
7. K-Theory of Projective Bundles
7.1. The Canonical Resolution of Regular Sheaves on PE
7.2. The Projective Bundle Theorem
7.3. Exercises
8. Work of Swan on Quadric Hypersurfaces
8.1. Hypersurfaces in Projective Spaces
8.2. Canonical Resolution for Projective Schemes
8.2.1. Truncation on R−1(X)
8.3. Quadratic Spaces
8.3.1. Clifford algebra
8.4. Canonical Resolution and Minimal Resolution
8.5. The Clifford Sequence
8.5.1. C(q)-action on the Exterior Algebra
8.5.2. Comparison of resolutions
8.6. K-Theory of Quadric Hypersurfaces
8.6.1. Graded interpretation
8.7. The Affine Case
8.7.1. Special case of q = q1 − T2
8.8. Algebraic and Topological K-Theory of Spheres
9. Epilogue: K-Theory
9.1. Introduction to the Epilogues
9.2. Waldhausen K-Theory
9.2.1. Exactness and Functorial properties
9.2.2. Agreement with Quillen K-theory
9.2.3. The chain complex categories
9.2.4. Expected results
9.3. K-Theory of Complicial Exact Categories
9.4. Negative K-Theory
9.4.1. Cofinality and idempotent completion
9.4.2. Negative K-theory spectrum of exact categories
9.4.3. Negative K-theory of complicial exact categories
9.5. K-Theory of Schemes
9.5.1. Quasi-projective schemes
10. Epilogue: Hermitian K-Theory
10.1. Hermitian K-Theory of Exact Categories
10.1.1. Exact categories with weak equivalences and duality
10.2. dg Categories with Weak Equivalences and Duality
10.3. Hermitian K-Theory of dg Categories
10.3.1. Shifted dualities in dg categories
10.3.2. The GW spectrum of dg categories
10.4. Nonconnective Hermitian GW-Theory
10.4.1. The category Sp of symmetric spectra
10.4.2. The Bispetra BiSp
10.4.3. The Karoubi GW-spectra
10.4.4. Karoubi GW theory for quasi-projective schemes
10.4.5. Gersten complex for the Karoubi GW-groups
10.4.6. Further generality for regular schemes
10.5. Nori Homotopy Obstructions
11. Epilogue: Triangulated Categories
11.1. Basic Definitions
11.2. Triangulated Witt Groups
11.2.1. Localization
11.2.2. Derived categories of exact categories and agreement
11.3. Derived Witt Groups of Schemes
11.4. Revisit Chow–Witt Groups
Appendices
Appendix A. Category Theory and Exact Categories
A.1. Main Definitions
A.1.1. Classical and standard examples
A.1.2. Pullback, Pushforward, kernel, and cokernel
A.1.3. Equivalence and adjoint functors
A.2. Additive and Abelian Categories
A.2.1. Abelian categories
A.3. Frequently Used Lemmas
A.3.1. Pullback and Pushforward Lemmas
A.3.2. The Snake Lemma
A.4. Exact Categories
A.5. Localization and Quotient Categories
A.5.1. Calculus of fractions
A.5.2. Quotient of abelian categories
A.5.3. Quotient of exact categories
A.6. Exercises
Appendix B. Homotopy Theory
B.1. Elements of Topological Spaces
B.1.1. Compactly generated topologies
B.2. Homotopy
B.2.1. Relative homotopy groups
B.2.2. Excision: Dold–Thom Theorem
B.3. Fibrations
B.4. Construction of Fibrations
B.5. The Quasi-Fibrations
B.6. Exercises
Appendix C. CW Complexes
C.1. Elements of CW Complexes
C.2. Product of CW Complexes
C.3. Frequently Used Results
C.3.1. A triangle of fibrations
C.4. Exercises
References
Index

Citation preview

ALGEBRAIC K-THEORY The Homotopy Approach of Quillen and an Approach from Commutative Algebra

World Scientific

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ALGEBRAIC K-THEORY The Homotopy Approach of Quillen and an Approach from Commutative Algebra

SATYA MANDAL University of Kansas, USA

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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World TAIPEI Scientific CHENNAI TOKYO

HONG KONG

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Control Number: 2023015302 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Cover photo: Daniel Quillen Photo credit: Cynthia Cohen ALGEBRAIC K-THEORY The Homotopy Approach of Quillen and an Approach from Commutative Algebra Copyright © 2023 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-126-938-7 (hardcover) ISBN 978-981-126-939-4 (ebook for institutions) ISBN 978-981-126-940-0 (ebook for individuals) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/13236#t=suppl Desk Editors: Balasubramanian Shanmugam/Rok TingTan Typeset by Stallion Press Email: [email protected] Printed in Singapore

Dedicated to my mother!

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Preface

In this book, I take a pedagogic approach to Algebraic K-theory. I tried to find the shortest route possible, with complete details, to arrive at the homotopy approach of Quillen [Q] to Algebraic K-theory, with a simple goal to produce a self-contained and comprehensive pedagogic document in Higher Algebraic K-theory, which is accessible to upper-level graduate students as well as a broader mathematics community. With the intent to launch this project, I gave a course on Algebraic K-theory in Fall 2017. The lecture notes for the course constituted the first draft of this manuscript, and I could cover only about half of what is in this current version of the manuscript. While preparing this manuscript I always kept these students in my mind. In all sincerity, throughout I remained mindful of what it may take to have a meaningful communication with these students. I made a rule for myself that I would provide the full background that these students would require. Most of the students were in their second year. Two of them were bold enough to concede that they did not take a course in general topology. So, the background on Homotopy theory (Appendix B) begins with a section (Section B.1) on the basic jargon of Topology. I tried to provide complete proofs, and left little that a student may get stuck on and abandon. However, I was able to trim some of the materials that appear to be almost integral parts of the required background. In particular, I avoided singular homology and the plus construction. Homology theory is not needed for such a course in Quillen K-theory. The homotopy approach to Algebraic K-theory, in its early part, sprouted out of the study of projective modules, which has been vii

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my specialization. I was always fascinated by the homotopy approach of Quillen [Q] to Algebraic K-theory, and I faced a literature gap, in the pedagogic sense. I am sure I was not alone in that. I overcame this gap, and eventually published in K-theory [M]. I concluded that a sizable readership would benefit if I wrote down the roadmap that I took. My prior experience of publishing a book [M2] gave me the confidence that this would be a doable endeavor. In this book I wrote down whatever literature gap I faced, to build this bridge. In fact, I was able to crystallize the required information, out of mountains of literature [GJ, Ha, H, Hu, S, St, W], to name some. A small portion of this book includes my own research contributions, on Nori Homotopy and Chow–Witt groups (Sections 9.5.1 and 10.5). However, strictly speaking, this manuscript is to be treated as a purely pedagogic document. The targeted readership of this book is the upper level graduate students and those academics who simply like to read good mathematics and sometimes decide to teach the same. The book concludes (Section 10.5) by relating the Nori homotopy obstruction set π0 (LO(P )) [MM], for projective A-modules P to split off a free direct summand, with the Chow–Witt groups  n (A, det P ) of Barge–Morel [BM], where A is a regular ring, conCH taining a field κ with 1/2 ∈ κ, and rank(P ) = n. The Chow-Witt  n (A, det P ) is the cohomology, at degree n, of the Gersten group CH GW -complex [M]. This reconnects the homotopy approach of Quillen to K-theory to projective modules. I submit that my work [M, M5] in Section 10.5, provides an antithesis to any skepticism that Quillen K-theory cannot generate any new research. From the point of view of commutative algebra, the research in Quillen K-theory was never explored. This book fills the literature gap that the commutative algebra community faced, which was solely responsible for this non occurrence. I expect that this book will generate more research in Quillen K-theory from the viewpoint of commutative algebra. Among the available literature I would like to mention two books [Wc2, Sv] and a survey article [Sm1]. I benefited enormously from all of them. These are more suited for specialists. My approach in this book has been fully pedagogic. Barring some exceptions, none of the topics reported in Chapters 8–11 appeared in any book. Ample literature is available on classical K-theory [B,Mi,GM,R,Sv,Wc2], so it is not discussed in this book, except in Chapter 4 on Agreement,

Preface

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because that is not the point of this project. This takes us to the heart of the matter quicker. For motivational purposes, readers are referred to the books of Milnor [Mi] and Rosenberg [R]. The book of Swan [Sw3] is an important landmark, as well. The book of Bass [B] provided the early foundation of classical K-theory. As the subtitle of the book indicates, my targeted readership includes the Algebra and the Algebraic geometry community. The nature of the topics and methods span out so broadly (e.g. Chapters 5–8), it brings commutative algebra, algebraic geometry, and K-theory under the same umbrella. The book is written in a commutative algebra style. I am hopeful that a faculty member in any mid-level university will be able to pick up this book and teach a graduate course in Algebraic K-Theory. There is a harmony and a melody in the way the results and the methods build up in this theory, from chapter to chapter. There is a complete story to tell in Algebraic K-theory. I am hopeful that this will be evident in this exposition. I divide the materials in this book into three parts. The original goal of this project was to provide an exposition, fully faithful to the pedagogic approach, of the paper of Quillen [Q], along with its agreement with classical K-theory. At some point, the inclusion of the work of Swan on K-theory of quadrics [Sw1] became an integral part of the project, in my mind. I consider these (Chapters 2–8) as the main body of this book. Chapters 2, 3, 5, 6, and 7 provide expositions of the paper [Q], and Chapter 4 is on agreement of classical K-theory and Quillen K-theory, also due to Quillen [Gd1]. Chapter 8 is an exposition of [Sw1]. I deviate a little from [Sw1] by assuming 2 is invertible in the base ring and by avoiding Azumaya algebras. In Chapters 9–11 I provide some Epilogues on the progress made since the paper [Q] was published. My original thinking was that there should be two more books on these, with the same pedagogic approach, one on K-theory and one on Hermitian K-theory. However, it became clear that an abrupt termination at the end of Chapter 8 would seem unnatural. In fact, it would definitely be beneficial to point out to readers how much of the subsequent developments is based on the foundation already built in [Q], and background and techniques already established in the remaining 11 chapters. The approach in these Chapters (9–11) differs from the rest of the chapters. These are more of an overview, and I do

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not attempt to give complete proofs, as I do in the other chapters. In Chapter 9, I give an overview of Waldhausen K-Theory [Wh,TT] and the homotopy approach to the Negative K-Theory, which is mostly an exposition of some of the papers of Schlichting. In Chapter 10, I give an exposition of homotopy approach to Hermitian K-theory, which is also an exposition of some other papers of Schlichting. In Chapter 11, I provide an exposition of the Witt groups of triangulated categories along with basic definitions regarding triangulated categories. The Quillen K-theory still remains the foundation of Algebraic K-theory. The Waldhausen K-theory is a natural progression of the Quillen K-theory. An exact sequence is determined by its inflation M → N , barring a choice of the cokernel. So, the extra information in an exact sequence, beyond its inflation, is superfluous. In Waldhausen K-theory the “cofibrations” take the place of inflations. The remainder in Waldhausen K-theory is an improvisation of Quillen K-theory. Similar comments can be made regarding developments in the Hermitian K-theory. In other words, subsequent developments (Chapters 9–11) are a natural progression of the work of Quillen [Q]. This justifies my decision to explain Quillen K-theory thoroughly, and to give only an overview on the Epilogue chapters. Readers get a chance to work out the details in these chapters, improvising the methods learned in the preceding chapters. The background, preliminaries, and tools needed to deal with the main body and the Epilogues (Chapters 2–11) are developed in Chapters A, B, C, and 1. A part of these materials is classical. In order to get to the heart of the matter as quickly as possible, I had to make a decision to put the Chapters A, B, and C in the Appendix. However, I want to caution readers that these contain the basic jargon and tools that are absolutely necessary for the rest of the book. The other rationale for this decision is that, some of it being classical, some readers will know some of these chapters, and some parts are used much later in the book. I take some credit for being able to crystallize everything that is needed in these four chapters. It is particularly so because not everything in these four chapters is classical. Some comes directly from [Q], as it falls within the context of the respective chapters. This includes Section A.4 on Exact categories, Section A.5 on Localization and Quotient Categories, Section B.5 on Quasifibrations, Section 1.3 on Bisimplicial sets, and others.

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I provide a set of exercises and problems. To deal with some of the problems, readers may need to do some extra work, while exercises should be within the reach of the intended readership. I comment on prerequisites and related issues of instructions. The author being a commutative algebraist, the book was written from an algebraic viewpoint. Accordingly, it is assumed that readers will have a reasonable familiarity with commutative algebra (and scheme theory). Many in algebra being less prepared with the jargon of Topology, familiarity with basic Topology will not be a required prerequisite. A complete and required background information on Topology is developed in Appendix B. We avoid Homology theory because it is not needed for our purpose. Appendix B is devoted mainly to Homotopy theory, which is absolutely essential for our purposes. It is less likely that intended readership in Algebra or K-theory will be familiar with the Dold–Thom theory on Excision and Quasifibrations (Sections B.2.2 and B.5). It would be a stretch to assume that the average readership would be familiar with Appendix C, on CW complexes. The final result in Appendix C is the theorem of Whitehead (C.3.8), which is one of the most frequently used tools in this book. I cover category theory in Appendix A. For completeness, and convenience of readers, I include basic jargon and facts on categories, abelian categories, pullback and pushforward, and other topics. The most basic object that we work with in this theory is exact categories, defined by Quillen [Q]. These are covered in Sections A.4 and A.5. The book is fully self-contained, for the intended readership. A prepared audience may skip some of what is in these appendix chapters. However, an unprepared audience must go through all of it, while some may choose to skip some of the detailed proofs. Instructors may like to apply proper judgment and cover the materials in these appendix chapters, as and when needed. The idea in this book is to provide all the necessary tools for the algebra community, so that the homotopy approach to K-theory may look like another tangible theory. For a prepared reader in Algebra, the gradient is properly set, for a climb from the most basic concepts, like homotopy, to the Epilogue chapters. This should be viewed more like an upper-level graduate course in Algebra. Depending on the preparedness of the audience, in a one semester course, a feasible goal may be to finish up to Chapter 8.

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About the Author

Satya Mandal joined University of Kansas in 1988 and currently holds the position of a professor. He completed his graduate studies in Indian Statistical Institute, Kolkata and Tata Institute of Fundamental Research, Mumbai. He completed his PhD under the supervision of Amit Roy in Tata Institute and got his degree in 1985. He served as a Wissenschaftliche Hilfskraft (Research Assistant) in Universit¨at Regensburg, Germany, during the year 1983– 1984, under the supervision of Earnst Kunz. He further served as a Postdoctoral Fellow, during the year 1987–1988, in Mathematical Sciences Research Institute (MSRI), Berkeley. The author is a self-made academic. He devoted the early part of his research career in specializing in Classical K-Theory and Projective Modules. In the later years, his interest shifted to the algebra of Quillen K-theory. The author has published approximately 50 papers including his other book published in 1997. He gave numerous lectures in national and international conferences. He served his department in the capacity of the Chairman of the department. He has been active with experimentations with online pedagogy, since the inception of Internet around 2000.

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Acknowledgment

I communicated and consulted with many, in person or by email, on this project. Among them are Sankar P. Dutta, Daniel Grayson, Peter May, M. P. Murthy, Madhav V. Nori, Marco Schlichting, Richard Swan, Charles Weibel, and others. I am truly thankful to all of them for their patience, sincerity, and inputs. Let me also acknowledge some of the teachers who trained me. Among them are Professors Ashok Maitra (truly a legend), Amiya Mukherjee, B. V. Rao at Indian Statistical Institute, and Professors Amit Roy (my advisor), R. C. Cowsik, N. Mohan Kumar, M. S. Raghunathan, K. Santaram, R. R. Simha, R. Sridharan, and Balwant Singh at Tata Institute. The whole manuscript was professionally copyedited by Dr. Tim Lantz. I am immensely indebted to him for his patience with the manuscript and for his impressive contributions. Funding for this came from some internal grants from KU and from an NSA grant. I would like to take this opportunity to express my gratitude to the State of Kansas and students at my university for supporting my position for more than three decades by tax dollars and tuition. Numerous internal resources from the University of Kansas, postdoctoral support, and travel and visitor support, were the keys for this project to be conceived and completed. I express my gratitude to my extended family for the mutual loving support we share. I am thankful to my recently departed fatherin-law Injachan and brother-in-law Samir for all that they did for the family. My loving thanks to my wife Elsit, my daughter Nila and also to Arunima, Sulangna, Debmalya, and Nairit for their inspirational presence and support. xv

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Contents

Preface

vii

About the Author

xiii

Acknowledgment

xv

1.

Simplicial Sets 1.1 1.2

1.3 1.4 1.5 2.

1

Simplicial Sets . . . . . . . . . . . . Geometric Realization . . . . . . . . 1.2.1 The CW Structure on |K| . 1.2.2 Simplicial Spaces . . . . . . Bisimplicial Sets . . . . . . . . . . . The Homotopy Groups of Simplicial 1.4.1 Higher Homotopy Groups . Exercises . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . Sets . . . . . .

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Classifying Spaces of Categories 2.1

2.2

2.3

The Classifying Spaces of Categories . . . . 2.1.1 Properties of the classifying spaces . 2.1.2 Directed and filtering limit . . . . . . 2.1.3 A key lemma on quasifibrations . . . Exact Sequences of Homotopy Groups . . . . 2.2.1 The Theorem A . . . . . . . . . . . . 2.2.2 The Theorem B . . . . . . . . . . . . 2.2.3 Fiberd and cofiberd category version of Theorems A and B . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . xvii

2 6 9 15 18 23 28 32 35

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35 38 41 44 47 48 53

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3.

Algebraic K-Theory

Quillen K-Theory 3.1 3.2 3.3 3.4

3.5 3.6

3.7 3.8 4.

Quillen’s Q-Construction . . . . . . . . . . . . . . 3.1.1 Admissible layers . . . . . . . . . . . . . . K0 (E ) and π1 (BQE , 0) of Exact Categories E . . Higher K-Groups of Exact categories . . . . . . . Exact Sequences and Filtrations . . . . . . . . . . 3.4.1 Additivity theorem . . . . . . . . . . . . . 3.4.2 Some examples . . . . . . . . . . . . . . . The Resolution Theorem . . . . . . . . . . . . . . 3.5.1 Extension closed subcategories . . . . . . . D´evissage and Localization in Abelian Categories 3.6.1 Semisimple Objects in abelian categories . 3.6.2 Quillen’s localization theorem . . . . . . . K-Theory Spaces and Reformulations . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

The Agreement with Classical K-Theory 4.1 4.2 4.3 4.4 4.5 4.6 4.7

4.8 4.9 5.

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Symmetric Monoidal Categories . . . . . . . . S −1 S-Construction . . . . . . . . . . . . . . . The Projection Functors . . . . . . . . . . . . The X -Coordinate Functor . . . . . . . . . . Split Exact Categories . . . . . . . . . . . . . 4.5.1 S −1 S and QE . . . . . . . . . . . . . . Cofinality . . . . . . . . . . . . . . . . . . . . . Agreement of Modern and Classical K-Theory 4.7.1 The Whitehead Group . . . . . . . . . 4.7.2 The Agreement of K1c (A) and K1 (A) . 4.7.3 The Agreement of K2c (A) and K2 (A) . Hc -Spaces . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . .

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K-Theory of Rings 5.1 5.2 5.3 5.4

K-Theory of Graded Rings Homotopy Invariance . . . Filtered Rings . . . . . . . Exercises . . . . . . . . . .

61 67 70 74 79 84 87 92 94 106 110 112 123 125 128 131 135 145 153 159 171 177 177 179 183 187 193 195

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195 203 207 215

Contents

6.

G-Theory of Schemes 6.1

6.2

6.3 6.4

6.5

6.6 7.

Preliminary Results . . . . . . . . . . . . . . . 6.1.1 Closed subschemes and the localization sequence . . . . . . . . . . . . . . . . . Pullback and Pushforward . . . . . . . . . . . 6.2.1 Pullback maps . . . . . . . . . . . . . . 6.2.2 Pushforward maps . . . . . . . . . . . 6.2.3 A projection formula . . . . . . . . . . G-Theory of Affine and Projective Bundles . . Filtration by Support . . . . . . . . . . . . . . 6.4.1 Gersten conjecture . . . . . . . . . . . 6.4.2 The Chow groups . . . . . . . . . . . . ˇ Cech Cohomology Tools . . . . . . . . . . . . ˇ 6.5.1 Application of Cech cohomology . . . . 6.5.2 A spectral sequence . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . .

219 . . . 220 . . . . . . . . . . . . .

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K-Theory of Projective Bundles 7.1 7.2 7.3

8.

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The Canonical Resolution of Regular Sheaves on PE . . . . . . . . . . . . . . . . . . . . . . . . . . 287 The Projective Bundle Theorem . . . . . . . . . . . 298 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 301

Work of Swan on Quadric Hypersurfaces 8.1 8.2 8.3 8.4 8.5

8.6

225 226 226 230 240 243 249 255 263 267 271 276 282

Hypersurfaces in Projective Spaces . . . . . . Canonical Resolution for Projective Schemes . 8.2.1 Truncation on R−1 (X) . . . . . . . . . Quadratic Spaces . . . . . . . . . . . . . . . . 8.3.1 Clifford algebra . . . . . . . . . . . . . Canonical Resolution and Minimal Resolution The Clifford Sequence . . . . . . . . . . . . . . 8.5.1 C(q)-action on the Exterior Algebra . . 8.5.2 Comparison of resolutions . . . . . . . K-Theory of Quadric Hypersurfaces . . . . . . 8.6.1 Graded interpretation . . . . . . . . . .

303 . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

303 306 313 317 322 324 331 334 337 344 349

xx

Algebraic K-Theory

8.7 8.8 9.

Epilogue: K-Theory 9.1 9.2

9.3 9.4

9.5 10.

The Affine Case . . . . . . . . . . . . . . . . . . . . 350 8.7.1 Special case of q = q1 − T 2 . . . . . . . . . . 359 Algebraic and Topological K-Theory of Spheres . . 362 Introduction to the Epilogues . . . . . . . . Waldhausen K-Theory . . . . . . . . . . . . 9.2.1 Exactness and Functorial properties . 9.2.2 Agreement with Quillen K-theory . . 9.2.3 The chain complex categories . . . . 9.2.4 Expected results . . . . . . . . . . . . K-Theory of Complicial Exact Categories . . Negative K-Theory . . . . . . . . . . . . . . 9.4.1 Cofinality and idempotent completion 9.4.2 Negative K-theory spectrum of exact categories . . . . . . . . . . . . . . . 9.4.3 Negative K-theory of complicial exact categories . . . . . . . . . . . . . . . K-Theory of Schemes . . . . . . . . . . . . . 9.5.1 Quasi-projective schemes . . . . . . .

369 . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

369 370 380 381 385 387 392 399 399

. . . . 402 . . . . 407 . . . . 409 . . . . 411

Epilogue: Hermitian K-Theory

419

10.1 Hermitian K-Theory of Exact Categories . . . . . . 10.1.1 Exact categories with weak equivalences and duality . . . . . . . . . . . . . . . . . . . 10.2 dg Categories with Weak Equivalences and Duality 10.3 Hermitian K-Theory of dg Categories . . . . . . . . 10.3.1 Shifted dualities in dg categories . . . . . . . 10.3.2 The GW spectrum of dg categories . . . . . 10.4 Nonconnective Hermitian GW -Theory . . . . . . . 10.4.1 The category Sp of symmetric spectra . . . . 10.4.2 The Bispetra BiSp . . . . . . . . . . . . . . 10.4.3 The Karoubi GW -spectra . . . . . . . . . . 10.4.4 Karoubi GW theory for quasi-projective schemes . . . . . . . . . . . . . . . . . . . . 10.4.5 Gersten complex for the Karoubi GW-groups . . . . . . . . . . . . . . . . . . 10.4.6 Further generality for regular schemes . . . . 10.5 Nori Homotopy Obstructions . . . . . . . . . . . . .

419 426 432 442 444 448 452 452 458 462 467 469 476 479

Contents

11.

xxi

Epilogue: Triangulated Categories 11.1 Basic Definitions . . . . . . . . . . 11.2 Triangulated Witt Groups . . . . 11.2.1 Localization . . . . . . . . 11.2.2 Derived categories of exact and agreement . . . . . . . 11.3 Derived Witt Groups of Schemes . 11.4 Revisit Chow–Witt Groups . . . .

491 . . . . . . . . . . . . . . . . . . categories . . . . . . . . . . . . . . . . . .

. . . . 491 . . . . 498 . . . . 504 . . . . 508 . . . . 511 . . . . 515

Appendices Appendix A. Category Theory and Exact Categories A.1 Main Definitions . . . . . . . . . . . . . . A.1.1 Classical and standard examples . . A.1.2 Pullback, Pushforward, kernel, and cokernel . . . . . . . . . . . . A.1.3 Equivalence and adjoint functors . A.2 Additive and Abelian Categories . . . . . . A.2.1 Abelian categories . . . . . . . . . A.3 Frequently Used Lemmas . . . . . . . . . . A.3.1 Pullback and Pushforward Lemmas A.3.2 The Snake Lemma . . . . . . . . . A.4 Exact Categories . . . . . . . . . . . . . . A.5 Localization and Quotient Categories . . . A.5.1 Calculus of fractions . . . . . . . . A.5.2 Quotient of abelian categories . . . A.5.3 Quotient of exact categories . . . . A.6 Exercises . . . . . . . . . . . . . . . . . . Appendix B. Homotopy Theory B.1 Elements of Topological Spaces . . . . B.1.1 Compactly generated topologies B.2 Homotopy . . . . . . . . . . . . . . . . B.2.1 Relative homotopy groups . . . B.2.2 Excision: Dold–Thom Theorem B.3 Fibrations . . . . . . . . . . . . . . . . B.4 Construction of Fibrations . . . . . . . B.5 The Quasi-Fibrations . . . . . . . . . . B.6 Exercises . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . .

521 521 524

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

527 531 533 535 537 537 541 548 551 552 554 557 558

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

563 563 570 573 578 584 593 597 601 616

xxii

Algebraic K-Theory

Appendix C. CW Complexes C.1 Elements of CW Complexes . C.2 Product of CW Complexes . . C.3 Frequently Used Results . . . C.3.1 A triangle of fibrations C.4 Exercises . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

619 619 626 628 642 643

References

645

Index

653

Chapter 1

Simplicial Sets

Triangulation of topological spaces are among the foundational pillars of classical mathematics. The standard n-simplex Σn ⊆ [0, 1]n+1 n+1 such is defined  to be the set of all points (t0 , t1 , . . . , tn ) ∈ [0, 1] that ti = 1. So, they are, essentially, the points, closed intervals, triangles, tetrahedrons, and so on. A triangulation of a topological space X is a process of dissecting X into simplexes, where all the faces of an n + 1-dimensional simplex σ fit along the existing n-simplexes. Triangulable spaces X are convenient to study, because it is easier to compute the (co)Homology and homotopy groups. In a sense, the theory of Simplicial Sets does the reverse engineering to this process of tringulation. The n-simplexes of a simplicial set K and the gluing rules are given, and its geometric realization |K| is to be constructed. Needless to say that it comes with a lot of modern twists to the classical theory, and is a very rich theory by its own rights. In a sense, Quillen’s homotopy approach to K-theory is built on the foundation of the geometry of simplicial sets. We begin this chapter by introducing and developing the basic concepts of simplicial sets and the geometric realizations. The other main goal of this chapter is to prove Quillen’s theorem on geometric realization of Bisimplicial sets, in Section 1.3.

1

2

Algebraic K-Theory

1.1.

Simplicial Sets

Definition 1.1.1. For integers n ≥ 0, denote the set [n] := {0, 1, 2, . . . , n}. Let Δ be the category whose objects are {[n] : n = 0, 1, 2, . . .} and the morphisms are the order-preserving maps. So, M orΔ ([m], [n]) := Δ([m], [n]) := {f : [m] −→ [n] : f (i) ≤ f (j) ∀ 0 ≤ i ≤ j ≤ m}. This category is referred to as the Simplicial category, or sometimes the Δ-category. 1. Δ has two distinguished subcategories Δ+ , Δ− whose objects are the same as that of Δ. The morphisms of Δ+ are the order-preserving injective maps. The morphisms of Δ− are the order-preserving surjective maps. 2. For integers n ≥ 1, and 0 ≤ i ≤ n, define the morphisms in Δ+ di : [n − 1] −→ [n] by  k if k < i i d (k) = k + 1 if i ≤ k ≤ n − 1

by skipping i.

Also, for integers n ≥ 1, and 0 ≤ i ≤ n − 1, define the morphisms in Δ− si : [n] −→ [n − 1] by  k if k ≤ i i s (k) = k − 1 if i < k ≤ n

by identifying i, i + 1.

3. The following are the basic cosimplicial identities: dj di = di dj−1 sj di = di sj−1 = id = di−1 sj sj si = si−1 sj

∀i < j ∀i < j i = j, j + 1 ∀ i>j+1 ∀i > j

(1.1)

Simplicial Sets

3

The following is a useful lemma. Lemma 1.1.2. Let f ∈ M orΔ ([n], [m]). Then f can be written (uniquely) as  0 ≤ j1 < j2 < · · · < jl < n f = dik dik−1 · · · di1 sj1 sj2 · · · sjl with 0 ≤ i1 < i2 < · · · < ik ≤ m Proof. Left to the readers! [Mc] Definition 1.1.3. Given a category C , a Simplicial Object in C is a contravariant functor Δ −→ C . Likewise, a covariant functor Δ −→ C is called a Cosimplicial Object in C . So, a simplicial set is a contravariant functor K : Δ −→ Sets. Such a simplicial set K is given by the following data: (1) A sequence of sets Kn := K[n]. An n-simplex is an element x ∈ Kn , and we write dim x = n. An element x0 ∈ K0 is called a vertex. (2) The face maps di : Kn −→ Kn−1 , which are dual to the maps di above. (3) The degeneracy maps si : Kn−1 −→ Kn , which are dual to the maps si above. (4) The face and degeneracy maps are subject to the simplicial identities di dj = dj−1 di

∀i < j

di sj = sj−1 di

∀i < j

= id

i = j, j + 1

= sj di−1

∀ i > j +1

si sj = sj si−1

(1.2)

∀i > j

These identities are obtained by reversing the arrows in (1.1). Conversely, because of (1.1.2), any association n → Kn , satisfying the simplicial identities (1.2), determines a simplicial set. A map τ : K −→ L of simplicial sets is a natural transformation. Such a map is given by a sequence of set theoretic maps

4

Algebraic K-Theory

τn : Kn −→ Ln commuting with the face and degeneracy maps: Kn o τ

si



Ln o

Kn−1 

si

τ

Ln−1

Kn τ

di



Ln

/ Kn−1 

di

τ

/ Ln−1

The simplicial sets, together with the maps of simplicial sets, form a category, to be denoted by SSet. For x ∈ Kn , images of x under arbitrary iterations of face maps di s are called faces of x, and images of x under iterated degeneracies are called degeneracies of x. Since 0-iteration is included, x is both a face and degeneracy of itself. A simplex x ∈ Kn is called nondegenerate if it is a degeneracy of itself only. Given any simplex x ∈ Km , it is the degeneracy of a unique nondegenerate simplex y ∈ Kn for some n ≤ m. So, any x ∈ K0 is nondegenerate, but its images s(x) ∈ K1 are not. The following are some of the usual definitions in the category SSet. Definition 1.1.4. Let K ∈ SSet. A simplicial subset of K is a map L −→ K in SSet such that Ln −→ Kn is one to one for all n ≥ 0. In other words, Ln ⊆ Kn , and the inclusion map L −→ K is a map of simplicial sets. Given simplicial subset L ⊆ K, the quotient K L is defined as   Kn K = , i.e., by identifying Ln to one point. L n Ln For K ∈ Obj(SSet), and a subset A ⊆ ∪Kn , the simplicial subset K(A) ⊆ K, generated by A, is defined to be the smallest simplicial subset of K, containing A. So, K(A) consists of all the iterated degeneracies and faces of simplexes in A. For K, L ∈ SSet, the product K × L ∈ SSet is defined by ⎧ (K × L)n := Kn × Ln ⎪ ⎨ ∀n∈N f ∈ M orΔ ([m], [n]) (K × L)(f ) := K(f ) × L(f ) : ⎪ ⎩ Kn × Ln −→ Km × Lm It is worthwhile to point out that the product is the diagonal of the bifunctor K×L : Δ×Δ −→ Sets, sending ([m], [n]) → Km ×Ln . Such bifunctors are called bisimplicial sets, to be discussed in Section 1.3.

Simplicial Sets

5

The following are examples of some objects in SSet that may be considered among the basic building blocks of the category SSet. Example 1.1.5. Fix an integer n ≥ 0. 1. The representable functor Δ(−, [n]) : Δ −→ Sets

given by

Δ[n]([k]) = Δ([k], [n])

is a simplicial set. This is denoted by Δ[n] := Δ(−, [n]) ∈ SSet. For integers k ≥ 0, it can be described by Δ[n]([k]) = {(0 ≤ n1 ≤ n2 ≤ · · · ≤ nk ≤ n)} A nondegenerate k simplex is a strictly increasing sequence 0 ≤ n1 < n2 < · · · < nk ≤ n. For k ≥ n + 1, all k-simplexes of Δ[n] are degeneracies. 2. The boundary ∂Δ[n] of Δ[n] is defined to be the smallest simplicial subset of Δ[n], whose nondegenerate k simplexes are those ∼ of Δ[n], except the identity 1n : [n] −→ [n]. 3. For integers 0 ≤ r ≤ n, the r-horn Λr [n] ∈ SSet is defined to be the smallest simplicial subset of Δ[n], whose nondegenerate k-simplexes = {ι ∈ Δ([k], [n]) : ι is injective,

ι = 1n , ι = dr }

So, Λr [n] ⊆ ∂Δ[n] has one fewer nondegenerate simplex, namely the face dr opposite of the vertex r. Δ[n] ∈ SSet. It has two nondegener4. Define the n-sphere, S n := ∂Δ[n] n ate simplexes,  ∈ S0 and σn ∈ Snn . Cosimplicial sets (objects) are defined by reversing the arrows. Definition 1.1.6. A cosimplicial set is a covariant functor K : Δ −→ Set. This is given by a sequence of sets K n , and for each f ∈ M orΔ ([m], [n]) in Δ, there is a map K(f ) : K m −→ K n satisfying obvious laws of compositions. With di := K(di ) and si := K(si ), a cosimplicial set is determined by cosimplicial identities (1.1). Likewise, a covariant functor K : Δ → Top is called a cosimplicial space. For any category C , cosimplicial objects are defined likewise.

6

Algebraic K-Theory

The most natural example of a cosimplicial set (in fact, a space) would be the following. Example 1.1.7. As usual N = {0, 1, 2, . . .}. Let e0 , e1 , e2 , . . . ∈ RN be the standard basis of RN . Define, ⎧  n Σn = { ni=0 λi ei : ∀ i 0 ≤ λi ≤ 1 and ⎪ i=0 λi = 1} ⎪ ⎨ m Σ(f ) : Σ −→ Σn is defined by ∀ f ∈ M orΔ ([m], [n]),  ⎪    ⎪ ⎩ Σ(f ) ( m λi ei ) = n f (i)=k λi ek i=0 k=0 Then Σn ⊆ [0, 1]n+1 inherits the usual topology, and Σ(f ) is a map in Top. Therefore, Σ : Δ −→ Top is a covariant functor, or a cosimplicial space. The space Σn is called the Standard n-simplex. 1.2.

Geometric Realization

Given a simplicial set K, we would define its geometric realization |K|. Definition 1.2.1. Let K ∈ SSet. Let Σ be the cosimplicial space, as in (1.1.7). With discrete topology on Kn , consider the disjoint (disconnected) union of spaces: X (K) :=

∞

Kn × Σn

n=0

We define a relation ∼ on X (K). For (τ, y) ∈ Km × Σm and (σ, x) ∈ Kn × Σn , define (σ, x) ∼ (τ, y) if ∃ f ∈ M orΔ ([m], [n])

K(f )(σ) = τ,

Σ(f )(y) = x.

The relationship ∼ is reflexive and transitive, but not necessarily symmetric. Let  denote the equivalence relation on X (K), generated by ∼. Note, this equivalence relation can be described using the face and degeneracy maps only.

Simplicial Sets

7

The geometric realization of K is defined to be [Mi2] ∞ Kn × Σn X (K) = n=0 with quotient topology (B.6.8). |K| :=   (1.3) So, a subset U ⊆ |K| is open, if and only if its inverse image in Kn × Σn is open. Since Kn has discrete topology, U is open if and only if its inverse image in σ × Σn is open, ∀ σ ∈ Kn , ∀ n ≥ 0. It follows immediately, ∀n≥0

|Δ[n]| = Σn

(1.4)

To describe the topology, for σ ∈ Kn , consider the natural maps Φσ : Σn −→ |K|, given by Φσ (t) = [(σ, t)]. Then a set theoretic map f : |K| −→ Z is continuous if and only if f Φσ is continuous, for all σ ∈ K. The following is on the behavior of geometric realizations as a functor. Lemma 1.2.2. This association K → |K| (1.2.1), defines a functor | − | : SSet −→ Top Further, this functor has a right adjoint, defined as follows: S : Top −→ SSet

by S(Y )n = M orTop (Σn , Y )

(This functor S is called the singular functor.) Proof. It is easy to see that both | − | and S are functors. What remains is to define the natural bijections: ∼

Φ : M orTop (|K|, Y ) −→ M orSSet (K, S(Y )) ∀ K ∈ Obj(SSet), Y ∈ Obj(Top) For f ∈ M orTop (|K|, Y ) and ∀ n ≥ 0, consider the commutative triangle of maps in Top, as follows: / |K|

Kn × Σn A

J

Q T)  Y

f

This gives a map Φ(f )n : Kn −→ M orTop (Σn , Y ) = S(Y )n

Define Φ : M orTop (|K|, Y ) −→ M orSSet (K, S(Y )) {Φ(f )n : n ≥ 0}.

by

Φ(f ) =

8

Algebraic K-Theory

It is obvious that Φ is natural with respect to both the coordinates. Conversely, given F ∈ M orSSet (K, S(Y )), we have a sequence of maps Fn : Kn −→ S(Y )n = M orTop (Σn , Y ) This defines a map Fn : Kn × Σn −→ Y in Top One checks that Fn factors through a continuous map Ψ(F ) : |K| −→ Y . Now, Ψ defines a map Ψ : M orSSet (K, S(Y )) −→ M orTop (|K|, Y ) It is clear that Φ and Ψ are inverse of each other. The proof is complete. The following is on geometric realizations and pushforward diagrams in SSet. Lemma 1.2.3. Let A fL



fK

L _ _ ιL_/ K

/K  ι  K 

AL

be a pushforward diagram in SSet (or in the category of simplicial spaces). Then the diagram of the geometric realizations: |A| |fL |



|L|

|fK |

/ |K| |ιK |

/ |K

|ιL |



A

L|

is a pushforward diagram in Top.

Simplicial Sets

9

Proof. Clearly, the diagram of geometric realization commutes. Consider the commutative diagram of continuous maps. |A| |fL |

|fK |



|L|

/ |K|

/ |K



|ιL |

|ιK | A L|

β



α

 ,X

n An Ln ) × Σ We have |K A L| = ∼ Define,      α([(σ, t)])   L −→ X by ϕ([(σ, t)]) = ϕ : K   β([(σ, t)])



(Kn

A

if (σ, t) ∈ Kn × Σn if (σ, t) ∈ Ln × Σn

One checks that ϕ is well defined and continuous. It is clear that ϕ |ιK | = α and ϕ |ιL | = β. Uniqueness of such a map ϕ is obvious. 1.2.1.

The CW Structure on |K|

Given a simplicial set K ∈ SSet, the geometric realization |K| ∈ Top. In fact, |K| has a CW structure, which we describe in this section. First, we define the skeletons of |K|. Notation 1.2.4. Suppose K ∈ SSet. For integers n ≥ 0, let Kn (K) ⊆ K denote the simplicial subset generated by the set ∪ni=0 Ki . We would often write Kn := Kn (K) when K is understood. We would establish that |K| has a CW structure, |Kn (K)| ⊆ |K| being the n-skeleton. For completeness, we prove the following, which are some set theoretic issues.

10

Algebraic K-Theory

Lemma 1.2.5. Let K ∈ SSet and n ≥ 1 be an integer. Let (σ, x), (σ  , x ) ∈ Kn−1 × Σn−1 be distinct, (τ, y) ∈ Kn × Σn , such that (σ, x) ∼ (τ, y) ∼ (σ  , x ) in |Kn |

“∼” are induced by face or degeneracy maps.

Then either τ ∈ Kn−1 or there is a chain of equivalences, of length two, (σ, x) ∼ (τ  , y  ) ∼ (σ  , x )

where (τ  , y  ) ∈ Kn−2 × Σn−2

and the equivalences are induced by face or degeneracy maps. Proof. Suppose one of the two equivalences is induced by a degeneracy. Assume (σ, x) ∼ (τ, y) is given by a degeneracy si : [n−1] −→ [n]. In this case, τ = K(si )(σ) ∈ Kn−1 . So, we assume both relations are induced by face maps. So,  K(di )τ = σ Σ(di )(x) = y i j ∃ d , d ∈ M orΔ ([n − 1], [n]) K(dj )τ = σ  Σ(dj )(x ) = y Denote the faces in M orΔ ([n − 2], [n − 1]) by δi . If i = j, then σ = σ  and x = x . So, we assume i < j. We have ⎧ j i i j K(δj−1 )K(di ) = K(di δj−1 ) ⎪ ⎨ K(d δ ) = K(δ )K(d ) = K(δj−1 )K(di )τ = K(δj−1 )(σ) Σ(di )(x) = y ⎪ ⎩ K(δi )K(dj )τ = K(δi )(σ  ) Σ(dj )(x ) = y ⎧ Let ⎪ ⎨ With y = (y0 , y1 , . . . , yn ) we have yi = yj = 0.  n−2 y = (y0 , . . . , yi−1 , yi+1 , . . . , yj−1 , yj+1 , . . . , yn ) ∈ Σ ⎪ ⎩ and τ  = K(δj−1 )(σ) = K(δi )(σ  ) ∈ Kn−2 We have  Σ(di )(Σ(δj−1 )(y  )) = Σ(di )(x) = y =⇒ Σ(δj−1 )(y  ) = x Σ(dj )(Σ(δi )(y  )) = |dj |(x ) = y =⇒ Σ|δi |(y  ) = x

Simplicial Sets

11

It follows ⎧ ⎪ (τ  , y  ) = (K(δj−1 )(σ), y  )  (σ, x) ⎪ ⎪ ⎪ ⎨ given by δj−1 : [n − 2] −→ [n − 1] (τ  , y  ) = (K(δi )(σ  ), y  )  (σ  , x )] ⎪ ⎪ ⎪ ⎪ ⎩ given by δi : [n − 2] −→ [n − 1] So, we have (σ, x) ∼ (τ  , y  ) ∼ (σ  , x ), with (τ  , y  ) ∈ Kn−2 ×Σn−2 . This completes the proof. Proposition 1.2.6. Suppose K ∈ SSet. For integers m ≤ n, consider the following commutative diagram of geometric realizations: |Km |m 

 ιmn / n |K  | _

ιm

'



ιn

|K|

For all integers m ≤ n, the maps ιmn , ιn , ιn are injective. Proof. to prove, for integers n≥ 1, the  map ι := ι(n−1)n :  n−1  It is enough  −→ |Kn | is injective. Let ω, ω  ∈ Kn−1 , with ι(ω) = ι(ω  ). K We can write ω = [(σ, x)], ω  = [(σ  , x )] for some (σ, x), (σ  , x ) ∈ Kn−1 × Σn−1 . There is a chain of equivalences (σ, x) = (τ0 , y0 ) ∼ (τ1 , y1 ) ∼ · · · ∼ (τr , yr ) = (τ, y) where (τi , yi ) ∈ Kni × Σni and the relation (τi−1 , yi−1 ) ∼ (τi , yi ) is given by a degeneracy  map or a face map. We use induction on r, to prove ω = ω  in Kn−1 . For r = 2, it follows from (1.2.5). If i = 1, . . . , r − 1, then by induction, it follows τi ∈ Kn−1, for some  / Kn−1 , and hence ni ≥ n for all ω = ω  in Kn−1 . So, we assume τi ∈ i = 1, . . . , r − 1. Let N = max{ni : i = 1, . . . , r − 1}. Now, again, we use induction on N . If N = n, it follows (τ1 , y1 ) = · · · = (τr−1, yr−1 ), and hence it follows from case r = 2 that ω = ω  in Kn−1 . Now,  assume  N −1  N ≥ n + 1. By the same argument it follows ω = ω in . So, the proof is complete by induction on N . K

12

Algebraic K-Theory

Notation 1.2.7. Let K ∈ SSet and σ ∈ Kn . Let qn and Φσ denote the maps, as in the commutative diagram Σn

H

t→(σ,t)

K

⎧X n(K) :=  K × Σk ⊆ X (K) k k≤n ⎪ ⎪ n ⎪ ∂Σ denotes the boundary of Σn ⎪ ⎨ n n

/ Kn × Σn

N

Further,

qn Q S Φσ V X+    |Kn |

Also, ρσ : Σm −→

/

X n (K) 

|K|

Int(Σ ) denotes the interior of Σ 

∂Σn −→ Kn−1 

would denote the natural map.

The following lemma is intuitive. Lemma 1.2.8. Let K ∈ SSet, and n ≥ 0. Then the map p : X n (K) −→ |Kn | is a homeomorphism.  Proof. Clearly p is a well-defined surjective and continuous map. It also follows from (1.2.5) that p is bijective. We prove that p is an n open map. Let U ⊆ X (K) be open. For τ ∈ (Kn (K))k ⊆ Kk , we have m ≤ n, σ ∈ Km (nondegenerate and unique), and a degeneracy s ∈ M orΔ ([k], [m]), such that τ = K(s)σ. We have a commutative diagram Σ(s) Σk Σm oOO −1 −1 OOO and Φ−1 OOΦ τ p(U ) = Σ(s) Φσ pU σ O ρσ OOO Φ OOO  τ  = Σ(s)−1 ρ−1 U is open. ' n ∼ X (K) n / |K |  p

The proof is complete. Lemma 1.2.9. Let K ∈ SSet, and σ ∈ Kn be a nondegenerate n-simplex, and Φσ be as in (1.2.7). Write eσ = Φσ (Int(Σn )). Then n the restriction (Φσ )|Int(Σ ) :  Int(Σ ) −→ eσ is a homeomorphism.  nn−1 n . Consequently, Further, Φσ (∂(Σ )) ⊆ K ⎛    |Kn | = Kn−1  ⎝

 σ∈Kn , nondegenerate



We Prove (1.2.9) Φσ (∂Σn ) ⊆ Kn−1  ⎪ ⎪ ⎪ ⎪ ϕσ := (Φσ )|∂Σn : ⎩and denote  

⎞ eσ ⎠

is a disjoint union.

Simplicial Sets

13

 Proof. Note Int(Σn ) = {(t0 , t1 , . . . , tn ) : 0 < ti < 1 ∀ i, ti = 1}. n It follows that the restriction (ρσ )|Int(Σn ) : Int(Σ ) −→ ρσ (Int(Σn )) is bijective and open, hence is a homeomorphism. By (1.2.8), (Φσ )|Int(Σn ) : Int(Σn ) −→ eσ is a homeomorphism. Now, suppose t = (t0 , t1 , . . . , tn ) ∈ ∂(Σn ). So, ti = 0 for some i. Assume t0 = 0. Then t = d0 (t1 , . . . , tn ). So,   Φσ (t) = [(σ, t)] = [(K(d0 )σ, (t1 , . . . , tn ))] ∈ Kn−1  The proof is complete. Proposition 1.2.10. Let K ∈ SSet, and Kn ⊆ Kn be the set of all nondegenerate n-simplexes. Then for all integers n ≥ 1, the following 

 n  ∂Σ σ∈Kn



ϕσ

_

i





/ Kn−1  _

is a pushforward diagram in Top

ι(n−1)n



 σ∈Kn

Σn



/

Φσ

where



ϕσ := (Φσ )|∂Σn .

|Kn |

Proof. We check that the diagram satisfies the universal property of pushforward in Top. Extend the above diagram as follows: 

  σ∈Kn 

_

i



∂Σn



ϕσ



/ Kn−1  _ ι(n−1)n



 σ∈Kn

Σn



 / |Kn |

Φσ 

gσ

f

E

E h E

E

E"  -Z

   In this diagram, f, gσ are given, such that f ( ϕσ ) = ( gσ )i. We need to prove that there is a unique map h, such that the outer triangles commute. When it exists,  the uniqueness of h follows, because |Kn | is a union of the images of Φσ and ι(n−1)n . Use the partition

14

Algebraic K-Theory

(1.2.9) of |Kn | to define h, as follows:    f (ω) if ω ∈ Kn−1  h(ω) = gσ (t) if ω = Φσ (t) ∈ eσ , σ ∈ Kn Since (1.2.9) provides a partition, h is well defined. The upper right triangle commutes by definition. The lower left triangle commutes because gσ and f agree on ∂Σn . n To check continuity, recall by (1.2.8) p : X (K) −→ |Kn | is a homeomorphism. We need to check, ∀ k ≤ n, σ ∈ Kk , the composition hΦσ is continuous. If k ≤ n − 1, then hΦσ = f Φσ , which is continuous. If k = n and σ ∈ Kn , then also hΦσ = gσ is continuous. Suppose σ ∈ Kn − Kn . Then σ = K(s)τ for some s ∈ M orΔ ([n], [m]), with m ≤ n − 1. Then the diagram Σ(s)

Σn Φσ

'

|Kn |

w

/ Σm Φτ

commutes. So, hΦσ = hΦτ Σ(s) = f Φτ Σ(s) is continuous.

The proof is complete. The following is the main result [Mi2, Thm. 1] on CW structures on the geometric realizations. Theorem 1.2.11. Let K ∈ SSet. Then the geometric realization |K| has a CW complex structure. With notation as in (1.2.7), in this CW structure, |Kn | would be the n-skeleton, and the patching maps are {ϕσ : σ ∈ Kn , nondegenerate}. Further, the cell decomposition is given in (1.2.9). Proof. It follows from (1.2.10) |Kn | is obtained by pushforward as required. Now, let C ⊆ |K| be such that C ∩ |Kn | is closed in |Kn | for all n. Again, it follows that the preimage of C in σ × Σn is closed, for all n and σ ∈ Kn . So, C is closed in |K|. The converse is also obvious. The proof is complete. The following is a basic result on the geometric realizations. Corollary 1.2.12. Suppose K is a simplicial set and L ⊆ K is a simplicial subset. Then the map |L| −→ |K| is injective. Further, |L| is a closed subset of |K|. Proof. This follows immediately from cell decomposition (1.2.9).

Simplicial Sets

1.2.2.

15

Simplicial Spaces

We recall the definition of simplicial spaces. Definition 1.2.13. The category of Simplicial Topological spaces would be denoted by SimTop. So, an object of SimTop is a contravariant functor Δ −→ Top,

Δop −→ Top

equivalently, a covariant functor

∀ K, L ∈ SimTop, M orSimTop (K, L) = Set of all natural transformations τ : K −→ L For such a τ : K −→ L, the maps τn : Kn −→ Ln are required to be continuous. Note that there is a forgetful functor SimTop −→ SSet. On the other hand, any set X with its discrete topology can be considered a topological space. In this way, there is a natural functor SSet −→ SimTop. For K ∈ SimTop, its geometric realization |K| is defined exactly as in (1.2.1), while the skeletal structure |K|n is defined with some extra care, as follows [Mp, p. 100]. Definition 1.2.14. Let K ∈ SimTop. As in (1.2.1), X (K) =

∞

Kr × Σr ,

and

|K| =

r=0

X (K) 

where  is as in (1.2.1). The geometric realization of K is defined to be the set |K| with the quotient topology. Now, let X n (K) = |K|n =

n

Kr × Σr ,

r=0 X n (K)



By (1.2.6, 1.2.8),

and let

with quotient topology. |K| =

∞ 

|K|n as a set.

n=0

It follows that a subset A ⊆ |K| is closed if and only if |K|∩A is closed in |K|n for all n ≥ 0. However, without further hypothesis, |K|n may

16

Algebraic K-Theory

not be closed in |K|. Set theoretically, nothing has changed from (1.2.1). So, all the set theoretic aspects discussed above (Section 1.2) remain valid. We remark that [Mp] works exclusively in the subcategory U ⊆ Top, of all compactly generated T2 -spaces. So, K = {Kn } ∈ SimTop, considered in [Mp], are actually simplicial objects in U , meaning Kn ∈ U . For subsequent references, we introduce the following notation. Notation 1.2.15. Let K ∈ SSet. The quotient map would be denoted by Φn : Kn × Σn −→ |K| , and may be referred to as a characteristic map. i For n ≥ 1, let Ln := Ln (K) := ∪n−1 i=0 K(s )Kn−1 ⊆ Kn , and let L0 be the empty set. So, Ln is the set of degeneracies, which is also called the Latching set. Subsequently, it should be taken as a standing hypothesis that Ln is closed in Kn for all n.

Lemma 1.2.16. Let K ∈ SimTop. Then 1. The image Φn (Kn × Σn ) is closed in |K|n . 2. If Ln is closed in Kn , then |K|n−1 is closed in |K|n . Consequently, if Ln is closed in Kn for all n, then |K|n is closed in |K|, for all n. n k Proof. For k ≤ n, it is easy to see that Φ−1 k Φn (Kn × Σ ) = Kk × Σ . n n−1 ) = (Kn − Ln ) × (Σn − This establishes (1). Also, Φ−1 n (|K| − |K| n ∂Σ ), which is open. This establishes (2). The proof is complete. We quote the following from [P], which we used in (1.2.16).

Lemma 1.2.17. Let K ∈ SimTop, and assume Kn is a T2 -space ∀ n ≥ 0. Then the geometric realization |K| is a T2 space. Proof. It is similar to the proof of (C.1.10), while much more tedious. Lemma 1.2.18. Let K ∈ SimTop, and Ln be closed in Kn . Then |K|n = |K|n−1 ∪ Φn (Kn × Σn ) is the union of two closed sets.

Simplicial Sets

17

Further, with Bn = (Ln × Σn ) ∪ (Kn × ∂Σn ), the diagram Bn _

/ |K|n−1 _





Kn × Σn

Φn

is a pushforward diagram in Top.

/ |K|n

In particular, if |K|n−1 is compactly generated, and Kn is locally compact, then |K|n is compactly generated. Proof. It is immediate, because |K|n−1 and Φn (Kn × Σn ) are closed in |K|n , by (1.2.16). We use (1.2.17) to prove |K|n is a weak T2 space. Remark 1.2.19. For K ∈ SimStop, the geometric realizations |K| enjoy most of the usual topological properties, as in the case of simplicial sets. We left some of that as exercise (1.5.5). While |K|n behave like skeletons, as such there is no CW structure on |K|n . We state the following [Mp] on CW structure. Proposition 1.2.20. Let K ∈ SimTop. Assume, for all n, Kn is a CW complex and the face and degeneracy maps K(di ), K(si ) are cellular maps. Then the geometric realization |K| has a CW structure, with one n + m-cell for each m-cell in Kn − Ln (K). Proof. In the general case, the proof would be an extension of the proof (1.2.11), which is the case when K ∈ SSet. Our interest in this is limited to the case when K ∈ SSet. We skip the proof for the general case [Mp, p. 103]. For completeness, the following lemma would be helpful. Lemma 1.2.21. Let K ∈ SimTop, and X be a compactly generated T2 space. Assume Ln (K) is closed in Kn , and Kn is locally compact (T2 ) space ∀ n. Let X × K denote the simpicial space n → X × Kn . Then k(X × |K|) ∼ = |X × K| is homeomorphic. n Proof. It follows from definition (1.2.14) that |X × K|n ∼ = X × |K| . n n Write Y = X × |K| , Y = X × |K|. Note X × Ln (K) is closed in X × Kn . So, Y n is closed in Y . Also, since X × Kn is compactly

18

Algebraic K-Theory

generated (B.6.7), by induction, Y n is compactly generated (1.2.18). Consider the commutative diagram k(X × |K|)

v

|X × K|

s

6 p m

 / X × |K|

where the horizontal and vertical arrows are known to be continuous, we want to prove that the broken arrow is a homeomorphism. Suppose A ⊆ Y = X × |K| is compactly closed. Let C ⊆ Y n be compact in Y n . Then A ∩ C is closed in Y . So, A ∩ Y n ∩ C is closed in Y . So, A ∩ Y n is compactly closed in Y n . Therefore, A ∩ Y n is closed in Y n . Hence, A is closed in |X × K|. What remains to be shown is, if A ⊆ |X × K| is closed, then it is compactly closed in X × |K|. Let C ⊆ X × |K| be compact. Then C ⊆ C1 × C2 , with C1 , C2 compact. It follows C2 ⊆ |K|n for some n ≥ 0 (1.5.5). We have A ∩ (X × |K|n ) is closed in X × |K|n . So, A ∩ C is closed in X × |K|n , which is closed in X × |K|. Therefore, A ∩ C is closed in X × |K|. So, A is compactly closed. The proof is complete.

1.3.

Bisimplicial Sets

In this section, we discuss a lemma on bisimplicial sets. Definition 1.3.1. A bisimplicial set T is a functor T : Δo × Δo −→ Set. For ([m], [n]) ∈ Obj(Δo × Δo ) we write Tmn = T([m], [n]). Similarly, a functor T : Δo × Δo −→ Top. is called a bisimplicial space Immediate examples of bisimplicial sets are obtained by combining two simplicial sets K, L ∈ SSet. We would mainly be interested in bisimplicial sets only. However, for the rest of this section, it would be convenient to treat Bisimplicial Sets T as bisimplicial topological spaces, with discrete topology on Tmn . Given a bisimplicial space T, we define several simplicial spaces as follows:

Simplicial Sets

19

1. First, the diagonal DT : Δo −→ Top, is defined by (DT)n := (DT)([n]) := Tnn . Then DT ∈ SimTop. We call it the diagonal of T. For a fixed integer m ≥ 0, define RTm• : Δo −→ Top, by RTm• ([n]) = Tmn . So, RTm• ∈ SimTop. Likewise, for a fixed integer n ≥ 0, define LT•n : Δo −→ Top, by LT•n ([m]) = Tmn . So, LT•n ∈ SimTop. If T is a bisimplicial set, then DT, RTm• , LT•n ∈ SSet. 2. Now, define two functors  (LRT)m := (LRT)[m] := |RTm• | o LRT, RLT : Δ −→ Top by (RLT)n := (RLT)[n] := |LT•n | Therefore, LRT, RLT ∈ SimTop are simplicial topological spaces. We will compare the geometric realizations of DT, LRT, RLT ∈ SimTop, in Lemma (1.3.2). The following lemma is from the paper of Quillen [Q, p. 94]. Lemma 1.3.2. Let T : Δo × Δo −→ Top be a bisimplicial space and DT, LT, RT be as above (1.3.1). Then there are natural homeomorphisms: |LRT| o

∼

|DT|

∼

/ |RLT|

Proof. We establish only the first homeomorphism. We have  ⎧ Tmm ×Σm ⎪ ⎨ |DT| = m≥0 ∼ ⎪ ⎩

|LRT| =



m≥0 |RTm•

∼

|×Σm





=

m≥0

n ≥ Tmn ×Σ ×Σm ∼

∼

There is a natural map ϕ : |DT| −→ |LRT| where ϕ([(τ, t)]) = [((τ, t)], t)] ∀ (τ, t) ∈ Tmm × |Δ[m]| Now, one proves that ϕ is a homeomorphism. This is done in two stages. For integers r, s ≥ 0, let Ω[r, s] := Δ[r] × Δ[s] be the bisimplicial set. So, Ω[r, s]([p], [q]) = Δ([p], [r]) × Δ([q], [s])

∀ p, q ≥ 0

20

Algebraic K-Theory

Fix S ∈ Top and integers r, s ≥ 0. Let Tr,s,S := Ω[r, s]× S be defined by Tr,s,S mn = Ω[r, s]([m], [n]) × S = Δ([m], [r]) × Δ([n], [s]) × S o Δo × Δ J

L

The diagram

Ω[r,s]

O

Q T V +

Tr,s,S

/ Set 

×S

commutes.

Top

We establish the lemma for such a bisimplicial set Tr,s,S . We have    Δ([m], [r]) × Δ([m], [s]) × S × Σm ∼ DTr,s,S  = m≥0 −→ S×Σr ×Σs ∼ Similarly, for a fixed integer m ≥ 0, we have    Δ([m], [r]) × Δ([n], [s]) × S × Σn RTr,s,S  = n≥0 m• ∼ ∼ −→ Δ([m], [r]) × S × Σs Therefore,   LRTr,s,S  = =

 

m≥0 |RTm• |

× Σm

∼ Δ([m], [r]) × S × Σs × Σm ∼ m≥0 −→ S × Σr × Σs ∼

This establishes that ϕ is a homeomorphism, in this case. Now, we establish the same for general bisimplicial spaces T. 1. Fix integers r, s ≥ 0. ∀ (f, g, τ ) ∈ Δ([m], [r]) × Δ([n], [s]) × Trs εrs (f, g, τ ) := T(f, g)(τ ) ∈ Tmn This defines maps (natural transformation) of bisimplicial sets, εrs : Ω([r, s]) × Trs = Δ[r] × Δ[s] × Trs −→ T

Simplicial Sets

21

We note the following: (a) εrs is a map of bisimplicial spaces. (b) εrs is determined by the equation εrs (1[r] , 1[s] , τ ) = τ , while 1[r] is the nondegenerate simplex of Δ[r]. (c) Therefore, the images under εrs are the faces and degeneracies of elements in Trs , of the third coordinate. 2. Combining, we get a surjective map

ε := εrs : Ω[r, s] × Trs −→ T r≥0,s≥0

r≥0,s≥0

3. For (f, g) ∈ Δ([r], [r  ]) × Δ([s], [s ]), we have two maps of bisimplicial spaces: (f,g,1)

Δ[r] × Δ[s] × Tr s

/ Δ[r  ] × Δ[s ] × Tr s ,

(1,1,T(f,g))

Δ[r] × Δ[s] × Trs

In fact, the diagram Ω[r, s] × Tr s (1,1,T(f,g))

(f,g,1)

/ Ω[r  , s ] × Tr s



Ω[r, s] × Trs



commutes.

εr  s

/T

εrs

4. Taking the coproduct, we have a commutative diagram  

 (r,s) Ω[r, s] × Tr  s

(1,1,T(f,g))



(f,g,1)

/



(r  ,s ) Ω[r

 , s ] ×

Tr  s

ε



(r,s) Ω[r, s]

× Trs

ε

(1.5)

 /T

The coproduct is taken over the set I = {(f, g) ∈ Δ([r], [r  ]) × Δ([s], [s ]) : r, r  , s, s ≥ 0, } One checks, that this is a pushforward diagram of bisimplicial spaces. To check this, one needs to check that for all integers,

22

Algebraic K-Theory

p, q ≥ 0, the (p, q)-component of the above diagram is a pushforward in Top:

where ε , ε” denote the restrictions of ε. In fact, both ε , ε” are surjective maps. It follows from construction of pushforward (in Set) that the above rectangle is a pushforward diagram. 5. Therefore, it is established that the diagram (1.5) is a pushforward diagram of bisimplicial sets. Hence, it follows that the diagonals D of the diagram (1.5) are also pushforward diagrams. Further, it follows from (1.2.3) that the LR of the above diagram (1.5) are also pushforward diagrams. Now, consider the commutative diagram

We need to prove that the lower right hand map ϕ is a bijection. Since the other three are bijections, by the properties of pushforward, so is the fourth one. The proof is complete. The following corollary is of some importance to us. Corollary 1.3.3. Let K, L ∈ SSet. Then |K × L| ∼ = k(|K| × |L|) are homeomorphic. Proof. Let T([m], [n]) = Km × Ln . Then T is a bisimplicial set. Then RTm• ([n]) = Km × Ln . So, the geometric realization |RTm• | = Km × |L|. Note, |L| is a compactly generated T2 -space, by (1.2.11,

Simplicial Sets

23

C.1.10, C.1.11). By (1.2.21), |LRT| = k(|K| × |L|). By (1.3.2), |K × L| = |DT| ∼ = |LRT| = k(|K| × |L|) The proof is complete. In fact, Corollary 1.3.3 works in a much further generality, which we state in two steps. Corollary 1.3.4. Suppose K, L ∈ SimTop, such that Ln (K), Ln (L) are closed in Kn , Ln , for all n ≥ 0. Further, assume that Kn , Ln are compactly generated, locally compact T2 spaces, for all n ≥ 0. Then |K × L| ∼ = k(|K| × |L|) are homeomorphic. Proof. Same as the proof of (1.3.3). Use Exercises 1.5.5 and B.6.7. Remark 1.3.5. Corollary 1.3.4 is valid in further generality [Mp, p. 103]. In fact, a more elementary set theoretic proof is possible, while tedious. 1.4.

The Homotopy Groups of Simplicial Sets

For K ∈ SSet, the easiest way to define homotopy groups of K is πn (K, v0 ) := πn (|K| , v0 )

∀ n ≥ 0,

∀ v ∈ K0

(1.6)

There is a desire to define πn (K, v0 ) fully combinatorially, without any reference to Topological Homotopy Theory. This works, to an extent [H, Chapter 3], mainly when K is fibrant (1.4.1). In the context of Quillen K-Theory, this may not be of much use, because the simplicial sets we consider (nerve of a category) may not be fibrant. For our purpose, the above definition (1.6) would suffice. While we avoid going into deeper details of the homotopy theory of Simplicial Sets, we provide a flavor. We define the Kan extension property, which is a key concept in this theory. Definition 1.4.1. A simplicial set K, is said to satisfy the Kan extension condition if ∀ n ∈ N, 0 ≤ k ≤ n, and maps f : Λk [n] −→ K in SSet; f extend to a map ϕ ∈ M orSSet (Δ[n], K) as in the

24

Algebraic K-Theory

commutative diagram f

Λk [n]  _



Δ[n]

{ϕ

r

/K E 

Such a simplicial set K is also called Fibrant. More generally, a map g : K −→ L is called a fibration, if g has the right lifting property, as in the diagram Λk [n]  _

 {

Δ[n]

f

{

{ { ϕ F

/K {= g



/L

Meaning, if the outer diagram commutes in SSet, then there is a map ϕ such that the inner triangles commute in SSet. In particular, a simplicial set K is fibrant ⇐⇒ the terminal map K −→ Δ[0] is a fibration. So, a fibration in SSet is analogous to Weak fibrations (B.3.1) in Top. Recall that every object X ∈ Top is fibrant. The following provides a contrast. Example 1.4.2. The simplicial set Δ[1] does not satisfy Kan extension condition. Before we proceed to demonstrate this, we give a complete list of nondegenerate simplexes (which we denote with  ), in the respective simplicial sets: ⎧  ⎪  0  ⎨ (Δ[2])0 = {(0), (1), (2)} (Λ [2])0 = {(0), (1), (2)}  (Δ[2])1 = {(0, 1), (1, 2), (0, 2)} ⎪ (Λ0 [2])1 = {(0, 1), (0, 2)} ⎩ (Δ[2]) = {(0, 1, 2)} 2 Define f ∈ M orSSet (Λ0 [2], Δ[1]) as follows: 

f (0, 1) = (0, 1) f (0, 2) = (0, 0)

Λ0 [2]  _

and consider 

Δ[2]

f

/ Δ[1]

Simplicial Sets

25

So, f is defined on a set of generators of Λ0 [2], and extends to Λ0 [2]. We show f does not extend to Δ[2]. In deed, f (0) = f (d1 (0, 1)) = d1 (f (0, 1)) = d1 (0, 1) = 0, Likewise, f (1) = 1, f (2) = 0. Now, suppose there is an extension F : Δ[2] −→ Δ[1], of f . Write F (0, 1, 2) = (i, j, k) with 0 ≤ i ≤ j ≤ k ≤ 1. Then ⎧ ⎪ ⎨ F (1, 2) = F (d0 (0, 1, 2)) = d0 (i, j, k) = (j, k) 1 = F (1) = F (d1 (1, 2)) = d1 (F (1, 2)) = d1 (j, k) = j ⎪ ⎩ 0 = F (2) = F (d (1, 2)) = d (F (1, 2) = d (j, k) = k 0 0 0 This contradicts that j ≤ k. Example 1.4.3. Let Y ∈ Obj(Top), and SY ∈ SSet be the singular functor (1.2.2). Then S(Y ) has the Kan extension property. Consequently, for K ∈ SSet, by adjunction (1.2.2), we obtain an universal arrow K −→ S |K|

in SSet, while S |K| is fibrant.

(One may say S |K| is a fibrant replacement of K.) Proof. Let f ∈ M orSSet (Λk [n], SY ) be any arrow. By adjunction (1.2.2) M orTop (|Λk [n]|, Y ) ∼ = M orSSet (Λk [n], SY ) Corresponding to f , there is a continuous map f˜ : |Λk [n]| −→ Y .  n k   Recall |Δ[n]| = Σ . Further, Λ [n] → |Δ[n]| has a retraction   r : |Δ[n]| → Λk [n] (see 1.5.3). Hence, f˜r : |Δ[n]| −→ Y . By adjunction, there is an arrow F ∈ M orSSet (Δ[n], SY ) corresponding to f˜r, completing the commutative diagram f

Λk [n]  _



Δ[n] The proof is complete.

q

y

/ SY D 

∃ F

26

Algebraic K-Theory

Definition 1.4.4. Let K ∈ SSet be fibrant. For x, y ∈ K0 , we say that x, y are homotopic, and write x ∼ y, if there is z ∈ K1 such that d1 z = x, d0 z = y. Further, let  be the equivalence relations on K0 , generated by ∼. Define, π0 (K) =

K0 

(1.7)

Lemma 1.4.5. Let K ∈ SSet. Then there is a natural bijection π0 K ∼ = π0 |K|. Proof. A vertex v ∈ K0 determines a point in |K|, which we denote by the same notation v ∈ |K|. For a point x ∈ |K|, let [x] ∈ π0 |K| denote the path component of x. We have  Kn × Σn . Consider the diagram |K| = n  K0 p0





where p0 (v) = [v]

π0 K _ _p _/ π0 |K| It follows that p0 factors through a surjective map p, as in the diagram. To see that p is injective, let [v] ∈ π0 K, with v ∈ K0 . Let K([v]) be the subset of K, consisting of σ ∈ K, such that σ has a vertex in [v] (hence all the vertices in [v]). Clearly, K([v]) is a simplicial subset of K, and K is a disjoint union of K([v]). So, |K| = [v]∈π0 K |K([v])| is a disjoint (disconnected) union. This established that p is injective. The proof is complete. Irrespective of whether ∼ is an equivalence relation or not, π0 K and π0 |K| coincide (1.4.5). However, for higher homotopy groups, one may require further caution and need to work with fibrant objects. Lemma 1.4.6. Let K ∈ SSet be fibrant. Then homotopy of vertices is an equivalence relation on K0 . We denote the set of equivalence classes by π0 K.

Simplicial Sets

27

Proof. We prove reflexivity, symmetry and transitivity of ∼, as follows. 1. (Reflexivity): Let x ∈ K0 . We have the commutative diagrams, di

[0] ∀ i = 0, 1 1

/ [1] & 

s0

K0

s0

So,

/ K1 ' 

1

di

=⇒

K0

[0]

d0 (s0 (x)) = x = d1 (s0 (x)) 2. (Symmetry): Suppose x ∼ y ∈ K0 . So, ∃ e ∈ K1 d1 e = x, d0 e = y. We give a complete list of nondegenerate simplexes, in the respective simplicial sets: Δ[1] =



⎧  ⎪ ⎨ 0, 1, 2 0, 1, 2 0, 1  0  Δ[2] = d2 := (0, 1), d1 := (0, 2), Λ [2] = (0, 1) d1 , d2 ⎪ ⎩ d := (1, 2), (0, 1, 2) 0

Note s0 x, e ∈ K1 . Nondegenerate edges d1 , d2 ∈ Λ0 [2] generate Λ0 [2]. The association  d1 → s0 x defines an arrow f : Λ0 [2] −→ K in SSet. d2 → e For clarity, we represent Λ0 [2] ⊆ Δ[2] as follows: 1>

1 d2

0

Λ [2] :=

⊆ Δ[2] :=

0

2

d1

>> >>d0 >> >

d2

0

d1

The map f : Λ0 [2] −→ K is given by 1 d2

0

→ d1

2



e 

x

 

y

s0 x

x

2

28

Algebraic K-Theory

Since K is fibrant, f extends to F : Δ[2] −→ K. Write σ := F (1[2] ) ∈ K2 and ˜e = d0 σ = F (d0 ). So, the diagrams above extend to 1>

>> >>d0 >> >

d2

0

→ x

2

d1



e 

 

y? ?

??˜e ?? ??

s0 x

=⇒ d0˜e = y, d1˜e = x x

3. (Transitivity): Suppose x ∼ y ∼ z. By symmetry, we assume z ∼ y. So, ∃ e2 , e0 ∈ K1 , d1 e2 = x, d0 e2 = y = d1 e0 , d0 e0 = z. The nondegenerate edges d0 , d2 generate Λ1 [2]. The association  d0 → e0 defines an arrow f : Λ1 [2] −→ K in SSet d2 → e2 For clarity, the association is displayed as follows: 1>

>> >>d0 >> >

d2

0

→ x

2

     e2

y? ?

?? e0 ?? ??

z

Since K is fibrant, f extends to F : Δ[2] −→ K. Let σ = F (1[2] ) and e1 = F (d1 ) = d1 (σ). We have 1 d2

0

d1

>> >>d0 >> >

→ 2

y ?  ???e0   ??  ?  e2

x

e1

=⇒ d1 (e1 ) = x, d0 e1 = z z

The proof is complete. 1.4.1.

Higher Homotopy Groups

We define higher homotopy groups for fibrant objects K in SSet. First, we define homotopy.

Simplicial Sets

29

Two vertices of 0, 1, lead to two maps 0, 1 : Δ[0] → Δ[1] in SSet. Definition 1.4.7. Suppose K, L ∈ SSet and f, g : K −→ L are two maps in SSet. We say that f is homotopic to g, if ∃ H : K × Δ[1] −→ L in SSet H(x, 0) = f (x), H(x, 1) = g(x) ∀ x ∈ Kn , n ∈ N where H is a map of simplicial sets, and H(−, 0) and H(−, 1) are defined as follows: K

∼

O

Q

/

K × Δ[0]

1K ×0

/

K × Δ[0]

S U H W X H(−,0) Z - L

K and

∼

O

Q

/

K × Δ[0]

1K ×1

/

K × Δ[0]

S U H W X H(−,1) Z - L

Further, for simplicial subsets A ⊆ K, two maps f, g : K −→ L in SSet are said to be homotopic, relative to A, if there is a homotopy ⎧ ∀σ∈K ⎪ ⎨ H(σ, 0) = f (σ) H(−, 1) = g(σ) ∀σ∈K H : K×Δ[1] −→ L ⎪ ⎩ H(σ, η) = f (σ) = g(σ) ∀ (σ, η) ∈ A × Δ[1] To define higher homotopy groups of simplicial sets, we introduce the following lemma, together with notation and definitions. Lemma 1.4.8. Let K ∈ SSet, v ∈ K0 and n ≥ 1 be an integer. Let Fn (K, v) = {f ∈ M orSSet (Δ[n], K) : f (∂Δ[n]) = v} ∼

(Recall, M orSSet (Δ[n], K) −→ Kn given by f → f (1[n] ). So, an n-simplex σ ∈ Kn is in Fn (K, v), if all its faces collapse to v.) For f, g ∈ Fn (K, v), define f ∼ g, if f is homotopic to g, relative to ∂Δ[n]. So, f ∼ g, if ⎧ ⎪ ⎨H(σ, 0) = f (σ) ∀ σ ∈ Δ[n] H : Δ[n]×Δ[1] −→ K H(σ, 1) = g(σ) ∀ σ ∈ Δ[n] ⎪ ⎩H(σ, η) = v ∀ (σ, η) ∈ ∂Δ[n] × Δ[1]

30

Algebraic K-Theory

If K ∈ SSet is fibrant, then ∼ is an equivalence relation on Fn (K, v). Proof. We skip the proof. See [H, Thm. 3.3.1] and others. Definition 1.4.9. Let K ∈ SSet be fibrant, v ∈ K0 and n ≥ 1 be an integer. With notation, as in (1.4.8), define πn (K, v) =

Fn (K, v) ∼

the set of all equivalence classes.

We also quote the following [H, p. 85] without proof. Lemma 1.4.10. Let K ∈ SSet be fibrant, v ∈ K0 be a vertex, n ≥ 0. For σ ∈ Fn (K, v), [σ] = [v] ∈ πn (K, v) if and only if there is a simplex τ ∈ Kn+1 such that dn+1 τ = σ and di τ = v for all i ≤ n. The following is an important correspondence. Lemma 1.4.11. Let K ∈ SSet be fibrant and v ∈ K0 be a vertex. Then there is a natural bijection ∼

 : πn (K, v) −→ πn (|K| , v)

∀

n≥0

Therefore, πn (K, v) is a group for n ≥ 1 and is abelian for n ≥ 2. Outline of the Proof. The case n = 0 was proved above (1.4.5). For n ≥ 1, we only define the natural map. Let f ∈ Fn (K, v). That means f : Δ[n] −→ K such that f |∂Δ[n]) = v. So, the geometric realization |f | is a map |f | : (|Δ[n]| , |∂Δ[n]|) −→ (|K| , v). Define 0 : Fn (K, v) −→ πn (|K| , v)

by (f ) = [|f |]

One notes that 0 respects homotopy and the 0 factors through a map  : πn (K, v) −→ πn (|K| , v) Now, one checks that  is bijective. A complete proof is available in [H, Prop. 3.6.3]. Remark 1.4.12. Suppose p : K −→ L is a map of fibrant K, L ∈ SSet and v ∈ K0 is a vertex. Then p induces a map of the homotopy

Simplicial Sets

31

groups (or sets) πn (K, v) −→ πn (L, p(v)), making these associations functorial: ⎧ ⎪ ⎨ π0 : SSet −→ Sets π1 : SSet −→ Gr ⎪ ⎩ πn : SSet −→ Ab n ≥ 2 We proceed to develop a long exact sequence of homotopy groups. Definition 1.4.13. Let L, K ∈ SSet be fibrant, p : L −→ K be a fibration in SSet, and v ∈ L0 , a vertex. Let [g] ∈ πn (K, p(v)), with g ∈ Fn (K, p(v)). Consider the diagram v

Λn [n] γ

Δ[n − 1]

 z

dn

/ Δ[n]

z

z

z

/L z= p



/K

g

In this diagram, the rectangle commutes. Since p is a fibration, g lifts to a map γ in SSet. By definition pdn γ = dn pγ = dn g = p(v). Let F = p−1 (pv) be the fiber of p over p(v). Consider the diagram Δ[n − 1]

JJ dn γ JJ JJ JJ JJ % &/

F 

Δ[0]

L 

p(v)

p

/K

So, dn γ lies in F . Being pullback of p, F is fibrant. Further, dn γ ∈ Fn−1 (F, v). So, [dn γ] ∈ πn−1 (F, v). Define ∂ : πn (K, p(v)) −→ πn−1 (F, v)

by

∂([g]) = [dn γ]

One checks that [dn γ] is independent of the choice of the representative g and the lift γ. So, ∂ is a well-defined map. Lemma 1.4.14. Let p : L −→ K be a fibration of fibrant simplicial sets L, K and v be a vertex of L. Let i : F −→ L be the fiber of p over p(v). Then the sequence

32

Algebraic K-Theory

···

∂

/ πn (F, v)

/ πn (L, v)

/ πn (K, p(v))

∂

/ πn−1 (F, v)

/ πn−1 (L, v)

/ πn−1 (K, p(v))

···

···

···

∂

/ π1 (F, v)

/ π1 (L, v)

/ π1 (K, p(v))

∂

/ π0 (F, v)

/ π0 (L, v)

/ π0 (K, p(v))

(1.8)

is exact. Proof. By (1.4.11), πn (F, v) = πn (|F | , v), πn (L, v) = πn (|L| , v), πn (K, pv) = πn (|K| , pv). It is known (see [H, Cor. 3.6.2], [GJ]) that the geometric realization |p| : |L| −→ |K| is a weak fibration. Now, the exactness of (1.8) follows from the exactness of (B.20). The proof is complete.

1.5.

Exercises

Exercise 1.5.1. Let K ∈ SSet. For integers n ≥ 0, prove M orSSet (Δ[n], K) ∼ = Kn . Exercise 1.5.2. Let K, L ∈ SSet. Let B ⊆ K be the set of generators of K, consisting of the nondegenerate simplexes. Let f : B −→ L be a set theoretic map such that (1) ∀ σ ∈ B, dim σ = dim f (σ), (2) for σ, τ ∈ B, if K(δ1 )σ = K(δ2 )τ for some face maps δ1 , δ2 in Δ− , then L(δ1 )f (σ) = L(δ2 )f (τ ). Then f extends to a unique map F ∈ M orSSet (K, L). Exercise 1.5.3. Interpret the geometric realizations of the examples in ( 1.1.5), and prove. In particular, (1) |Δ[n]| is homeomorphic to the is homeomorphic standard n-simplex Σn ∼ = I n (see 1.1.7), (2)  0|∂Δ[n]|  n n n−1 ∼ ∼   , (2) Λ [n] is homeomorphic to to the boundary ∂Σ = ∂I = S J n−1 := (I n−1 × 1) ∪ (∂I n−1 × I).

Simplicial Sets

33

Exercise 1.5.4. Let X ∈ SSet and  ∈ X1 be an 1-simplex. Then there is a path γ() : I −→ |X|, such that γ(0) = d0 () and γ(1) = d1 (). Exercise 1.5.5. Let K ∈ SimTop. Assume Kn is Hausdorff and Ln is closed in Kn , for all n ≥ 0. Then 1. Prove if Y ⊆ |K| is compact, then Y ⊆ |K|n for some n. (Hint: The proof of (C.1.8) works.) 2. If Kn is compactly generated Hausdorff, for all n ≥ 0, then |K| is compactly generated. (Hint: We need to prove that |K|n is compactly generated. Note, Kn × Σn is compactly generated. Use the pushforward diagram in (1.2.18).) 3. Prove |K| is Hausdorff. (Hint: [P], Proof would be similar to (C.1.10).) The following is from [GG], which we formulate as exercises. Before that, we give the following definition. Definition 1.5.6. There is a convenient way to look at the Δ category (see [GG]). Its objects are defined to be nonempty  finite totally ordered sets. For A, B ∈ Obj(Δ), define AB := A B ∈ Obj(Δ) to be the disjoint union, and a < b, ∀a ∈ A, b ∈ B. It is clear, this category is the same as Δ (i.e., equivalent). For integers, m, n ≥ 0, we have [m][n] = [m+n+1]. Given A, B ∈ Obj(Δ), there are natural arrows AB −→ A, AB −→ B, A −→ AB, B −→ AB. Let F : X −→ Y be a map of simplicial sets, and ρ ∈ Ym . Define   (ρ | F )([n]) = (σ, τ ) ∈ X([n]) × Y ([m][n]) : τ|[m] = ρ, F (σ) = τ|[n] (1.9) Exercise 1.5.7. Let F : X −→ Y be a map of simplicial sets, and ρ ∈ Y ([m]). Prove that (ρ | F ) ∈ SSet. Hint: See [GG]. Exercise 1.5.8. Let X ∈ SSet and ρ ∈ Xn . Prove that the geometric realization |(ρ | 1X )| is contractible. Hint: See [GG].

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Chapter 2

Classifying Spaces of Categories

To each small category C , we associate a topological space BC ∈ Top, to be called the classifying space of C in this chapter. Then we adapt all the topological and homotopy properties of BC to the categories C . First, we introduce the following notation. Notation 2.0.1. Recall a category C is called a small category if Obj(C ) is a set. Most of what follows would make sense only for small categories or those categories that are equivalent to a small category. For this reason, unless stated otherwise, subsequently all the categories are assumed to be equivalent to a small category. 1. The category of small categories and functors will be denoted as Cat. We will often work with categories C such that there is an ∼ equivalence F : Cs −→ C where Cs is in Cat. Such a category, C , is said to have a set of isomorphism classes of objects. 2. The category of CW complexes would be denoted as CW. The category of pointed CW complexes would be denoted as CW• . Also, the category of pairs of CW complexes would be denoted as CWPair. 2.1.

The Classifying Spaces of Categories

We define the nerve of a category. Definition 2.1.1. Suppose C is a small category. To any such category, we associate a simplicial set N C ∈ SSet, as follows: 35

Algebraic K-Theory

36

1. Let the set of vertices N C0 be the set of all objects in C . For integers n ≥ 1, an n-simplex is a sequence of composable arrows σ:

X0

f1

f2

/ X1

f3

/ X2

/ Xn−1 fn

/ ···

/ Xn

(2.1) and N Cn denotes the set of all n-simplexes. 2. The face maps di : N Cn −→ N Cn−1 for i = 0, n are given by sending an n-simplex σ, as in (2.1), by dropping Xi and replacing

Xi−1

fi−1

/ Xi

fi

/ Xi+1

with

Xi−1

fi fi−1

/ Xi+1

Further, d0 and dn are obtained by dropping X0 , Xn , respectively. 3. The degeneracy maps si : N Cn −→ N Cn+1 are given by sending an n-simplex σ, as in (2.1), by replacing Xi−1 with Xi−1

fi−1

fi−1

/ Xi

fi

/ Xi+1

1

/ Xi _ X _ i _/ Xi

fi

/ Xi+1

The simplicial set N C ∈ SSet is called the nerve of C . Definition 2.1.2. Let C be a small category. The Classifying space BC of C is defined to be the geometric realization |N C | ∈ Top. Therefore, BC := |N C |. So, objects X ∈ Obj(C ) correspond to a vertex in BC . For X ∈ Obj(C ) and integers n ≥ 0, define πn (C , X) := πn (BC , X) to be called the Homotopy groups (or set, when n = 0) of C . Remark: Without further conditions, there is no reason that the nerve N C would be a Fibrant as a simplicial set. The following obvious lemma would be very useful. Lemma 2.1.3. Suppose C is a category and f : X −→ Y is a morphism. Then there is a path γ := γ(f ) : [0, 1] −→ BC such that γ(f )(0) = X and γ(f )(1) = Y . Proof. Follows from Lemma 1.5.4. Following are two useful examples of classifying spaces. Example 2.1.4. Let I := {0, 1} be the partial ordered set. Then BI = [0, 1]. Note, the nerve N (I) = Δ[1].

Classifying Spaces of Categories

37

Example 2.1.5. Suppose G is a group. Let G denote the category with one object  and M or(, ) = G. Then  G if i = 1 πi (G, ) = 0 if i = 1 ∼

Here the isomorphism G −→ πi (G, ) sends g ∈ G to the loop corresponding to g :  −→  (see Lemma 1.5.4). Proof. This is very classical and we give a proof. Write K = N G. So, Kn = Gn , and an n-simplex looks like 

g1

/

g2

/ ···



gn

/

with gi ∈ G

(2.2)

We show K ∈ SSet is fibrant. It is clear any map Λ0 [1] −→ K, Λ1 [1] −→ K, extends to a map Δ[1] −→ K. Now let g : Λ2 [2] −→ K be any map. The nondegenerate simplexes of Λ2 [2] are (0, 2), (1, 2). The map g is represented as follows: 2 |> | || || |_| _ _ _ 0 →g02

aBB BB→g12 BB BB _ _ _/ 1

Λ2 [2]

misses the dotted arrow. Let

−1 g01 = g12 g02

The map g : Λ2 [2] −→ K is determined by (1, 2) → g12 , (0, 2) → g02 . This extends to a map Δ[2] −→ K by sending 1[2] → (g01 , g02 ). We extend maps g : Λ [2] −→ K similarly. Now, for n ≥ 3, let g : Λn [n] −→ K be a map. We have g((0, 1)) = g1 ∈ K1 , and g(d0 ) ∈ Kn−1 is given by 1

g2

/ 2

g3

/ ···

/ n−1 gn

/ n

Now, g extends to a map Δ[n] −→ K by sending 1[n] → (g1 , g2 , . . . , gn ). More generally, for  = 0, 1, . . . , n, and for a map g : Λ [n] −→ K, all the edges gi,i+1 ∈ G are available. So, any such map extends to a map Δ[n] −→ K• . This completes the proof that K is fibrant. ∼ By (1.4.11), πn (K, ) −→ πn (|K| , ). Use the notation, as in Section 3.3. Fix n = 1, Fn (K, ) is the set of all σ ∈ Xn such that di σ =  for all i = 0, . . . , n. It checks easily that (2.2), Fn (K, ) is a singleton. Hence, πn (|K| , ) ∼ = πn (K, ) = 0, for n = 1.

Algebraic K-Theory

38

In the case n = 1, F1 (K, ) ∼ = G, we have the commutative diagram G

ι

/ / π1 (K, ) 

''

write 

ι(g) = [g]

πn (|K| , ) Suppose ι(g) = 0, meaning [g] = []. By unwinding the definition of equivalence (1.4.8), it follows g = e is the identity. Alternately, by (1.4.10), there is a 2-simplex σ := 0

g1

/ 1

g2

/ 2



d2 σ = g, d0 σ = d1 σ = 

This means σ := 0

g1 =g

/ 1 g2 =e / 2

and d1 σ = eg = e =⇒ g = e

So, ι is injective. The proof is complete. 2.1.1.

Properties of the classifying spaces

In this section, we summarize some of the basic properties of the classifying spaces. Proposition 2.1.6. Let C , D be two small categories and F : C −→ D be a functor. Then F induces a continuous map B(F ) : BC −→ BD. This association  C → BC defines a functor B : Cat −→ Top F → B(F ) Proof. Follows from construction. Proposition 2.1.7. Let C be a small category. Then there is a (cel∼ lular) homeomorphism BC −→ BC o , where C o is the dual (opposite) category. This homeomorphism is not induced by any natural functor.

Classifying Spaces of Categories

39

Proof. Follows from construction. Proposition 2.1.8. Let C , D be two small categories. It follows eas∼ ily that N (C × D) −→ N (C ) × N (D) is a bijection. So, there is a continuous bijection B(C × D) −→ BC × BD. It ∼ follows from (1.3.3) that the map B(C × D) −→ k(BC × BD) is a homeomorphism, where k indicates compactly generated topology. Consequently, if BC or BD is locally compact, then B(C × D) −→ BC × BD is a homeomorphism (B.6.7). This would be the case if one of them has only finitely many distinct skeletons (1.2.18). In particular, B(C × I) ∼ = BC × [0, 1] are homeomorphic, with notation as in (2.1.4). Proposition 2.1.9. Let C , D be two small categories and F, G : C −→ D be two functors. Suppose θ : F −→ G is a natural transformation. Then θ induces a homotopy H : BC × I −→ BD



H(0) = B(F ), H(1) = B(G)

Proof. For a morphism f ∈ M orC (X, Y ), we have the commutative diagram F (X) θX

F (f )



G(X)

/ F (Y ) 

G(f )

θY

/ G(Y )

With I = {0, 1}, define a functor H : C × I −→ D  ⎧ H(X, 0) = F (X), H(X, 1) = G(X) ⎪ ⎪ ⎪ ⎨ H(f, ι) = F (f ) : F (X) −→ F (Y ) ⎪ H(f, ι) = G(f ) : G(X) −→ G(Y ) ⎪ ⎪ ⎩ H(f, ι) = θY F (f ) = G(f )θX

∀ X ∈ Obj(C ) ∀ f ∈ M orC (X, Y ), ι ∈ M orI (0, 0) ∀ f ∈ M orC (X, Y ), ι ∈ M orI (1, 1) ∀ f ∈ M orC (X, Y ), ι ∈ M orI (0, 1)

By (2.1.6, 2.1.8), there is a homotopy H := H : BC × [0, 1] ≡ B(C × I) −→ BD It follows that H(−, 0) = BF and H(−, 1) = BG. The proof is complete.

40

Algebraic K-Theory

Corollary 2.1.10. Let F : C −→ D, G : D −→ C be two functors of small categories, and F be left adjoint to G. Then B(F ) : BC −→ BD is a homotopy equivalence. Proof. There is a natural transformation 1C −→ GF as follows, with A ∈ Obj(C ), X ∈ Obj(D): ηAX

M orD (F A, X) ∼ / M orC (A, GX)  ∼ M orD (F A, F A) −→ M orC (A, GF A) So, ∼ M orD (F GX, X) −→ M orC (GX, GX) Let θ(A) = η(1F A ) : A −→ GF A. Then θ : 1C −→ GF is a natural transformation. By (2.1.9), 1BC  B(G)B(F ) are homotopic. Likewise, B(F )B(G)  1BC are homotopic. So, B(F ) is a homotopy equivalence. Corollary 2.1.11. Suppose C is a category with an initial object or a final object. Then BC is contractible. Proof. Assume C has an initial object 0. Let 0 denote the full subcategory of C consisting of the zero object. Let F : 0 −→ C be the inclusion functor, and G : C −→ 0 be the constant functor. It follows that F is a left adjoint to G. Hence, B0 −→ BC is a homotopy equivalence, by (2.1.10). Definition 2.1.12. From now on, terminology and concepts from Topology are borrowed and adapted freely to the Category Theory. A category C is said to have a “topological property” if the classifying space BC has the same property. For example, 1. Two functors F, G : C −→ D are called homotopic if B(F ) and B(G) are homotopic. 2. A functor F : C −→ D is called a homotopy equivalence if BC −→ BD is a homotopy equivalence. 3. A category C is called contractible if BC is contractible.

Classifying Spaces of Categories

41

4. A commutative diagram of categories C

/D





E BC 

BE 

l

is called Homotopy Cartesian if

/ D / BD 

is so, as in (B.5.1).

/ BD 

/D / D  is a In this case, if E  is contractible, we say C homotopy fibration. Consequently, a long exact sequence of homotopy groups/sets follows (B.5.5).

2.1.2.

Directed and filtering limit

In this section, we establish some definitions in category theory before adapting them to classifying spaces. Definition 2.1.13. A (nonempty) partially ordered set (P, ≤) is called a directed set, if ∀ i, j ∈ P there is a k ∈ P such that i ≤ k, j ≤ k. A small nonempty category I is defined to be a filtering category, if 1. ∀ i, j ∈ Obj(I) there are two arrows: i

f



j

g

kH

(2.3)

42

Algebraic K-Theory

2. Given two arrows λ, β : i −→ k, there is an arrow γ : k −→ κ such that γλ = γβ. Diagramatically, i

λ β

/k γ

' 

γλ=γβ

κ

It follows that a directed set is a filtering category. Definition 2.1.14. Suppose I is a filtering small category. A functor A : I −→ Set, also written as {Ai : i ∈ I}, is called a filtering family of sets. Also, a functor Θ : I −→ Cat is called a filtering family of small categories. With Ci = Θ(i), such a family may be written as {Ci : i ∈ I}. Proposition 2.1.15. Suppose I is a filtering category and Θ : I −→ Cat is a filtering family of small categories, indexed by I. We will write Ci = Θ(i). A (small) category C , together with a family {θi : Ci −→ C ∀ i ∈ I} of functors, is defined to be the limit of Θ if this family {θi } satisfies the universal property in the sense of the following diagram: Ci A

∀ f ∈ M orI (i, j)

AA θ ψi AAi AA # _Ψ _ _/ D Θ(f ) C < ~? ~~ ~ ~~  ~~ θj ψi

(2.4)

Cj

In this diagram, the left hand-triangle commutes. Further, given a family of the commutative diagram of the outer triangles, there is a unique functor, Ψ, such that all the triangles commute. We write C = limI Ci or C = limΘ Ci . Since we are working with small categories, this is, indeed, a limit in the category of sets, as described subsequently. In particular, such

Classifying Spaces of Categories

43

limits exist and are defined up to equivalences of categories. The limit C is constructed as follows:  ⎧ i∈I Obj(Ci ) ⎪ ⎨ Obj(C ) = limI Obj(Ci ) = ∼ (2.5) Denote θi : Obj(Ci ) −→ Obj(C ) ⎪ ⎩ M orC (X, Y ) := limi∈I,θi (Xi )=X,θi (Yi )=Y M orCi (Xi , Yi ) 1. It follows that the nerve N (C ) = limI N (Ci ). 2. Let Xi ∈ Obj(Ci ) be a family such that X = θi (Xi ). Then θi induces the maps πn (Ci , Xi ) −→ πn (C , X)

∀ n≥0

Proposition 2.1.16. With notation as in (2.1.15), we have lim πn (Ci , Xi ) = πn (C , X) I

∀ n≥0

Proof. We have N C = limI N Ci . The rest of the proof follows from the corresponding result on simplicial sets (2.1.17), as follows. Proposition 2.1.17. Let I be a filtering category and {K(i) : i ∈ I} be a filtering family of simplicial sets. Let K = limI K(i) and v = limI vi ∈ K, where vi ∈ K(i)0 are vertices ∀ i ∈ I. Then lim πn (|K(i)| , vi ) = πn (|K| , v) i∈I

∀n≥0

Proof. Let f (i) : K(i) −→ K denote the canonical maps and |f (i)| : |K(i)| −→ |K| be the map of geometric realizations. Fix n ≥ 0. The map limi∈I πn (|K(i)| , vi ) −→ πn (|K| , v) is induced by the maps |f (i)|. Let γ : (Sn , ) −→ (|K| , v) be a map representing an element [γ] ∈ πn (|K| , v). Since γ(Sn ) ⊆ |K| is compact, by (C.1.8) and (1.2.11) there is a finite subcomplex F ⊆ K such that F is generated by finitely many nondegenerate simplexes and γ(Sn ) ⊆ |F|. There is an i0 ∈ I, large enough, and a finite

44

Algebraic K-Theory

subcomplex F(i0 ) ⊆ K(i0 ) such that F(i0 ) −→ F is a bijection. So, |F(i0 )| −→ |F| is a homeomorphism. So, γ lifts to a map γ˜ : (Sn , ) −→ (|F(i0 )| , vi0 ) ⊆ (|K(i0 )| , vi0 )). Therefore, [˜ γ] ∈ πn (|K(i0 )| , vi0 ) maps to [γ] ∈ πn (|K| , v). This establishes that the map limi∈I πn (|K(i)| , vi ) −→ πn (|K| , v) is surjective. Now suppose [γ] ∈ limI πn (|K(i)| , vi ) maps to zero in πn (|K| , v), where γ is represented by a map γ : (Sn , ) −→ (|K(i0 )| , vi0 ) for some i0 . So, [|f (i)| γ] = 0 and there is a homotopy H : Sn × I −→ |K| such that H(−, 0) = cv , the constant map and H(−, 1) = |f (i)| γ. Since H(Sn × I) is compact, by the same argument above, for some i1 ∈ I ˜ : Sn × I −→ |K(i1 )| from large enough, H lifts to a homotopy H the constant map to γi1 , where γi1 is the image of γ. So, [γi1 ] = 0 ∈ πn (|K(i1 )| , vi1 ). So, [γ] = [γi1 ] = 0 ∈ limI πn (|K(i)| , vi ). So, the map limi∈I πn (|K(i)| , vi ) −→ πn (|K| , v) is injective. The proof is complete. Corollary 2.1.18. Use the notation as in (2.1.15). Assume ∀f : i −→ j maps, the map BCi −→ BCj is a homotopy equivalence. Then BCi −→ BC is a homotopy equivalence ∀i ∈ I. Proof. (It would clearly be obvious if the homotopies BCi −→ BCj were assumed to be compatible.) However, for a fixed i0 ∈ I, by ∼ (2.1.16), πn (BCi0 , vi0 ) −→ πn (BCi , v) are isomorphisms, for all vi0 , and n ≥ 0. Therefore, by Whitehead’s theorem (C.3.8), BCi0 −→ BC is a homotopy equivalence. The proof is complete. Corollary 2.1.19. Suppose I is a filtering category. Then I is contractible. Proof. We need to prove that BI is contractible. In fact, I = limi∈I (I/i). Since I/i has a final object, B(I/i) is contractible. By (2.1.18), BI is contractible. The proof is complete. 2.1.3.

A key lemma on quasifibrations

The following is a key needed to prove the Theorem A, B (2.2.2, 2.2.3) in [Q]. Lemma 2.1.20. Let I be a small category and ι : I −→ Top be a functor. For objects i in I, we write Xi := ι(i). Let N I be the nerve

Classifying Spaces of Categories

45

of I. Subsequently, a p-simplex σ ∈ N Ip is denoted by f0

σ = i0

/ i1

f1

/ ip ,

/ ···

and

σ(j) := ij

(2.6)

 For integers p ≥ 0, write Xp := σ∈N Ip Xσ(0) . Then X := {Xp : p ≥ 0} is an object in SimTop. So, a p-simplex in X is a pair (σ, z), with σ ∈ N Ip , and z ∈ Xi0 . We have the 0th face d0 (σ, z) = (d0 σ, ι(f0 )(z)), and for i ≥ 1, the ith face di (σ, z) = (di σ, z). There is a natural map γ : {Xp } −→ N I, in SimTop, defined by γ(σ, z) = σ. Assume ∀ f : i −→ j ∈ M orI , the induced arrow ι(f ) : Xi −→ Xj , is a homotopy equivalence. Then the geometric realization |γ| : |X | −→ BI is a quasifibration. Proof. We denote the geometric realization XI := |X |, and g := |γ|. Let Fp be the p-skeletons of BI. Let N Ip denote the set of all nondegenerate simplexes in N Ip . We have a commutative diagram of pushforward (co-cartesian) diagrams as follows: 

 σ∈N Ip

|∂Δ[p]| × Xσ(0)

_

ψp

SSS SSS SSS SSS SS  )

 σ∈N Ip



 σ∈N Ip



|Δ[p]| × Xσ(0)

_

|∂Δ[p]|

Ψp

SSS SSS SSS SSS SS )



 σ∈N Ip

|Δ[p]|

/ g−1 F p−1 _ GG GG gp−1 GG GG GG # ϕp / Fp−1 _  / g−1 Fp GG GG gp GG GG GG #  / Fp

Φp

(2.7) The front of the cube is the pushforward diagram defining the p-skeleton Fp (see 1.2.1, 1.2.11). For the back side, refer to (1.2.14). It is clear that g −1 Fp = XIp . At least set theoretically, the back side is a push forward diagram. One checks that, in this case, the back side is also a pushforward in Top, while it does not match exactly with the diagram in (1.2.18). The Proposition B.5.12 (Dold–Thom) applies to the map g. So, it would suffice to prove that the restriction map gp : g −1 Fp −→ Fp , of g, is a quasifibration, for all p ≥ 0. We do this by induction on p.

Algebraic K-Theory

46

The mid-point Op = (1/(p + 1), 1/(p + 1), . . . , 1/(p + 1)) ∈ |Δ[p]| is called its barycenter. Let V1 ⊆ Fp be the open subset obtained by removing all the barycenters in the p-cells, and U1 = g−1 V1 . Let V2 = Fp \ Fp−1 and U2 = g−1 V2 . So, V1 , V2 is an open cover of Fp , while V1 ∩ V2 is the image of the lower horizontal map, barycenters removed. To prove that the restriction map gp : g−1 Fp −→ Fp is a quasifibration, we use Proposition B.5.11 (Dold–Thom), with respect to this open cover V1 , V2 . We need to prove that the restriction maps (labels added) f1 : U1 −→ V1 , f2 : U2 −→ V2 , and f12 : U1 ∩ U2 −→ V1 ∩ V2 are quasifibrtions. Let (|Δ[p]|)int := |Δ[p]| \ ∂ |Δ[p]| denote the interior of |Δ[p]|. Note, f2 , f12 are, respectively, equivalent to the maps   int × X int σ(0) −→ σ∈N Ip (|Δ[p]|) σ∈N Ip (|Δ[p]|)   int \ {O }) × X int \ {O } p p σ(0) −→ σ∈N Ip ((|Δ[p]|) σ∈N Ip (|Δ[p]|) Since, projection maps are quasifibration, f2 , f12 are quasifibrations. What remains to be established is that f1 is a quasifibration. There is a strong deformation retraction μ : |Δ[p]|\{Op } −→ ∂ |Δ[p]|, corresponding to the retraction r : Dp \ {0} −→ Sp−1 sending x → x x . Use the diagram (2.7). Then μ induces two deformation retracts ρ, η, as in the following commutative diagram: g−1 Fp−1  gp−1



Fp−1





/ U1 

η

f1

/ V1

/ g −1 Fp−1 

ρ

gp−1

(2.8)

/ Fp−1

Fix n ≥ 0. Let b ∈ Fp , F := f1−1 b, and x ∈ F . First assume b ∈ Fp−1 . We have πn (U1 , F, x) ∼ = πn (Fp−1 , F, x), πn (V1 , b) ∼ = πn (Fp−1 , b). −1 ∼ Further, πn g Fp−1 , F, x = πn (Fp−1 , b), by induction. Therefore, πn (U1 , F, x0 ) ∼ = πn (U1 , b), as required. Now, let b ∈ V1 \ Fp−1 . So, b = Φp (σ, y) for some σ ∈ N Ip , y ∈ |Δ[p]| \ ∂ |Δ[p]|, and y = Op . Also x = Φp (σ, y, ξ) for some ξ ∈ Xi0 . Use the description of σ, as in (2.6). It follows, F ∼ = (σ, z) × Xi0 . We write μ(y) = z = (z0 , . . . , zp ). Let {j : zj = 0} = {j0 , j1 , . . . , jq }, with j0 < j1 < · · · < jq . Write z  = (zj0 , . . . , zjq ) ∈ |Δ[q]|. Let τ = / ij / ··· / ij . Then τ ∈ NI  , z  ∈ |Δ[q]| \ ∂ |Δ[q]|. ij0 q 1 q

Classifying Spaces of Categories

47

We have c := Φq (τ, z  ) ∈ Fq ⊆ Fp−1 . We have G := f1−1 c ∼ = (τ, z  ) ×  Xij0 , c = ρ(b), η(x) = ϕp (τ, z , fi0 ij0 (ξ)), where fi0 ij0 : Xi0 −→ Xj0 is the composition map. Consider the commutative diagram (σ, y) × Xi0 fi0 ij

0

∼

/F η|F



(τ, z  ) × Xij0

∼

 /G

By hypothesis, fi0 ij0 is a homotopy equivalence, and hence so is η|F : F −→ G. Therefore, η : (U1 , F, x) −→ (g−1 Fp−1 , G, η(x)) is a homotopy equivalence. For integers n ≥ 0, consider the commutative diagram πn (U1 , F, x) 

πn (U1 , b)

η∗ ∼

∼ ρ∗





/ πn g −1 Fp−1 , G, η(x) 



/ πn (Fp−1 , c)

The second vertical map is an isomorphism, by induction. Also, ρ∗ , η∗ are isomorphisms. So, the first vertical map is an isomorphism. The proof is complete.

2.2.

Exact Sequences of Homotopy Groups

Perhaps [Q, Theorems A, B] provide the foundation of Quillen K-Theory. Theorem A (2.2.2) provides a sufficient condition for a functor to be a homotopy equivalence, and Theorem B (2.2.3) provides a Long Exact sequence of homotopy groups. In analogy to the homotopy fiber F (f, b), for a continuous map f : X −→ B, and b ∈ B (B.4.5), we introduce the following definitions. Definition 2.2.1. Let F : C −→ D be a functor. For Y ∈ Obj(D), we define two categories.

Algebraic K-Theory

48

1. Define the category Y /F as follows: 

Obj Y /F = {(X, u) : X ∈ Obj C , u : Y −→ F (X) ∈ M orD (Y, F (X))} M orY /F ((X1 , u1 ), (X2 , u2 )) = {ϕ ∈ M orC (X1 , X2 ) : u2 = F (ϕ)u1 }

u1

Y

/ F X1

Meaning the diagram &

u2



F (ϕ)

commutes.

F X2

Further, for arrows v : Y −→ Z, in D, there are functors  v∗ : Z/F −→ Y /F

sending

(X, u) → (X, uv)

(2.9)

ϕ ∈ M or ((X1 , u1 ), (X2 , u2 )) → ϕ

2. Define the category F/Y as follows: 

Obj F/Y = {(X, u) : X ∈ Obj C , u : F (X) −→ Y ∈ M orD (F (X), Y )} M orF /Y ((X1 , u1 ), (X2 , u2 )) = {ϕ ∈ M orC (X1 , X2 ) : u2 F (ϕ) = u1 } .

F X1 F (ϕ)

Meaning the diagram

u1



F X2

u2

 /Y

commutes.

Also, for arrows v : Y −→ Z, in D, there are functors  v∗ : F/Y −→ F/Z

sending

(X, u) → (X, vu) ϕ ∈ M or ((X1 , u1 ), (X2 , u2 )) → ϕ

(2.10)

3. With F = 1C , we denote Y /C := Y /1C and C /Y := 1C /Y . 4. Remark: These categories Y /F , F/Y are fairly analogous to the homotopy fibers (B.4.5) F (f, y) of continuous maps f : X −→ Y and y ∈ Y . The functors (2.9, 2.10) are analogous to similar maps between homotopy fibers of continuous maps f : X −→ Y , obtained by adjoining paths when there is one. 2.2.1.

The Theorem A

The following is the statement of celebrated [Q, Theorem A].

Classifying Spaces of Categories

49

Theorem 2.2.2 (Theorem A). Let F : C −→ D be a functor such that Y /F is contractible, ∀ Y ∈ Obj(D). Then F is a homotopy equivalence. Dually (see Proposition 2.1.7), if F/Y is contractible, ∀ Y ∈ Obj(D), then F is a homotopy equivalence. Proof. (Intuitively, the proof would be an application of the theorem of Whitehead (C.3.8), if it was applicable.) Let S(F ) be the category, described as follows: 1. The objects are Obj(S(F )) = {(X, Y, v) : X ∈ Obj(C ), Y ∈ Obj(D), v ∈ M orD (Y, F X)}

2. For (X, Y, v), (X  , Y  , v  ) ∈ Obj(S(F )), define M orS(F ) ((X, Y, v), (X  , Y  , v  )) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

=

⎪ ⎪ ⎪ ⎪ ⎪ ⎩

w

Y (u, w) ∈ M orC (X, X  ) × M orD (Y  , Y ) :

v



F X

o

/Y 

Fu

v

commutes

FX

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

There are functors:  Do

o

p2

S(F )

p1

/C



p1 (X, Y, v) = X p1 (u, w) = u p2 (X, Y, v) = Y p2 (u, w) = w

Let T (F ) be the bisimplicial set, namely the functor T (F ) : Δo × Δo −→ Sets



T (F ) p, q =

⎧ ⎪ ⎪ ⎪ Yp ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ X0

fp

u1

/ Yp−1

/ ···

/ Y0

/ X1

/ ···

/ Xq−1

u2

T (F )pq :=

defined by

v0

uq

/ F X0 / Xq

⎫ ⎪ ⎪ in D ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ in C ⎭

So, T (F )pq ⊆ (N D)p+1 × (N C )q . For morphisms (f, g) in Δo × Δo , define T (F )(f, g) accordingly. Treat the simplicial set N (C ) (the

Algebraic K-Theory

50

nerve) as a bisimplicial set, by (p, q) → N (C )q . Now consider the map of bisimplicial sets: τ1 : T (F ) −→ N (C )

T (F )pq −→ N (C )pq = N (C )q

is the projection

(2.11) Given an ordered pair in T (F )pp , as described above, for 0 ≤ k ≤ p, let vk = F (uk ) · · · F (u1 )v0 f1 · · · fk : Yk −→ F Xk . (X0 , Y0 , v0 )

(u1 ,f1 )

/ (X1 , Y1 , v1 )

/ ···

So,

(Xk , Yk , vk ) ∈ S(F ),

and

/ (Xp , Yp , vp ) ∈ N (S(F )) . p

This shows that the nerve N (S(F )) is the diagonal p → T (F )pp (i.e. in bijection). The first part of the following commutative diagram leads to its geometric realization, as follows: N S(F )

∼

/ {T (F )pp }

N (p1 )

*



BS(F )

∼

/ |{T (F )pp }|

τ1

NC

B(p1 )

*



|τ1 |

BC

Now, for a fixed q   So, T (F )pq = X• ∈N (C )q N (D/F X0 )op  o |p → T (F )pq | = X• ∈N (C )q B(D/F X0 ) Since D/F X0 has a final object, namely, 1 : F X0 −→ F X0 , B(D/F X0 )   is contractible. So, for all q ≥ 0, there is a homotopy equivalence   B(D/F X0 )o −→  = N Cq |p → T (F )pq | = X• ∈N (C )q

X• ∈N (C )q

with discrete topology on N (C )q . Write Tq = |p → T (F )pq |. So, T = {Tq } is a simplicial topological space. It follows from above, ∀ q ≥ 0, the map Tq −→ N (C )q

is a homopoty equivalence, with discrete topology on N (C )q .

Claim: |T | = |(q → |p → T (F )pq |)|  BC is a homotopy equivalence.

The claim follows, at least intuitively, because Tq −→ N (C )q is a homotopy equivalence. However, we give a proof using Lemma 2.1.20,

Classifying Spaces of Categories

51

with I = C , and the functor C : X → B(D/F X)o . Note B(D/F X)o is contractible, hence they are homotopically equivalent. Therefore, Lemma 2.1.20 is applicable, and |T | −→ BC is a quasifibraton, with the fibers being contractible. It follows immediately that the map |T | −→ BC is a weak equivalence. Since both |T |, BC are CW complexes, it follows from Whitehead’s Theorem (C.3.8) this map is a homotopy equivalence. So, the claim is established. We have the commutative diagram ∼

BS(F )

/ |{T (F )pp }| ∼

B(p1 )

*



/ |T |

|τ1 |

BC

u

In this diagram, the second horizontal homeomorphism is obtained by Lemma 1.3.2. The right diagonal map is a homotopy equivalence, by the claim established above. Now, it follows that B(p1 ) is a homotopy equivalence. Now consider the functor p2 : S(F ) −→ D o . Using similar arguments, we establish that B(p2 ) is a homotopy equivalence. As before, treat N (D o ) as a bisimplicial set by (p, q)) → N (D o )p . As in (2.11), consider the projection map to the Y -coordinate: τ2 : T (F ) −→ N (D o )

sending T (F )pq → N (D o )p

(2.12)

As before, the first of the following commutative diagram leads to the second, as follows: N S(F )

∼

N (p2 )

/ {T (F )qq } )



BS(F )

∼

/ |{T (F )qq }|

τ2

N Do

B(p2 )

*



|τ2 |

BD o

In fact, for any fixed Y• ∈ N (D o ), we have {(x• , y• ) ∈ T (F )pq : y• = Y• }  N (Y0 /F )  For a fixed p,

 So, T (F )pq = Y• ∈N (D)op N (Y0 /F )q  |q → T (F )pq | = Y• ∈N (D)op B(Y0 /F )

Algebraic K-Theory

52

By hypothesis, B(Y0 /F ) is contractible. Therefore, for fixed p, the map  B(Y0 /F ) −→ N (D 0 )p |q → T (F )pq | = Y• ∈N (D 0 )p

is a homotopy equivalence, ∀ p, with discrete topology on N (D 0 )p . Write Fp = |q → T (F )pq |, and F = {Fp }. So, Fp −→ N (D 0 )p is a homotopy equivalence, ∀ p. Arguing as before, using Lemma 2.1.20, it follows |F| = |(p → |q → T (F )pq |)|  BD o

is a homotopy equivalence.

Now, we have the commutative diagram ∼

BS(F )

/ |{T (F )qq }| ∼ *

B(p2 )



/ |F|

|τ2 |

u

BD o

As before, the second horizontal homeomorphism is obtained by Lemma 1.3.2. Thus, we established that the right diagonal map is a homotopy equivalence. So, B(p2 ) is a homotopy equivalence. The above inferences on F apply to the identity functors as well. So, we have the following commutative diagram: Do o Do o

p2



π2

p1

S(F )

/C

f

S(id)



π1

(2.13)

F

/D

This induces the following corresponding commutative diagram of the classifying spaces: B (D o ) o B (D o ) o

B(p2 )

B (S(F )) 

B(π2 )

B(p1 )

B(f )

B (S(id))

/ B (C ) 

B(π1 )

B(F )

(2.14)

/ B (D)

All the horizontal maps are homotopy equivalences. Hence, B(f ) is a homotopy equivalence. Consequently, B(F ) homotopy equivalence. The proof is complete.

Classifying Spaces of Categories

53

The Theorem B

2.2.2.

The [Q, Theorem B] provides the tool needed to use the long exact sequences in Homotopy theory (B.21) to develop the same for categories and functors. Let F : C −→ D be a functor. Fix Y ∈ Obj(D). Consider the commutative diagram, /C

j

Y /F

( D

Fj

 where

F

j(X, v) = X ∀ (X, v) ∈ Obj(Y /F ) j(u) = u ∀ u ∈ M orY /F ((X1 , v1 ), (X2 , v2 ))

So, B(F j) = B(F )B(j). Let CY : Y /F −→ D be the constant functor Y . Therefore, cY := B(CY ) : B (Y /F ) −→ B (D) is the constant map. Define a natural transformation CY −→ F j

by v : CY (X, v) = Y −→ F j(X, v) = F X

By (2.1.9), this leads to a homotopy H : B (Y /F ) × [0, 1] −→ B (D)

cY → B (F j)

For z ∈ B (Y /F ), t → H(z, t) is a path starting from Y to B(F j)(z). Using the opposite path H(z, −), we obtain a canonical map B (Y /F ) −→ F (B(F j), Y ) sending z → H(z, −)

(2.15)

Now we state the celebrated Theorem B. Theorem 2.2.3 (Theorem B). Let F : C −→ D be a functor. Assume ∀ v ∈ M orD (Y, Z), the functors v ∗ : Z/F −→ Y /F , as defined in (2.9), are homotopy equivalences. Then ∀ Y ∈ Obj(D), the commutative diagram Y /F F

j



Y /D

/C 

j

/D

F

⎧ ⎪ ⎨ j(X, v) = X is a homotopy cartesian, where F  (X, v) = (F X, v) ⎪ ⎩   j (Y , v) = Y 

(2.16)

Algebraic K-Theory

54

We further have the following: 1. Note, ∀ Y ∈ Obj(D), the category Y /D has an initial object (Y, 1Y ). So, Y /D is contractible. Therefore, the upper right part of the cartesian square (2.16) Y /F

j

/C

F

/D

is a homotopy fibration.

2. For all X ∈ Obj(C ) and F X = Y , there is a long exact sequence:

/ πn+1 (D, Y )

···

/ πn Y /F, X ˜

/ πn (C , X)

/ πn (D, Y )

/ ···

˜ := (X, 1Y ) ∈ Obj(Y /F )). ending at π0 (D, Y ), where X 3. This theorem admits a dual formulation, replacing the categories Y /F by categories F/Y , etc., in the diagram (2.16). Proof. Let S(F ) denote the category, as defined in the proof of (2.2.2). Consider the projection functors  Do

o

p2

S(F )

p1

/C

where

p1 (X, Y, v) = X p1 (u, w) = u p2 (X, Y, v) = Y p2 (u, w) = w

As in the proof of (2.2.2), p1 is a homotopy equivalence. Also, from the proof of (2.2.2), the map B(p2 ) : BS(F ) −→ B(D o ) is the realization of the map 

B(Y0 /F ) −→ N (D 0 )p

Y• ∈N Dpo

By hypothesis, ∀ v ∈ M orD o (Y, Y  ), the maps B(Y /F ) −→ B(Y  /F ) are homotopy equivalences. So, Lemma 2.1.20 is applicable, with I := D o , to the functor Y → B(Y /F ) from D o −→ Top. By Lemma 2.1.20, we see that Bp2 is a quasifibration. So, for all Y ∈ Obj(D o ),

Classifying Spaces of Categories

55

the maps B(Y /F ) −→ F (B(p2 ), Y ) are weak equivalences. By Whitehead’s Theorem C.3.8, the map is a homotopy equivalence. Therefore,

∀ Y ∈ Obj(D o )

Y /F

/ S(F )





the diagram

pt

p2

is a homotopy cartesian.

/ Do

Y

Now, we have the following commutative diagram (compare with diagrams (2.13, 2.14)): / S(F ) p1 ∼

Y /F 



/ S(1D ) ∼ π1

Y /D 



F

/D

 π2



pt

f

/C

Y

 / Do

The combination of two rectangles on the left is a homotopy cartesian. Since two vertical maps in the lower left rectangle are homotopy equivalences, the upper left rectangle is a homotopy cartesian. Also, two horizontal maps on the right-hand rectangle are homotopy equivalences. Therefore, the combination of two upper rectangles is a homotopy cartesian. 2.2.3.

Fiberd and cofiberd category version of Theorems A and B

Given a functor F : C −→ D, and an object Y ∈ Obj(D), one can define the fiber category in a natural way (2.2.4). The goal of this section is to interpret Theorems A and B in terms of the fiber categories. Definition 2.2.4. Let F : C −→ D be a functor. For Y ∈ Obj(D), refer to the definitions (2.2.1) of the categories Y /F and F/Y . Further, define the fiber category F −1 Y as follows:  Obj F −1 Y = {X ∈ Obj C : F (X) = Y } M orF −1 Y (X1 , X2 ) = {ϕ ∈ M orC (X1 , X2 ) : F (ϕ) = 1Y }

Algebraic K-Theory

56

There are two natural (inclusion) functors Φ : F −1 Y −→ Y /F and Φ : F −1 Y −→ F/Y , as follows. The functor Φ := ΦY 2F : F −1 Y −→ Y /F is given by  Obj Φ : F −1 Y −→ Obj(Y /F ) X → (X, 1Y ) ϕ ∈ M orF −1 Y (X1 , X2 ) ϕ → ϕ Likewise, define the inclusion functor Φ := ΦF 2Y : F −1 Y −→ F/Y . We take the liberty to use the same notation Φ when there is no confusion. 1. The functor F is called prefiberd if ∀ Y ∈ Obj D the functor ΦY 2F : F −1 Y −→ Y /F has a right adjoint RY 2F : Y /F −→ F −1 Y . So, we have natural bijection ∼

η : M orF −1 Y (X1 , RY 2F (X2 , u)) −→ M orY /F ((X1 , 1Y ), (X2 , u)) Likewise, the functor F is called precofiberd if the functor ΦF 2Y : F −1 Y −→ F/Y has a left adjoint LF 2Y : F/Y −→ F −1 Y . So, we have natural bijections ∼

η : M orF −1 Y (LF 2Y (X1 , u), X2 ) −→ M orF/Y ((X1 , u), (X2 , 1Y )) 2. Now suppose F is precofibered. So, ΦF 2Y : F −1 Y −→ (F/Y ) has a left adjoint LF 2Y : (F/Y ) −→ F −1 Y for all Y ∈ Obj C . So, ∀v ∈ M orD (Y, Y  ), we have a commutative diagram F −1 Y v

ΦF 2Y

/ (F/Y )

 



ΦF 2Y 

F −1 Y  l

,



v∗

(F/Y  )

 where

2nd varetical v∗ is as in (2.10) and, the 1st v := LF 2Y  v∗ ΦF 2Y

LF 2Y

These functors v are called the cobase change maps. If F is prefiberd, define v  := RY  2F v ∗ ΦY  2F : F −1 Y  −→ F −1 Y , where v ∗ is as in (2.9). These functors v  are called the base change maps. Subsequently, we may use the same notation v ∗ for both v ∗ , and same notation v∗ for both v∗ . 3. If F is precofibered, then F is defined to be cofibered if (vw)∗ = v∗ w∗ whenever v, w are composable. If F is prefibered, F is defined to be fibered, likewise.

Classifying Spaces of Categories

57

The following would be the prefibered or precofibered version of Theorem A (2.2.2). Corollary 2.2.5. Suppose F : C −→ D is either prefibered or precofibered such that F −1 (Y ) is contractible, for all objects Y in D. Then F is a homotopy equivalence. Proof. We prove the first case. By Corollary 2.1.10, F −1 (Y ) −→ Y /F is a homotopy equivalence. Since F −1 (Y ) is contractible, so is Y /F . Now, the corollary follows from Theorem A (2.2.2). The following would be the prefibered or precofibered version of Theorem B (2.2.3). Corollary 2.2.6. Suppose F : C −→ D is a prefibered (resp. precofibered) functor. Assume, for every arrow u : Y −→ Y  in D that the base change functor u∗ : F −1 (Y  ) −→ F −1 (Y ) (resp. cobase change functor u∗ : F −1 (Y ) −→ F −1 (Y  )) is a homotopy equivalence. Then ∀ Y ∈ Obj(D) the diagram F −1 (Y )

/C



 /D

pt

is a homotopy cartesian. (So, the upper right sequence is a homotopy fibration.) Consequently, for any X ∈ Obj(F −1 (Y )), there is a long exact sequence as follows:

/ πi+1 (D, Y ) / πi F −1 (Y ), X ··· / πi (C , X)

/ πi (D, Y )

/ ···

Proof. Again, it follows from (2.1.10) that Y  /F −→ Y /F is a homotopy equivalence. Now, the corollary follows by an application of Theorem B (2.2.3).

2.3.

Exercises

Exercise 2.3.1. Let n = {0, 1, . . . , n} be the partially ordered set. Compute the classifying space Bn.

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58

Exercise 2.3.2. Let G be a group. Let G denote the one object category , and M or(, ) = G. Describe the classifying space BG and establish that  G if n = 1 πn (BG, ) = 0 if n = 1 Exercise 2.3.3. Give an explicit proof of (2.1.11). Exercise 2.3.4. Let A be a small category, and B, C ⊆ A be two full subcategories. 1. We say B and C are disjoint if M orA (B, C) = φ, for all B ∈ Obj(B), C ∈ Obj(C ). 2. Given two disjoint subcategories B, C ⊆ A , we say A is disjoint union of B, C if, further, Obj(A ) = Obj(B) ∪ Obj(C ). We write A = B C to mean this is a disjoint union. In this case, prove BA = BB ∪ BC is a disconnected union, meaning BB, BC are open and BB ∩ BC = φ. 3. Two objects A0 , A ∈ A are defined to be connected if there is a sequence of objects A0 , A1 , . . . , An−1 , An = A such that ∀i = 0, 1, . . . , n − 1 either M or(Ai , Ai+1 ) = φ or M or(Ai+1 , Ai ) = φ. This is an equivalence relation ∼A on Obj(A ). If all pairs of objects in A are connected, we say that A is a connected category. In this case, prove that BA is path connected. 4. Prove that A is a disjoint union A =



Bi

where

∀i Bi is connected, ∀i = j

Bi , Bj are disjoint.

i∈I

Further, prove BA =



BBi

i∈I

is a decomposition into path-connected components. Exercise 2.3.5. Formulate the dual version of Theorem A (2.2.2) by replacing Y /F by F/Y ; and derive it from Theorem A (2.2.2). Exercise 2.3.6. Formulate the dual version of Theorem B (2.2.3) by replacing Y /F by F/Y ; and derive it from Theorem B (2.2.3).

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59

Exercise 2.3.7. Under the notation and hypotheses of Theorem A (2.2.2), prove that the homotopy fiber F (BF, Y ) is null homotopic. Exercise 2.3.8. Under the notation and hypotheses of Theorem B (2.2.3), prove that the natural inclusion map |Y /F | → F (BF, Y ) is a homotopy equivalence. (Hint: Use (C.3.8).) The Theorems A and B (2.2.2, 2.2.3) constitute a central piece of argument in the subsequent chapters and, in general, in this theory. We provide two other subsequent versions of Theorems A, B from [Wh], [GG]. First, we give the versions from [Wh, Lemma 1.4.A, 1.4.B]. Let f : L −→ K be a map in SSet. For integers n ≥ 0 and yn ∈ Kn , define the simplicial set F (f, yn ), by the pullback diagram F (f, yn )

/L





Δn

y

f

(2.17)

/K

The following is a version of Theorem A (2.2.2). Exercise 2.3.9 (Theorem AW). With notation as above, suppose |F (f, yn )| is contractible for all n ≥ 0 and yn ∈ Kn . Then |f | : |K| −→ |L| is a homotopy equivalence. (Hint: See [Wh, Lemma 1.4.A] and apply Theorem A (2.2.2).) The following is a version of Theorem B (2.2.3). Exercise 2.3.10 (Theorem BW). Suppose f : K −→ L is a map of simplicial sets. Assume that for all arrows ι : [m] −→ [n] in Δ, and yn ∈ Ln , the induced maps |F (f, ι∗ yn )| −→ |F (f, yn )| are homotopy equivalences. Then the diagram |F (f, yn )|

/ |K|





|Δn |

y

|f |

is a homotopy cartesian, ∀n, yn ∈ Kn .

/ |L|

(Hint: See [Wh, Lemma 1.4.B] and apply (2.2.3).) The following are from [GG].

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Exercise 2.3.11 (Theorem AGG). Let F : X −→ Y be a map of simplicial sets. Recall the notation ρ | F from (1.9). Assume, for all ρ ∈ Y ([m]), the geometric realization |(ρ | F )| is contractible. Prove that the geometric realization |F | : |X| −→ |Y | is a homotopy equivalence. (Hint: See [GG].) Exercise 2.3.12 (Theorem BGG). Let F : X −→ Y be a map of simplicial sets. Recall the notation ρ | F from (1.9). Assume, for all ρ ∈ Y ([m]), and maps f : [n] −→ [m], the map (ρ | F ) −→ (f ∗ ρ | F ) is a homotopy equivalence. Then for all integers n ≥ 0 and ρ ∈ Y [n] the square |(ρ | F )|

/ |X|





|(ρ | 1Y )|

|F |

is a homotopy cartesian.

/ |Y |

Consequently, there is a long exact sequence of homotopy groups, with all appropriate choices of base points. (Hint: See [GG].)

Chapter 3

Quillen K-Theory

Given a small Exact Category E , Quillen [Q] associates a category QE to be called Quillen’s Q-category. The homotopy groups of BQE are defined to be the K-groups of E .

3.1.

Quillen’s Q-Construction

Given an exact category E , a new category QE is associated in [Q], as follows. Definition 3.1.1. Suppose E is a small exact category. Define the category QE as follows. 1. The objects of QE are the same as that of E . Given two objects X, Y ∈ QE , we define morphisms X −→ Y in QE , as follows. Consider pairs (p, i) of arrows in E , as in the diagram X oo

p

Z



i

/Y

 ∃

⎧ ⎨ conflations

⎩

K Z





/Z i

/Y

p

//X //C

in E .

(3.1) In other words, p is a deflation, and i is an inflation. Two such pairs ⎧ ⎨ X o o p Z  i / Y are defined to be isomorphic if  ⎩ X o o  Z  / Y p

i

61

Algebraic K-Theory

62

Z Ap }} AAA i } AA } AA }} ~ ~} }  τ X `A Y ` AA }> } AA } AA  }}}i p .} p

∼

∃ an isomorphism τ : Z −→ Z 



commutes.

Z

Note that such an isomorphism τ is unique because i is monic. A morphism X −→ Y in QE is an isomorphism class [(p, i)] of such pairs, as in (3.1). A diagram, as in (3.1), will be denoted by (Z, p, i). 2. (Compositions): Let X −→ Y and Y −→ Z be two morphisms in QE , represented by X o o p W   i / Y , Y o o q V   j / Z . The composition is represented by (U, pq  , ji ), as in the diagram U q

X

oo

p

 _ i_ _/ V  

 

 W



i

j

/Z

q

where U = V ×Y W is the pullback.

(3.2)

/Y

We remark, the composition is well defined and associative. It is assumed that the classes of such diagrams (3.2) form a set. We introduce the following special morphisms and notation. Given an inflation ι : X → Y in E , there is an arrow ι! : X −→ Y in QE represented by X o o 1X X   ι / Y . Such an arrow in QE is referred to as “injective” [Q]. Likewise, given a deflation p : Y  X in E , there is an arrow p! : X −→ Y in QE represented by  X o o p Y  1 / Y . Such an arrow in QE is referred to as “surY jective” [Q]. (Such references as “injective” and “surjective” to these arrows are not meant to be confused with the injective and surjective maps, in the usual sense (A.1.2)). We insert a few basic lemmas. Lemma 3.1.2. Let E be an exact category. Consider the commutative diagram

Quillen K-Theory i

U p



/Y p



X

i

63

/V

Suppose i is an inflation and p is a deflation. Then if V is a pushforward of i, p, then U is a pullback of i, p. Conversely, suppose i is an inflation and p is a deflation. Then if U is a pullback of i, p then V is a pushforward of i, p. Proof. We prove the first assertion. Consider the commutative diagram WB

g

Bh B

f

B

U



i

p

   

X

#



i

q

/Y /V

//C

p q

//C

In this diagram, the first rectangle is a pushforward. Note C = co ker(i) = co ker(i) coincide. Let f, g be given arrows such that pf = ig. We show that there is a unique arrow h, so that the diagram commutes. Uniqueness of h, if it exists, follows because i is monic. Now, note qf = qpf = qig = 0. Since U = ker(q), f = ih for some h. Again, iph = pih = pf = ig. Since E is exact i is monic. So, g = ph. So, U is the pullback. The latter assertion follows similarly, or by reversing the arrows. The proof is complete. Lemma 3.1.3. Let E be an exact category and u : X −→ Y be an arrow in QE . Then there is a unique decomposition u = i! p! , where i! : U −→ Y is an injective map corresponding to an inflation i : U → Y , and p! : X −→ U is a surjective map corresponding to a deflation p : U  X. Further, there is another unique decomposition u = p! i! , where i! : X −→ V is an injective map, the map corresponding to an inflation i : X → V , and p! : V −→ Y is a surjective map, the map corresponding to a deflation p : Y  V .

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64

Proof. Let u be represented by (U, i, p). It follows immediately, from the definition of composition, that u = i! p! , and the decomposition is unique. For the latter decomposition, consider the pushforward diagram U p



i



/Y  p  

 X  _ _ _/ V i

It follows from the definition of the composition that u = p! i! . Uniqueness of such a decomposition follows from (3.1.2). The proof is complete. Lemma 3.1.4. Let E be an exact category. Let p, ρ be deflations and i, ι be inflations in the following diagram: V |>  `A` AA | AAp |  i | AA |  . ||| A  τ p   XB Y  BB } BB }}  } B } ι BB  ~ }} ρ  ~} W

Then p! i! = ρ! ι! ⇐⇒ ∃ isomorphism τ

 the diagram commutes.

Proof. Follows from (3.1.2) We list the following observations: 1. Given a bicartesian square (i.e., pullback and pushout) U ζ



X



ι

/Y 

 β

η

=⇒

ι! ζ ! = η ! β!

(3.3)

/Z

2. Lemma. If a map is both injective and surjective, then it is an isomorphism.

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65

Proof. Suppose u = Θ! = Ψ! . Then we have the commutative diagram X oo

1X

X

 Θ / Y

 Φ

 

X oo

Ψ

Y

1Y

/Y

This shows that Θ = Ψ−1 . Now, suppose Θ is an isomorphism in E . Then we have a commutative diagram X oo

1X

X

 Θ / Y

Θ

X oo

  

Θ−1

Y

1Y

/Y

It follows from the diagram that Θ! is an isomorphism in QE and its inverse is (Θ−1 )! . The proof is complete. 3. Lemma. Every morphism in QE is a monomorphism. Proof. Since every map in QE is a composition of an injective followed by a surjective map, we prove that such maps are monomorphisms in QE . Suppose η : Y → W is an inflation, and β : Y  W is a deflation in E . Consider two arrows in QE :     v = X o o q Z  ζ / Y u = X o o p Z  ι / Y , Now suppose η! u = η! v. Then we have the commutative diagram η! u :

X oo

p

Z



ι

/Y



η

/W

/Y



η

/W

 ϕ

η! v :

X oo

q

  ζ

Z

where ϕ is an isomorphism in E . Since η is a monomorphism in E , ζϕ = ι and hence u = v. Likewise, we check that β ! u = β ! v implies u = v. The proof is complete.

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66

4. The Universal Property of QE : First, we have the following observations: (a) There is no natural functor from E to QE . This is because, for f ∈ M orE (X, Y ), there is no natural way to associate an arrow in M orQE (X, Y ). (b) However, given an inflation ι : X → Y , in E , we can associate ι! in QE . Given two composable inflations ι, ζ in E , we have (ζι)! = ζ! ι! in QE . Likewise, given a deflation p : X  Y in E , we can associate p! in QE . Given two composable deflations p, q in E , we have (pq)! = q ! p! in QE . Further, (IdX )! = (IdX )! = IdX . (c) Given the bicartesian square (3.3), where all the maps are admissible (i.e. inflation or deflation), then ι! ζ ! = η ! β! . Let C be a category. Let h : E −→ C provide an association of objects to objects, inflations and deflations to maps (which is not necessarily a functor), as follows: / QE

E

 h

(C

⎧ ⎨ X → hX ∈ Obj(C ) ι → h! (ι) : hX −→ hY ⎩ p → h! (p) : hY −→ hX

∀ X ∈ Obj(E ) ∀ inflations ι : X → Y ∀ deflations p : X  Y

If h respects composition of inflations and deflations (3.1), then (a) h : E− −→ C is a functor, where E− is the category whose objects are the same as those of E , and morphisms are the inflations of E , (b) h : E+ −→ C is a functor, where E+ is the category whose objects are the same as those of E , and morphisms are the deflations of E . Finally, if h! and h! respect both the conditions 4b and 4c, then there is a unique functor Θ : QE −→ C such that the diagram / QE

E h

' 

Θ

 commutes and is compatible with

C

The following is an obvious lemma.

ι → ι! p → p!

Quillen K-Theory

67

Lemma 3.1.5. For any exact category E , we have an isomorphism of categories QE = Q(E o ) induced by the contra variant functor E −→ E o . 3.1.1.

Admissible layers

Let E be an exact category. There is an alternate and useful way to describe the morphisms in QE , in terms of admissible layers. Since it is very useful, we consolidate it in this separate subsection. Definition 3.1.6. Let E be an exact category and M ∈ Obj(E ). As usual, an inflation K → M in E is also called an admissible monomorphism, and similarly, deflations are called admissible epimorphism. Fix an object M in E . Two inflations ι : K → M , ι : K  → M , respectively, and two deflations p : M  C, p : M  C  are defined to be equivalent or isomorphic if there is a commutative diagram K 

 *



K



/M F

ι

, ι

p

M

//C

respectively, p

''





,

C

where the vertical arrows are isomorphisms. It follows that these vertical isomorphisms are unique. An admissible sub-object K of M is defined to be an isomorphism class of admissible monomorphisms ι : K → M . Likewise, an admissible quotient C of M , is defined to be an isomorphism class of admissible epimorphisms p : M  C. There is a bijection between the admissible sub-objects of M and admissible quotient objects of M . Note, for an inflation ι : K → M ∼ and for an isomorphism τ : M −→ M , with τ = 1M , ι and τ ι do not represent the same admissible sub-object of M . When M is understood, an admissible sub-object ι : K → M would be denoted by [(K, ι)] or (K, ι), or K itself (by abuse of notation). 1. Given M ∈ Obj(E ), let S(M ) denote the set of all admissible subobjects of M . Given (K1 , ι1 ), (K2 , ι2 ) ∈ S(M ), define (K1 , ι1 ) ≤ (K2 , ι2 ), if there is a (unique) inflation ι : K1 → K2 , such that

Algebraic K-Theory

68

ι1 = ι2 ι. We would abbreviate K1 ≤ K2 . So, S(M ) is a partially ordered set and hence a category. Likewise, let I(M ) denote the set of all admissible quotients. Then I(M ) is also a partially ordered set. 2. An ordered pair ((K1 , ι1 ), (K2 , ι2 )) of admissible sub-objects in S(M ) is defined to be an admissible layer if (K1 , ι1 ) ≤ (K2 , ι2 ). We abbreviate and say (K1 , K2 ) is an admissible layer. 3. The set of all admissible layers in M is denoted by L(M ). As usual, L(M ) is also a partially ordered set, where ∀ (K1 , K2 ), (L1 , L2 ) ∈ L(M ) define (K1 , K2 ) ≤ (L1 , L2 ) ⇐⇒ L1 ≤ K1 ≤ K2 ≤ L2 Lemma 3.1.7. Let E be an exact category. 1. Let f ∈ M orQE (C, M ) be represented by

f =



C

oo

p

K



ι

∼

/

M



.

−→ C is Then ker(p) ∈ S(M ) is well defined, and p : an isomorphism. Further, (ker(p), K) is a well-defined admissible layer in S(M ), hence represents an arrow ker(p) −→ K in the partially ordered set S(M ). 2. Conversely, let (K0 , ι0 ), (K1 , ι1 )) be an admissible layer in S(M ). So, ι0 = ι1 ι, for (unique) inflations ι : K0 → K1 . Let of cokernel of ι. Then p(ι) : K1  C(ι) be a choice K ker(p)

f =

C(ι) o o

K1 

p(ι)

 ι1 / M

represents a morphism f ∈

M orQE (C(ι), M ). 3. With notation as in (3.1.7), the association f → (K, ker(p)), p) is a bijection ∼

where IsoE K1 K0

∼







M orQE (C, M ) −→ K1 K0 , C

((K1 , K2 ), ϕ) : (K1 , K2 ) ∈ L(M ), ϕ ∈ IsoE



K1 ,C K0



denotes the set of all the isomorphisms

−→ C.

 For another morphism, g = M o o the pullback diagram to define gf :

q

L



i

/N

 , consider

Quillen K-Theory

   K0 × M L  _ _ _/ K × M L  _ _ _/ L   

K0



 q   / K 



i

69

/N

q

/M

with

K0 = ker(p)

p

C Then the bijection sends gf → ((ker(pq  ), K ×M L), pq  ) = (K0 ×M L, K ×M L, pq  ). Proof. Obvious! Given a category C , there is a simple way to construct a category of arrows in C (see (3.1.9)). We describe the same for partially ordered sets, as follows. Definition 3.1.8. Let (S, ≤) be a partially ordered set. We define the category Arr(S) of arrows of S. Let Arr(S) = {(x, y) ∈ S × S : x ≤ y}. For (x1 , y1 ), (x2 , y2 ) ∈ Arr(S), define (x1 , y1 ) ≤ (x2 , y2 ) if x2 ≤ x1 ≤ y1 ≤ y2 . This defines a partial order relation on Arr(S). So, (Arr(S), ≤) is a category. Lemma 3.1.9. Let E be an exact category and M ∈ Obj(E ). Let L(M ) := Arr(S(M  )) be the category of admissible layers in M . For arrow f = C o o p K   ι / M , in QE , define F (C, f ) = (ker(p), K) ∈ L(M ). Then F : QE /M −→ L(M ) is a an equivalence of categories. Proof. As described in (3.1.7), this association (C, f ) → F (C, f ) is well defined on Obj(QE /M ). For an arrow u ∈ M orQE /M ((C1 , f1 ), (C2 , f2 )), by chasing the definition of compositions in QE , one can see F (C1 , f1 ) ≤ F (C2 , f2 ). So, F defines a functor. It follows from the comments in (3.1.7) that F is fully faithful and essentially surjective. Therefore, F is an equivalence of categories. The proof is complete.

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70

3.2.

K0 (E ) and π1 (BQE , 0) of Exact Categories E

For exact categories E , we compare the classical Grothendieck group K0 (E ) and π1 (BQE , 0). First, we define the Grothendieck group of an exact category. Definition 3.2.1. Suppose E is a small exact category. Let Z(E ) be the free abelian group generated by the objects of E . Let R(E ) be the subgroup of Z(E ) generated by the set 

 / / Z is a conflation. /Y Y − X − Z ∈ Z(E ) : X  Define the Grothendieck group

K0 (E ) :=

Z(E ) R(E )

For X ∈ Obj(E ), its image in K0 (E ) is denoted by [X]. We remark: 

/0 / / 0 implies 0 = [0] ∈ K0 (E ). 1. The conflation 0  ∼  /X π //Y 2. Suppose π : X −→ Y is an isomorphism. Then 0  is a conflation. Hence, [X] = [Y ] ∈ K0 (E ). 3. The map ι : Obj(E ) −→ K0 (E ) has the following universal property: Suppose G is an abelian group and β : Obj(E ) −→ G is a set theoretic map such that

β(Y ) = β(X)β(Z)

X

∀ conflations



/Y

/ / Z,

then there is a unique group homomorphism ι

Obj(E ) ψ : K0 (E ) −→ G

 β

/ K0 (E )   ψ commutes.   *G

We say that K0 (E ) has the above universal property in the category Ab of abelian groups. We will see next that K0 (E ) has the same universal property in the category Gr of groups. 4. The above definition of K0 (E ) makes sense whenever E has a set of isomorphism classes of objects, by replacing E by an equivalent small exact subcategory E  ⊆ E . In this case, let E be the set

Quillen K-Theory

71

of all isomorphism classes of objects in E , and Z(E) be the free abelian group generated by E. Let R(E) ⊂ Z(E) be defined as above. Then Z(E) . K0 (E ) = R(E) Lemma 3.2.2. Let E be a small exact category. Then the map ι : Obj(E ) −→ K0 (E ) has the following universal property: Given a group G (possibly non-commutative) and a set theoretic map ζ : Obj(E ) −→ G such that ζ(Y ) = ζ(X)ζ(Z)

X

∀ conflations



/Y

/ / Z,

then there is a unique group homomorphism ι

Obj(E ) ψ : K0 (E ) −→ G

 ζ

In particular ,

K0 (E ) :=

/ K0 (E )   ψ commutes.   *G

Z(E ) ∼ F (E ) = R(E ) N (E )

where F (E ) is the free (non-abelian) group generated by Obj(E ), and N (E ) is the normal subgroup generated by the set 

 / / X is a conflation . /Y W = Y (XZ)−1 : X  F (E ) Proof. Write G (E ) := N (E ) . (Recall elements of F (E ) are “words”.) Note that the map η : Obj(E ) −→ G (E ) has the universal property in the category Gr of groups. For X, Y ∈ Obj(E ), using the split conflation, it follows η(X)η(Y ) = η(X ⊕Y ) = η(Y ⊕X) = η(Y )η(X). Form this, it follows, G (E ) is abelian. Therefore, both G (E ) and K0 (E ) have the same universal property in the category Ab of abelian groups. Hence, G (E ) ∼ = K0 (E ). The following proposition establishes the key connection between classical K-theory to homotopy groups of BQE .

Proposition 3.2.3. Let E be a small exact category. The classifying space BQE is considered as a pointed topological space, the base point being a fixed zero object 0 of E . Then

Algebraic K-Theory

72

1. The classifying space BQE is (path) connected. So, π0 (BQE ) = 0. ∼ 2. There is an isomorphism K0 (E ) −→ π1 (BQE , 0). Proof. The following proof is drawn from [Sm1], which is more elaborate than the one in [Q]. Suppose X is an object of E or QE . The diagram 0

0oo



0X

/X

represents a map 0 −→ X

in

QE .

This gives a path from 0 to X. Since all the vertices of BQE are path connected to 0, QE is path connected. Hence, π0 (BQE ) = 0. (We comment that M orQ(E ) (0, X) is not necessarily a singleton.). With notation, as Lemma 3.2.2, K0 (E ) =

F (E ) . N (E )

Define a group homomorphism ϕ0 : F (E ) −→ π1 (BQE ), as follows. For f ∈ M orQE (X, Y ), the corresponding path from X to Y , in the classifying space B(QE ), is denoted by γ(f ). For X ∈ Obj(E ), there are two natural arrows in M orQE (0, X), as follows:  (0, 0, 0X ) =

0

oo

0



0X

/X



 ,

(X, 0, 1X ) =

0

oo

X



1X



/X .

By (2.1.3), these two arrows define two paths γ0X := γ((0, 0, 0X ), X γ1X := γ((X, 0, 1X ) in BQE , from 0 to X. So, X := γ X 0 γ1 is a loop X at 0, where γ X 0 denotes the path opposite to γ0 . Define ϕ0 : F (E ) −→ π1 (BQE )

by ϕ(X) = [ X ]

Now, consider a conflation X



i

/Y

p

//Z

X Now, X = γ X 0 γ1 : 0

which corresponds to γ0X

Y (XZ)−1 ∈ N (E ).

+

3 X is homotopic to

γ1X



γ(X, 1X , i)γ0X



γ(X, 1X , i)γ1X

γ0X

: 0

+

3X

γ(X,1X ,i)

/ Y.

γ1X

It follows, (X, 1X , i)(0, 0, 0X ) = (0, 0, 0Y ) ∈ M orQE (0, Y ) and (X, 1X , i)(0, 0, 1X ) = (X, 0, i) ∈ M orQE (0, Y ). So, [ X ] =

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[γ Y0 γ(X, 0, i)]. Likewise, [ Z ] = [γ(X, 0, i)γ1Y ]. So, [ X ][ Z ] = [ Y ]. Therefore, ϕ0 factors through a well-defined homomorphism ϕ : K0 (E ) −→ π1 (BQE , 0). Now, we prove that ϕ is surjective. Let ∈ π1 (BQE , 0). By (C.3.9), the map π1 BQE 1 , 0  π1 (BQE , 0) is surjective, where BQE 1 denotes the 1-skeleton. Therefore, is represented by a product of paths γn γn−1 · · · γ1 such that 1. γi is a path from Xi−1 to Xi , with Xi ∈ Obj(E ), and X0 = Xn = 0. 2. Further, (a) either γi = γ(fi ), in which case fi ∈ M orQE (Xi−1 , Xi ), (b) Or γi = γ(fi ), in which case fi ∈ M orQE (Xi , Xi−1 ). 3. In fact, for any nonzero object X in E , M orQE (X, 0) = φ. So, we can assume f1 : 0 −→ X1 , and fn : 0 −→ Xn−1 . Let gi = γ(Xi , 0, 1Xi ). Up to homotopy, we have γn γn−1 · · · γi · · · γ1 = (γn gn−1 ) · · · (g i γi gi−1 ) · · · (g 2 γ2 g1 ) (g 1 γ1 ) . We will prove that, each [(g i γi gi−1 )] is in the image of ϕ. There are two cases, 1. γi = γ(fi ), where fi : Xi−1 −→ Xi . So, fi is represented by Xi−1 o o

Ui 

p



ι

/ Xi .

We have gi−1 = γ(Xi−1 , 0, 1Xi−1 ), given by the arrow 0oo

1

Xi−1

So, γi gi−1 is given by the arrow 0oo

Ui 

 ι

/ Xi−1 . / Xi .

Therefore, g i γi gi−1 = γ(Xi , 0, 1Xi )γ(Ui , 0, ι) ∼ γ(Xi , 0, 1Xi )γ ((Ui , 1Ui , ι)o(Ui , 0, 1Ui )) ∼ γ(Xi , 0, 1Xi )γ(Ui , 1Ui , ι)γ(Ui , 0, 1Ui ) ∼ γ(Xi , 0, 1Xi )γ(Ui , 1Ui , ι)γ(0, 0, 0Ui )γ(0, 0, 0Ui )γ(Ui , 0, 1Ui ) ∼ γ(Xi , 0, 1Xi )γ ((Ui , 1Ui , ι)o(0, 0, 0Ui )) γ( Ui ) ∼ γ(Xi , 0, 1Xi )γ(Xi , 0, 0Xi )γ( Ui ) ∼ γ( Xi )γ( Ui )

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So, [(g i γi gi−1 )] = ϕ(Xi )−1 ϕ(Ui ) 2. γi = γ(fi )−1 , where fi : Xi −→ Xi−1 . This case is dealt similarly, as above. This establishes that ϕ is surjective. To prove that ϕ is injective, we define a map π1 (BQE , 0) −→ K0 (E ). As in (2.1.5), let K0 (E ) denote the category with one object  and M or(, ) = K0 (E ). Define a functor Φ : QE −→ K0 (E )

for f :=



p X oo

W

 ι

∀ X ∈ Obj(QE ) Φ(X) =  

/Y

∈ M orQE (X, Y ),

and,

Φ(f ) = [ker(p)].

We check that Φ is a functor. For another morphism   g := Y o o q V   ι / Z ∈ M orQE (Y, Z) we need to prove Φ(gf ) = [ker(p)] + [ker(q)]. 

/ / ker(p) is a confla/ ker(qp ) This follows, because ker(q)  tion. Therefore, Φ is a functor, which induces a map of the classifying spaces BQE −→ BK0 (E ). Hence, it induces a map

  η := π1 (BΦ) : π1 (BQE , 0) −→ π1 K0 (E ),  = K0 (E ). Now, for [X] ∈ K0 (E ), η(ϕ([X])) = η(γ(0, 0, X)−1 γ(X, 0, 1X )) = −[0] + [X] = [X]. Therefore, ϕ is a bijection. The proof is complete. 3.3.

Higher K-Groups of Exact categories

Now we define higher K-groups.

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Definition 3.3.1. Suppose E is a small exact category and 0 denotes the zero object. ∀n∈N

define Kn (E ) = πn+1 (BQE , 0)

The definition is independent of the choice of the zero object, because BQE is path connected. Further, the definition extends to any exact category E that has a set of isomorphism classes of objects. Lemma 3.3.2. Let F : E −→ D be an exact functor of exact categories. Then F induces a functor QF : QE −→ QD. The map QF sends an object X ∈ E to F X and a morphism W f :=



X



/Y

FW ∈ M orQE (X, Y )

→



/ FY ∈ M orQD (F X, F Y )



FX

Consequently, a continuous map BQF : BQE −→ BQD of pointed spaces is obtained. This induces homomorphisms of K groups Fn := Kn (F ) : Kn (E ) −→ Kn (D)

∀

n ≥ 0.

The following is a list of immediate consequences of (3.3.2). 1. For each (fixed) n ≥ 0, the association n → Kn (E ) is a functor from the category of small exact categories and exact functors to the category Ab of abelian groups. ∼ 2. If F : E −→ D is an equivalence of exact categories, then F induces a natural equivalence QE −→ QD. By (2.1.9), the induced ∼ maps Fn : Kn (E ) −→ Kn (D) are isomorphisms, ∀ n ≥ 0. Further, suppose F, G : E −→ D are two exact functors of exact categories. Let θ : F −→ G be a natural equivalence. Then θ induces a natural transformation (equivalence) QF −→ QG. Consequently, by (2.1.9), θ induces a homotopy between the maps BQF and BQG. Therefore, Kn (F ) = Kn (G) : Kn (E ) −→ Kn (D) for all n ≥ 0. (For two functors F, G : E −→ D of exact categories, a natural transformation θ : F −→ G need not induce a natural transformation from QF to QG. So, they need not induce homotopic maps.).

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3. For any exact category E , by (3.1.5), we have ∀n ≥ 0

Kn (E o ) = πn+1 (BQ(E o )) = πn+1 (B(QE )) = Kn (E ).

4. Suppose E , D are two exact categories. Then E ×D is also an exact category. It follows Q(E × D) = Q(E ) × Q(D). The natural bijective map ι : B(Q(E × D)) −→ B(Q(E )) × B(Q(D)) is continuous and is not necessarily a homeomorphism (see (2.1.8)). However, when the right side is endowed with compactly generated topology (see (2.1.8)), then ι is indeed a homeomorphism. Fix n ≥ 0 and consider the commutative diagram ∼ / πn ((BQ(E ) × (BQD), (0, 0)) RS TT  UV VW+ 

πn (BQ(E × D), 0)

πn (BQ(E , 0) ⊕ πn (BQD, 0)

Here the horizontal map is induced by ι. Since homotopy groups are determined by the compactly generated topology, the horizontal map is an isomorphism. Further, the vertical map is induced by the two projection maps, which is also an isomorphism, by usual topological arguments. So, it follows, ∀ n ≥ 0 the maps ∼

Kn (E ×D) −→ Kn (E )⊕Kn (D)

sending x → (pE )∗ (x)+(pD )∗ (x)

are isomorphisms, where pE , pD denote the projection functors. 5. Suppose E is an exact category. Then the direct sum functor E × E −→ E , sending (M, N ) → M ⊕ N , is an exact functor. This induces a group homomorphism ⊕∗ : Kn (E ) ⊕ Kn (E ) −→ Kn (E )

∀ n ≥ 0.

The map coincides with the direct sum map because of the commutative diagram Kn (E ) 1 i1



Kn (E ) ⊕ Ki (E ) O

⊕∗

i2

Kn (E )

1

# / Kn (E ) ;

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77

where i1 , i2 are induced, respectively, by the functors M → (M, 0) and M → (0, M ). Lemma 3.3.3. Suppose I is a small filtering category and j → Ej is a functor from I to the category of small exact categories. Let limj∈I Ej denote the filtered limit, which is an exact category. Then Q(limj∈I Ej ) = limj∈I (Q(Ej )). Hence, by (2.1.16),  Kn lim Ej = lim Kn (Ej ) ∀ n ≥ 0. j∈I

j∈I

Proof. It is obvious that Q(limj∈I Ej ) = limj∈I (Q(Ej )) is an exact category. The lemma is an immediate consequence of (2.1.16). Example 3.3.4. Let P(A) denote the category of all finitely generated projective (left) A-modules, where A is a ring with unity. Write Kn (A) := Kn (P(A))

∀n≥0

1. Any homomorphism A −→ B of rings induces an exact functor P(A) −→ P(B), sending P → B ⊗P . This induces a well-defined group homomorphism Kn (A) −→ Kn (B) ∀ n ≥ 0. So, ∀n ≥ 0, the association A → Kn (A) is a covariant functor from the category of rings with unity to the category of abelian groups. 2. It follows, ∀n≥0

Kn (A × B) ∼ = Kn (A) ⊕ Kn (B)

are naturally isomorphic. 3. Suppose I is a small filtering category and j → Aj is a functor from I to the category of rings with unity. It follows, from (3.3.3):  ∀n≥0 Kn lim Aj = lim Kn (Aj ). j∈I

j∈I

Example 3.3.5. Let X be a (noetherian) scheme. (For example X = Spec (A).)

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1. The category QCoh(X) of the quasi-coherent modules over X is an exact category. So, this gives rise to a K-theory. 2. The category Coh(X) of the coherent modules over X is also an exact category. So, this gives rise to a K-theory. This is often referred to as G-theory of X, and the K-groups are denoted by Gn (X). 3. Likewise, the category P(X) of locally free modules over X is an exact category (as in 3.3.4) . So, this gives rise to a K-theory. This is usually referred to as K-theory of X, and the K-groups are denoted by Kn (X). A comparison of the K-theory of these categories constitutes a big chunk of the literature. The classical K-theory mostly refers to the K-theory of the category P(A) of finitely generated projective A-modules, where A is a (commutative) ring (3.3.4). Classically, three groups were defined, which we (temporarily) denote by K0c (A), K1c (A), K2c (A). It seems odd to use superscript c to denote classical K-groups, because it predates everything we have discussed so far. However, they all coincide with Quillen K-groups. We have already established that K0c (A) agrees with Quillen K0 (A) (3.2.3). The coincidence of the other two is dealt with in Chapter 4. However, we state them here. Lemma 3.3.6. Let A be a (commutative) ring. Refer to Sections 4.7.1 and 4.7.2 for further details and notation. The classical K1 group of A is defined as follows (see 4.7): GL(A) . We have K1c (A)  K1 (P(A)). K1c (A) = EL(A) Proof. see (4.7.6). Lemma 3.3.7. Let A be a (commutative) ring. The classical K2 group of A, denoted temporarily by K2c (A), is defined as a subgroup of the so-called “Steinberg Group” St(A), given in [Mi, §5], and discussed in Section 4.7.3. We have, K2c (A)  K2 (P(A)) Proof. See Section 4.7.3.

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3.4.

79

Exact Sequences and Filtrations

Let E be an exact category, and let ε(E ) be the category of the exact sequences in E . More precisely, objects X, Y of ε(E ) are the exact sequences (i.e. conflations) in E , and a morphism f : X −→ Y is a commutative diagram in E : X := f



Y :=

/ K 

0

/M

ι

 / K  

0



//C

β

/ M



/0

(3.4)

ϕ

/ / C

/0

1. Given X ∈ Obj(ε(E )), as in (3.4), write s(X) := K, t(X) := M , and q(X) := C. Then s, t, q : ε(E ) −→ E define three functors. The images s(X), t(X), q(X) would, respectively, be called sub, total, and quotient objects of X. 2. In fact, ε(E ) has a structure of an exact category, where a sequence /Y / Z is declared exact (i.e. conflation) if the correX sponding vertical sequences, as in diagram (3.4), are exact. 3. It follows that s, t, q : ε(E ) −→ E are exact functors. Theorem 3.4.1. Let E be an exact category. Then the functor (s, q) : Q(ε(E )) −→ Q(E ) × Q(E ) is a homotopy equivalence. Proof. By Theorem A (2.2.2), it would be enough to show that (s, q)/(K, C) is contractible, for any (K, C) ∈ Obj (Q(E ) × Q(E )). Fix such a pair (K, C), and let C (K, C) := (s, q)/(K, C). So, Obj(C (K, C)) = (s, q)/(K, C) = {(X, u, v) : u ∈ M orQE (sX, K), v ∈ M orQE (qX, C)} More explicitly, u= v=



sX o o

U

qX o o

V

 

/K /C



, (3.5)

1. Let C  (K, C) be the full subcategory of C (K, C), consisting of objects (X, u, v) such that u is surjective (in the sense of  1 / Section 3.1), meaning u looks like u = sX o o K . K

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80

2. Let C ”(K, C) be the full subcategory of C (K, C), consisting of objects (X, u, v) such that u is surjective and v is injective (in the sense of Section 3.1), meaning u, v look like     1  1 / o o / v = u = sX o o C . qX qX K K Now the theorem follows from (3.4.2) as follows. The proof is complete. Lemma 3.4.2. Use the notation, as in the proof of (3.4.1). Then the inclusion functors C  (K, C) −→ C  (K, C) and C  (K, C) −→ C (K, C) have left adjoints. Consequently, both are homotopy equivalences (by (2.1.10)). Further, C  (K, C) is contractible, and hence so is C (K, C). Proof. We will abbreviate notations C := C (K, C), C  := C  (K, C), C  := C  (K, C). Given X = (X, u, v) in C , we will establish that there is a universal arrow χ : X −→ L(X ), with L(X ) ∈ C  (K, C). This means, given X1 ∈ C  and f : X −→ X1 in C (K, C), there is a unique morphism Φ, as in the commutative diagram χ

X f

/ L(X )   ∃ Φ unique  ' 

(3.6)

X1

By (A.1.15) this completes the proof that C  (K, C) −→ C (K, C) has objects E ∈ Obj(ε(C ) are denoted by E =  a left adjoint. The 

 ι / tE ζ / / qE . sE   in (3.5). Write X =   With X = (X, u, v) ∈ Obj(C ) and u as  ι /  i / ζ p  / / . Rewrite u = o o tX qX sX sX M K (compare with (3.5)). Consider the pushforward diagram

sX _ 



/ tX  _ u  β i     p / / M  _ _ _/ T K ι

ι

ζ

/ / qX

(3.7) ζ

/ / qX

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81

where T is the pushforward of i and ι, as shown. Define X =   ζ   ι / / / qX . We also denote i∗ X := X. Note, s(X) = M , T M t(X) = T , q(X) = qX. Further, p : K  s(X) induces the (surjective) map p! : M −→ K in QE . Now, define L(X ) := (X, p! , v). So, X is in C  . Note (i, β, 1qX )! : X −→ X defines an (injective) arrow in Qε(E ). It follows that (i, β, 1qX )! : X −→ L(X ) is an arrow in C (K, C). Define χ := (i, β, 1qX )! ∈ M orC (K,C) (X , L(X )). Now, let f : X = (X, u, v) −→ X1 be an arrow in C (K, C), as in (3.6). In particular, X1 = (X1 , u1 , v1 ), with u1 = j ! , for some deflation j : K  sX1 . We split the map f in QεE as f = h! g! : X −→ X1 , where h : X → X0 is an inflation and g : X1  X0 is a deflation in εE . The condition u = (sf )u1 is given by the diagram 8< K j

1   1 / sX sX 1  1 < p  sh  sh sf    sX   u sg / sX0  _ _1 _/ sX0D DD DD ∼ DDD   " i - sX

u

u



So, we can assume

sX0 = sX = M i = sg

So, the splitting of f is displayed in the following commutative diagram:  sX n _

K



{{ N i {{ sg { { }{{   / sX0  M ∼ OO ~> > p ~~~ sh ~~ ~~  // j

sX1

ι

/ tX _ 

ι1

ζ

sg

tg

/ tX0 OO



ζ1

/ tX1

/ / qX0 OO qh

th ι1

/ / qX _

ζ1

/ / qX1

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82

We construct a map X −→ X1 , as follows:

K

}

U

p

j



ζ / tXu  n U  β i )    sg tg / / M  D_ _ _ _ _ _ _ _/ T C   DD ι C tϕ ζ DD∼ C D sϕ=1 DD C!  !   sX0 _ _ _ ι_ _ _ _/ tX0 0 OO OO

sX u n

u

ι

((

sh

sX1

/ / qX n 1



/ / qX D

ζ0

/ tX1

ι1

D qϕ D D!  / / qX0 OO qh

th



qg

ζ1

/ / qX1

Here tϕ is defined by the pushforward property. The arrow ϕ = (sϕ, tϕ, qϕ) : X −→ X1 in εE is an inflation (because sϕ = 1, qϕ = qg are monic). Define Φ : X −→ X1 in QεE by combing ϕ and h. Note that qϕ = qg and hence qΦ = qf . It follows that Φ : L(X) = (X, p! , v) −→ (X1 , j ! , v1 ) and the diagram (3.6) commutes. To see uniqueness, let Ψ : L(X) −→ X1 be another arrow ! in  QεE commuting the diagram (3.6). Write Ψ = γ α! :=  α / oo γ ! X Y X1 . We have f = Φχ = Ψχ, which is h g! χ = γ ! α! χ. This equality is given by the isomorphism τ , as in the (partly) commutative diagram > X 0 `B ~~  ` BBB h ~ BB ~~ BB  / ~~~   χ /  p  τ X XA X1  AA || AA  | | α AAA  ||| γ  }|} ϕ



γ = τ h,

αχ = τ ϕχ

Y

It remains to be shown that α = τ ϕ. Since qχ = 1, it follows that qα = q(τ ϕ). Since Ψ : L(X) −→ X1 , we have j ! sΨ = p! . The following diagram shows sY ∼ = M.

Quillen K-Theory

M k

1

sα

83

/ M o o p K.    '  w w sγj

sY So, we can assume sY = M = sX0 , sα = 1, p = sγj. We expand the above diagram as follows: 

ζ / tXu  n U  i=sχ β=tχ ) u    sg tg _ _ _ _ _ _ _ _ / M T f C  8 _ D 8   p DD ι C tϕ ζ ppp DDsϕ=1 p p C DD pppp D!   C!   pppp  1 K MS _ _ _ ι _ _tα_ _/ tX0 0 { SS zz S sτ =1 zz {{ { zz {{ tτ   }z  }{{ z / tY h h M hh sh th ιY ζY

sX u n

U

j

ι

sγ

((

tγ

sX1

 ι1

/ tX1

/ / qX n 

qχ=1

/ / qX f D

ζ0

D qϕ D D!  / / qX0 qα SS

 / / qY h h

ζ1

qg

qh qγ

/ / qX1

Note that p is an epimorphism. We have s(τ h) = sγ =⇒ s(τ h)j = sγj = p =⇒ sτ ((sh)j) = sγj = p =⇒ sτ p = p =⇒ sτ = 1.

Further, (tτ )(tϕ)ι = (tτ )ι0 = ιY = (tα)ι ,

Also, τ ϕχ = αχ =⇒ t(τ ϕχ) = t(αχ).

Since T is a pushforward, we have t(τ ϕ) = tα. So, we have established that τ ϕ = α. This completes the uniqueness of Φ. Now, we prove that C  −→ C  has a left adjoint. Let X = (X, u, v) be an object in C  , where u = p! : sX −→ K is surjective and  v : qX −→ C is a morphism in QE , which is given by v := qX o o j M   i / C . Consider the diagram

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84

X := X=



0

/ sX 

0



/ sX

/T

/M





/ tX

/0

j

/ qX

/0

where T is the pullback. We also write j ∗ X := X. By a dual argument, similar to the above, it is established that there is a universal arrow X −→ L(X ), with L(X ) := (j ∗ X, u, i! ) ∈ Obj(C  ). This completes the proof that C  → C  has a left adjoint and hence is a homotopy equivalence. To prove C  is contractible, let 0 ∈ Obj (ε(E )) denote the zero conflation. Corresponding to the arrow K  0, let pK ∈ M orQE (0, K) denote the surjective map, and corresponding to the 0 → C, let iK ∈ M orQE (C, 0) denote the injective map. Consider the object I = (0, pK , iC ) ∈ C  . It is clear the I is an initial object in C  . So, C  is contractible. The proof is complete. 3.4.1.

Additivity theorem

In this section, we give several applications of Theorem 3.4.1. We define exact sequences of exact functors and admissible filtration of exact functors. Definition 3.4.3. Let E , D be two exact categories (small). 1. Let C = F un(E , D) be the category of all exact functors whose objects are exact functors and a morphism Φ : F −→ G is a natural transformation. 2. In fact, C = F un(E , D) is an exact category, where a sequence 0

/Φ

/Ψ

/Θ

/0

is declared exact (i.e. a conflation), if ∀ X ∈ Obj(E )

/ Φ(X)

0

/ Ψ(X)

/ Θ(X)

/0

is exact.

3. A sequence of natural transformations (morphisms), in C , 0 = Φ0 



/ Φ1 



/ Φ2 



/ ···

Φr = Φ

is defined to be an admissible filtration of Φ if ∀ X ∈ Obj(E ), ∀ i = 0, . . . , r − 1

Φi (X)



/ Φi+1 (X) are inflations in D.

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85

4. Given an admissible filtration, as above, the quotient morphisms Φi+1 Φi are defined, and determined up to isomorphisms. There is no reason why such a quotient would be exact. However, if exact ∀ i = 0, 1, . . . , r − 1, then

Φp Φq

Φi+1 Φi

is

is exact, for all 0 ≤ q ≤ p ≤ r.

The following is known as the Additivity theorem. Theorem 3.4.4. Let E , D be two exact categories. Let G, F, H : E −→ D be three exact functors, such that /G

0

/F

/H

∀n≥0

Then

/0

is also exact.

(3.8)

F∗ = G∗ + H∗ : Kn (E ) −→ Kn (D).

Proof. As before, let ε(D) denote the exact category of short exact sequences in D. Let s, t, q : ε(D) −→ D be the sub, total, and quotient functors. Then /s

0

/t

/q

/0

is exact.

First, we prove that t∗ = s∗ + q∗ : Ki (ε(D)) −→ Ki (D). Let f : D × D −→ ε(D) be the functor sending (K, C) to the split exact sequence /K

0

/K ⊕C

/C

/ 0.

Note (s, q)f = Id. By Theorem 3.4.1, the functor (s, q) : Q(ε(D)) −→ Q(D)×Q(D) is a homotopy equivalence. Hence, f : Q(D)×Q(D) −→ Q(ε(D)) is also a homotopy equivalence. Consider the commutative diagram of functors QD × QD f

f





Qε(D) (s,q)

/ Qε(D)

t



QD × QD

⊕

which is tf = ⊕(s, q)f Therefore, (BQt)(BQf ) = (BQ⊕)(BQ(s, q))(BQf )

 / QD

Since, BQf is a homotopy equivalence, BQt and (BQ⊕)(BQ(s, q)) are homotopic and hence induce the same morphisms on the homotopy

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86

groups. Therefore, ∀n≥0

Kn (t) = Kn (⊕)Kn ((s, q)) = Kn (s) + Kn (q) : Kn (D) −→ Kn (D).

With the notation of the theorem, this means ∀ n ≥ 0, Kn (D) −→ Kn (D). To prove the theorem, consider the exact functor Φ : E −→ ε(D) sending X →



/ G(X)

0

/ F (X)

t∗ = s∗ +q∗ :

/ H(X)

/0



Note tΦ = F , sΦ = G, and qΦ = H. Therefore, ∀ n ≥ 0, we have the commutative diagram Kn (E )

Φ∗

F∗ =G∗ +H∗

/ Kn (ε(D)) )



t∗ =s∗ +q∗

Kn (D)

The proof is complete. Corollary 3.4.5. Let Φi : C −→ D be exact functors, for i = 0, 1, . . . , n. Suppose 0

/ Φ0

/ Φ1

/ Φ2

/ ···

/ Φn

/0

is an exact sequence. Then n 

(−1)p (Φp )∗ = 0 : Ki (E ) −→ Ki (D).

p=0

Proof. Follows from (3.4.4), by induction. Corollary 3.4.6. Let Φ : C −→ D be an exact functor. Suppose Φ admits an admissible filtration 0 = Φ0 



/ Φ1  

Φp+1 Φp

/ Φ2  

/ · · · 

/ Φr = Φ

is exact ∀ p = 0, 1, . . . , r − 1. Then r   Φp : Kn (E ) −→ Kn (D) ∀ n ≥ 0. Φ∗ = Φp−1 ∗

such that

p=1

Proof. Follows from (3.4.4).

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3.4.2.

87

Some examples

Example 3.4.7. Let A be a commutative Noetherian ring and P(A) be the category finitely generated projective A-module. Then we write Ki (A) := Ki (P(A)). This example was discussed in (3.3.4). Given a finitely generated projective A-module P , the association Q → P ⊗ Q defines an exact functor P ⊗ : P(A) −→ P(A). This will induce homomorphisms as follows: [P ⊗ −] : Kn (A) −→ Kn (A)

∀n ≥ 0

In fact, short exact sequences 0

/ P−1

/ P0

/ P1

/0

in P(A)

induce exact sequences of exact functors: 0

/ P−1 ⊗ −

/ P0 ⊗ −

/ P1 ⊗ −

/ 0.

So, it follows, from (3.4.4), [P0 ⊗−]∗ = [P−1 ⊗−]∗ +[P1 ⊗−] : Kn (A) −→ Kn (A).

∀n ≥ 0

The same works for any Noetherian scheme X (more generally ringed spaces). The above example can be formulated for the category P(X) of locally free sheaves, exactly in the same manner. Before the next example, we give the following graded versions of Nakayama’s Lemma. Lemma 3.4.8.  Let S = S0 ⊕ S1 ⊕ S 2 ⊕ · · · be a graded commutative ring and S+ = i≥1 Si . Let M = n∈Z Mn be a finitely generated graded S-module. Assume M = S+ M . Then M = 0.

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Proof. Let m1 , m2 , . . . , mk be a set of homogeneous generators of M . We have ⎞ ⎛ ⎞⎛ ⎞ ⎛ a11 a12 · · · a1k m1 m1 ⎜ m2 ⎟ ⎜ a21 a22 · · · a2k ⎟ ⎜ m2 ⎟ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎝ ··· ⎠ = ⎝ ··· ··· ··· ··· ⎠⎝ ··· ⎠ mk ak1 ak2 · · · akk mk where aij ∈ S are homogeneous elements of positive degree. Let A = (aij ) be the k × k matrix, as above. So, ⎞ ⎛ ⎞ ⎛ 0 m1 ⎜ m2 ⎟ ⎜ 0 ⎟ ⎟ ⎜ ⎟ (I − A) ⎜ ⎝ · · · ⎠ = ⎝ · · · ⎠. mk 0 Multiplying by the Adj(I − A), we have det(I − A)mi = 0 ∀ i = 1, . . . , k. It follows that det(A) = 1 + f1 + · · · + fn , where fj ∈ Aj , for j = 1, . . . , n. Therefore, ∀ i = 1, . . . k

(1 + f1 + · · · + fn )mi = 0;

hence mi = 0.

So, M = 0. The proof is complete. The following is a graded analogue of the corresponding ungraded result.  Lemma n∈Z Pn ,  3.4.9. Let S and S+ be as in (3.4.8). Let P = Q = n∈Z Qn be finitely generated graded projective S-modules. Let  ∼ Q P S+ = i≥1 Si . Let ϕ0 : S+ P −→ S+ Q be an isomorphism of S0 ∼

modules. Then ϕ0 lifts to an isomorphism ϕ : P −→ Q of graded S-modules.

Proof. Similar to the proof of the corresponding ungraded result, using (3.4.8). Definition 3.4.10. Let S be as in (3.4.8). We recall a standard notation. For a graded S-module M = ⊕k∈Z Mk , and integer d, define M (d), by M (d)n = M (n + d), to be called the twisted module. Let Pgr(S) denote the category of finitely generated graded projective S-modules, P = n∈Z Pn , and graded homomorphisms (of degree zero). (One checks that P is graded projective if and only if P ⊕ Q ∼ = ⊕ni=1 S(di ) for some Q ∈ Obj(Pgr(S))).

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For n ≥ 0, let t : Kn (Pgr(S)) −→ Kn (Pgr(S)) denote the map induced by the functor P → P (−1). Then t has an inverse, given by P → P (1). This defines a Z[t, t−1 ]-module structure on the groups Kn (Pgr(S)). Proposition 3.4.11. Let S and S+ be as in (3.4.8). For integers n ≥ 0, the map Kn (S0 ) −→ Kn (Pgr(S)) induces an isomorphism ∼

Λ : Z[t, t−1 ] ⊗Z Kn (S0 ) −→ Kn (Pgr(S))

of Z[t, t−1 ] −modules.

Proof. As before, P(S0 ) denotes the category of the finitely generated projective S0 -modules. Fix integer n ≥ 0. We have natural functors ι : P(S0 ) −→ Pgr(S) sending Q −→ S ⊗S0 Q. Composing, the following commutative diagrams of functors and the induced homomorphisms on K-groups are obtained: / Pgr(S)

ι

P(S0 ) ιl

'



S(−l)⊗−

Kn (S0 ) × tl

Kn (ι)

, Kn (ιl )

Pgr(S)

Thus, we obtain the Z[t, t−1 ]-homomorphism

/ Kn (Pgr(S)) )



tl

Kn (Pgr(S)) (3.9)

Λ = (⊕l∈Z Kn (ιl )) : Z[t, t−1 ] ⊗Z Kn (S0 ) −→ Kn (Pgr(S)).

(3.10)

Consider S0 as a graded ring concentrated at degree zero, and Q ∈ Obj(P(S0 )) as an S0 -graded module concentrated at degree zero. Then ∀l ∈ Z, S(−l) ⊗S0 Q ∈ Pgr(S). Observe (S(−l) ⊗S0 Q)n = Sn−l ⊗S0 Q. In other words, S(−l)⊗S0 Q = Q⊕(S1 ⊗Q)⊕(S2 ⊗Q)⊕· · · , with the first term Q at degree l. For P = ⊕Pl ∈ Pgr(S), let P :=

P . Then P ∈ Pgr(S0 ). S+ P

So, P l =

Pl . (S+ P )l

We define functors  by T (P ) = P T : Pgr(S) −→ Pgr(S0 ) ∀ l ∈ Z, Tl : Pgr(S) −→ P(S0 ) by Tl (P ) = P l

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Since exact sequences in Pgr(S) split, T , Tl are exact functors. Define the homomorphism Θ as follows: Θ : Kn (Pgr(S)) −→ Z[t, t−1 ] ⊗Z Kn (S0 )

by

Θ(x) =



tl (Tl )∗ (x).

l∈Z

We need to justify that Θ is well defined. Note, Tl (P ) = 0 only for finitely many l ∈ Z. Let Gk ⊆ Pgr(S) be the full subcategory, so that Tl (P ) = 0 unless −k ≤ l ≤ k. Further, Θ is well defined on Kn (Gk ). Since Pgr(S) = ∪Gk , we conclude that Θ is well defined on Kn (Pgr(S)). We will prove that Θ is an inverse to Λ. By construction (3.9), the restriction of Λ on tr Kn (S0 ) is induced by the functor  ιr (Q) = Q ⊗S0 S(−r).

We have

Tl ιr (Q) =

0 Q

if l = r . if l = r

Therefore, ΘΛ(tr x) = Θ(ιr∗ (x)) =



tl (Tl )∗ (ιr∗ (x)) = tr (Tr ιr )∗ (x) = tr x.

l∈Z

This completes the proof that ΘΛ = 1. It remains to be shown that ΛΘ = 1. Let P∞ (S) ⊆ Pgr(S) denote the full subcategory of objects of the form ⊕l∈Z S(−l) ⊗S0 Ql , with Ql ∈ P(S0 ), and only finitely many Ql = 0. Note that P∞ (S) is closed under direct sum. We have a direct sum decomposition S ⊗ S0 P =



S(−l) ⊗S0 P l ∈ Obj(P∞ (S)).

(3.11)

l∈Z

Note P = S ⊗S0 P . Therefore, by (3.4.9), P is isomorphic (noncanonically) to S ⊗S0 T (P ). It follows that the inclusion functor P∞ (S) → Pgr(S) is essentially surjective. Consequently, QP∞ (S) → QPgr(S) is essentially surjective, hence a homotopy equivalence. So, we would replace Pgr(S) by P∞ (S) to prove ΛΘ = 1.

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For integers r ≥ 0, let Pr (S) ⊆ P∞ (S) be the full subcategory of objects of the form ⊕rl=−r S(−l) ⊗S0 Ql . Define the functors

Φk (P ) = ⊕l≤k S(−l) ⊗S0 P l . Φk : Pgr(S) −→ P∞ (S) by In fact, Φk is obtained by the following composition: / Pgr(S0 )

T

Pgr(S)

)

Φk



ϕk

 where

T (P ) = P ϕk (⊕l∈Z Ql ) = ⊕l≤k S(−l) ⊗S0 Ql )

P∞ (S)

(3.12) Therefore, Φk is exact, because so is T and ϕk . The restriction of Φr to Pk (S) defines a functor on Φr : Pk (S) −→ Pk (S) for all r ≤ k (for which we use the same notation Φr ).  Φ k ∀ P ∈ Pgr(S) Φk−1 (P ) = S(−k) ⊗S0 P k Φk k ∀ P = ⊕l∈Z S(−l) ⊗ Ql ∈ P∞ (S) Φk−1 (P ) = S(−k) ⊗S0 Q Further, the restriction (Φk )|Pk = 1Pk . Apply (3.4.6) to the restriction of Φk to Pk (S). k   Φr = Id : Kn (Pk (S)) −→ Kn (Pk (S)) . ∀n≥0 Φr−1 ∗ r=1

Since P∞ (S) = ∪Pk (S), it follows that   Φr = Id : Kn (P∞ (S)) −→ Kn (P∞ (S)) . ∀n≥0 Φr−1 ∗ r∈Z

(3.13) Here the functor 

Φr Φr−1



∗

: P∞ (S) −→ P∞ (S)

sends

P = ⊕l∈Z S(−l) ⊗ Ql → S(−r) ⊗S0 Qr .

For P = ⊕l∈Z S(−l)⊗Ql ∈ P∞ (S), we have Tl (P ) = Ql . Now is given by the composition P∞ (S) 

Φr Φr−1

/ P(S0 )

Tr



(



S(−r)⊗−

P∞ (S)

Kn (P∞ (S))

(Tr )∗

Φr Φr−1

 ∗

Φr Φr−1

/ Kn (S0 )

=⇒ 



)



x→tr Kn (ι)(x)

Kn (P∞ (S))



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So, P∞ (S)

ιTr

Φr Φr−1



/ P(S0 ) (





S(−r)⊗−



Φr Φr−1

∗

P∞ (S)

(x) = tr (ιTr )∗ (x)

Adding all these diagrams, over r ∈ Z, by (3.13), we have Kn (P∞ (S))

/ Kn (S0 ) ⊗ Z[t, t−1 ]

Θ

1

*



Λ

Kn (P∞ (S))

So, ΛΘ = 1. The proof is complete. 3.5.

The Resolution Theorem

The following from the Classical K-Theory motivates this section. Proposition 3.5.1. Let A be a Noetherian commutative ring. Let P(A) denote the category of finitely generated projective A-modules and let H(A) denote the category of finitely generated A-modules with finite projective dimension. Then K0 (P(A)) ∼ = K0 (H(A)). (Use Definition 3.2.1 only.) Proof. For definitions of the classical K0 , refer to Definition 3.2.1(4). We prove that the homomorphism K0 (ι) : K0 (P(A)) −→ K0 (H(A)) induced by the inclusion functor ι : P(A) −→ H(A) is an isomorphism. We define the inverse of this map K0 (ι). Let H denote the isomorphism classes of objects in H(A). For an object M in H(A), let M denote its isomorphism class in H and [M ] denote its class in K0 (H(A)) (and likewise for P(A)). For such an M , let 0

/ Pn

/ Pn−1

/ ···

/ P1

/ P0

/M

/0

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be a P-resolution of M . We say that P• is a P(A)-resolution of M . Now, define ϕ0 : H −→ K0 (P(A))

by

n  ϕ(M ) = (−1)i [Pi ]. i=0

We need to show that ϕ0 is well defined. Let Q• be another P(A)resolution of M (by augmenting zeros at the tails, we can assume that both P• and Q• have the same length n). It would be sufficient to prove that there is an exact sequence ···

/ P1 ⊕ Q2

/ P0 ⊕ Q1

/ Q0

/ 0.

To see this, note that by properties of projective modules, there is a map of chain complexes f• : P• −→ Q• , as in the following commutative diagram: 0

0

/ Pn 

/ Pn−1

fn

/ Qn



/ ···

/ P1

fn−1

/ Qn−1



/ ···

/ P0

f1



/ Q1

/M

/0

/M

/0

f0

/ Q0

Let C(f• ) denote the cone of f• (see [W, pp. 18–19]). So, C(f• )n = Qn ⊕ Pn−1 and the sequence 0

/ Q•

/ C(f• )

/ P• [−1]

/ 0.

is exact. This gives rise to a long exact sequence of homologies ··· H0 (P• )

/ H1 (P• ) / H0 (Q• )

/ H1 (Q• )

/ H1 (C(f• ))

/ H0 (C(f• ))

/

/ 0.

Since H0 (P• ) = H0 (Q• ) = M , and Hi (P• ) = Hi (Q• ) = 0, ∀ i ≥ 1, it follows Hi (C(f• )) = 0 ∀ i. This establishes the above, and hence ϕ0 is a well-defined set theocratic map. So, ϕ0 extends to a group

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homomorphism, as follows: H _ 

Z(H)

ϕ0

/ K0 (P(A)) > ϕ0

Z(H) Now, K0 (H(A)) = R(H) . We show that ϕ0 respects the relations R(H). To see this, suppose

/K

0

/M

/C

/0

is exact in H(A).

Given finite P(A)-resolutions P• of K and Q• of C (say of same length n), inductively, one can build a resolution 0

/ Qn ⊕ Pn

Q0 ⊕ P0

/ Qn−1 ⊕ Pn−1

/M

/ ···

/ Q1 ⊕ P1

/

/ 0.

Therefore, ϕ0 (M ) = ϕ0 (K) + ϕ0 (C). Hence, ϕ0 factors through a homomorphism: ϕ0

/ K0 (P(A)) r9 O ϕ rrr r r rϕ  r r 0  r / K0 (H(A)) Z(H)

H _

It is easy to see that K0 (ι) and ϕ are inverse of each other. The proof is complete. 3.5.1.

Extension closed subcategories

Suppose E is an exact category. For subsequent references, let /K

/M

/C

/0

be an exact sequence in E . (3.14) Recall that short exact sequences (3.14) are also known as “extensions” (see [HS, Chapter III]). (As always, assume that E has a set of isomorphism classes.) Refer to the definition (A.4.4) of an extension closed subcategory P ⊆ E of an exact category E . We have the following remarks: 0

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1. If P is an extension closed subcategory of E , then P is also an exact category, where a sequence in P is declared exact if and only if it is exact in E . Consequently, P → E is an exact functor. 2. As discussed before, for an example let A be a Noetherian commutative ring. Let Coh(A) be the category of finitely generated Amodules, and P(A) ⊆ Coh(A) be the full subcategory of finitely generated projective A-modules. Then P(A) ⊆ Coh(A) is an extension closed subcategory. 3. Further, if P is an extension closed subcategory of E , then Q(P) is a subcategory of Q(E ) while not necessarily a full subcategory. This is because an inflation ι : Q → P in E with Q, P ∈ Obj(P) need not be an inflation in P. In the above example, P(A) ⊆ Coh(A), for P, Q ∈ Obj(P(A)), an inclusion Q → P would not be an inflation in P(A), unless it splits. 4. For the convenience of subsequent discussions, for M ∈ E , an exact sequence 0

/ Pk

/ ···

d2

/ P1

d1

/ P0

d0

/M

/ 0 with Pi ∈ Obj(P)

(3.15) is called a P-resolution of M of length k. By an exact sequence, we mean dr : Pr  ker(dr−1 ) is a deflation in E for r = 0, 1, . . . , k. The goal of this section is to extend (3.5.1) and prove ∼ Kn (P(A)) −→ Kn (H(A)) for all n ≥ 0. The proof depends on the length of the resolutions (3.15). The following is the first step. Theorem 3.5.2. Suppose P is a full subcategory of an exact category E , which is extension closed. Assume the following: 1. Given a conflation (3.14) in E , if M ∈ Obj(P), then K ∈ Obj(P). 2. For any C ∈ Obj(E ), there is a conflation (3.14), with M ∈ Obj(P). Then the inclusion QP → QE is a homotopy equivalence. Conse∼ quently, Ki (P) −→ Ki (E ) ∀ i ≥ 0. (The conditions (1) and (2) are verbalized, respectively, as that P is closed under subobjects and any C ∈ Obj(E ) is the subquotient of an object in P.) Proof. Let C be the full subcategory of Q(E ), whose objects are those of P. Recall that Q(P) is not necessarily a full subcategory

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of Q(E ). However, we have the factorization Q(P)

g

'

/C 

f

Q(E )

We prove that both f, g are homotopy equivalences. First, we show g is a homotopy equivalence. As usual, we use Theorem A (2.2.2) and show that ∀ P ∈ Obj(P), the category g/P is contractible. Refer to (3.1.9) for the definition of the category L(P ) of admissible layers ((K, ιK ), (L, ιL )) in P , in the category E . We abbreviate (K, L) := ((K, ιK ), (L, ιL )). In fact, L(P ) is a partially ordered set. By hypothesis (1), it follows that K, L ∈ Obj(P). Let L(P, P) = {(K, L) ∈ L(P ) : L/K ∈ Obj(P)}. For (K0 , L0 ), (K1 , L1 ) ∈ L(P, P), we redefine ordering (K0 , L0 ) ≤P (K1 , L1 ) if and only if K1 ≤ K0 ≤ L0 ≤ L1 and K0 /K1 , L1 /L0 ∈ Obj(P). By reworking the arguments in (3.1.9), it follows that there ∼ is an equivalence of categories , g/P −→ L(P, P). We prove that L(P, P) is contractible. For (K, L) ∈ L(P, P). We have (K, L) ≤P (0, L) ≥P (0, 0). Define two functors  F : L(P, P) −→ L(P, P) by F (K, L) = (0, L) C : L(P, P) −→ L(P, P) by C(K, L) = (0, 0) We define two natural transformations:  τ1 : IdL(P,P) −→ F by the unique arrow (K, L) ≤P (0, L) by the unique arrow (0, 0) ≤P (0, L) τ2 : C −→ F By Proposition 2.1.9, there are homotopies: IdBQL(P,P) → BF and Constant → BF . So, IdBQL(P ),P is homotopic to the constant map. Hence, L(P, P) is contractible. Therefore, g/P is contactable. By Theorem 2.2.2, g is a homotopy equivalence. It remains to be proven that f : C −→ QE is a homotopy equivalence. To do this, we prove that M/f is contractible, ∀ M ∈ Obj(QE ). Fix M . The objects in M/f are pairs Y := (P, u), with P ∈ Obj(P) and  u ∈ M orQE (M, P ). Let  such a morphism u be represented as  o o / P , where j is a deflation and i is an inflaQ u= M j

tion in E .

i

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Write F (M ) = M/f . Let Fs (M ) ⊆ F (M ) be the full subcategory of objects (P, v) in F (M ), such that v = p! is surjective. We prove that the inclusion ι(M ) : Fs (M ) → F (M ) has a right adjoint. By (A.1.15), it suffices to show that ∀ Y ∈ Obj(F (M )) there is a universal arrow χY : R(Y ) −→ Y , with R(Y ) ∈ Obj(Fs (M )). Consider Y := (P, u) ∈ Obj(F (M )), with the notation as above. Let R(Y ) := (Q, j ! ). Then R(Y ) ∈ Obj(Fs (M )). Define χY := i! ∈ M orF (M ) (R(Y ), Y ). Then χY : R(Y ) −→ Y has the universal property, as in (A.1.15). While it requires routine checking, we outline it. Let X = (L, p! ) ∈ Fs (M ) and ϕ ∈ M orF (M ) (X, Y ). It follows ϕ = q! : L −→ Q is a surjective arrow, given by deflation q : Q  L with j = pq. Define Φ(ϕ) = q! ∈ M orFs (M ) (X, R(Y )). One checks χY Φ(ϕ) = ϕ. Note that any arrow in Fs (M ) is surjective. So, the uniqueness of the morphism Φ(ϕ) follows from the fact that χY Φ(ϕ) = ϕ is a decomposition of ϕ as a composition of an injective and a surjective arrow. This completes the proof that χY is universal. Hence, ι(M ) : Fs (M ) → F (M ) has a right adjoint. By (2.1.10), ι(M ) is a homotopy equivalence. So, it would be enough to prove that Fs (M ) is contractible. We will prove that Fs (M )op is contractible. Objects in Fs (M )op are pairs (P, p), where P ∈ Obj(P) and p : P  M are deflations. A morphism j : (P, p) −→ (Q, q) is a deflation j : P  Q such that p = qj. By hypothesis (2), there is a deflation p0 : P0  M , with P0 ∈ Obj(P). Fix such a deflation p0 . By hypothesis (1), ker(p0 ) ∈ Obj(P). Let (P, p) be an object in Fs (M )op . Consider the pullback diagram P ×M P0 p2

p1



P

/ / P0 

p

p0

//M

Since ker(p2 ) = ker(p0 ), ker(p2 ) ∈ Obj(P). By the extension closed property, it follows that P ×M P0 ∈ Obj(P). We define the following functors: ⎧ ⎨ Id : Fs (M )op −→ Fs (M )op Id(P, p) = (P, p) F : Fs (M )op −→ Fs (M )op F (P, p) = (P ×M P0 , pp2 ) ⎩ C : Fs (M )op −→ Fs (M )op C(P, p) = (P0 , p0 )

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So, p2 : F −→ Id and p0 p1 : F −→ C define two natural transformations. By (2.1.9), there are homotopies: BF → IdBFs (M )op . Therefore, F  is contractible. The proof is complete. Theorem 3.5.2 was formulated as an inductive step to the following (3.5.3). Corollary 3.5.3. Suppose E is an exact category and P is an extension closed full subcategory. For subsequent reference, let /K

0

/M

/C

/0

be a conflation in E . (3.16)

Assume the following: 1. For any conflation (3.16) in E , if C, M ∈ Obj(P), then K ∈ Obj(P). 2. Given a deflation j : M  P in E , with P ∈ Obj(P),  ∃

P P

 P, a deflation g: f : P  −→ M, a morphism

f

 P  ∈ Obj(P) and





M

j

g

commute.

//P

(The condition is valid if ∀ M ∈ Obj(E ), there is a deflation P   M , with P  ∈ Obj(P).) For integers n ≥ 0, let Pn denote the full subcategory of objects M in E that have a P-resolution of length ≤ n (see 3.15). So, P0 = P. Then 1. Pn is an extension closed full subcategory of E . Therefore, Pn is an exact category.  2. Write P∞ = Pn . Then P∞ is also an exact subcategory of E . 3. The following QP = QP0

∼

/ QP1 ∼

/ ···

∼

/ QPn ∼

/ ··· ∼

/ QP∞

are homotopy equivalences. In particular, ∼

∼

Kr (P) −→ Kr (Pn ) −→ Kr (P∞ )

∀ r ≥ 0, n ≥ 0.

Proof. The proof follows by an application of (3.5.2), to the inclusion Pn ⊆ Pn+1 , with additional help from the following Lemma 3.5.4. Note, since P ⊆ Pn , any C ∈ Obj(Pn+1 ) is a subquotient of an

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object in Pn . We use (3.5.4) to show that Pn is closed under subobjects, in Pn+1 . Suppose i : K → M is an inflation in E , and M ∈ Obj(Pn+1 ). By hypothesis, there is a deflation p0 : P0  M with P ∈ Obj(P). Let K0 = ker(p0 ). By (3.5.4(1)), K0 ∈ Obj(Pn ). So, M has a P-resolution P• of length n + 1. The pullback of P• gives a P-resolution of K of length n + 1. So, Pn is closed under subobjects, in Pn+1 . Thus, (3.5.2) applies to the inclusion Pn ⊆ Pn+1 , and QPn −→ QPn+1 is a homotopy equivalence. To see QP −→ QP∞ is a homotopy equivalence, use Whitehead Theorem (C.3.8). The proof is complete. Lemma 3.5.4. Suppose P, E , and Pn are as in (3.5.3). Suppose 0

/K

i

/M

j

/C

/0

is exact in E .

(3.17)

Under the hypothesis of (3.5.3), and for integers n ≥ 0, we have the following: 1. If M ∈ Obj(Pn ) and C ∈ Obj(Pn+1 ), then K ∈ Obj(Pn ). 2. If K, C ∈ Obj(Pn+1 ), then M ∈ Obj(Pn+1 ). (So, Pn is extension closed in E .) 3. If M, C ∈ Obj(Pn+1 ), then K ∈ Obj(Pn+1 ). Proof. (The proof is imitation of the standard arguments on projective dimension of modules M , over commutative Noetherian rings A (see [Mh].). We prove the case n = 0. To prove (1), for n = 0, assume M ∈ Obj(P0 ), C ∈ Obj(P1 ). Consider the diagram

0

0

/K /K

i

P1

P1





g

/ P1 ×C M

/ P0





/M

j

//0

f

/C

/0

where the last vertical line is a resolution of C, with P0 , P1 ∈ Obj(P), and P1 ×C M is the pullback of (j, f ). The middle vertical line and the middle horizontal line are conflations in E . Since the middle vertical line is a conflation, P1 ×C M ∈ Obj(P), by the extension closed

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property of P. Now, by hypothesis (1) of (3.5.3), K ∈ Obj(P). This establishes (1) for n = 0. Now we prove (2), in the case n = 0. So, we prove that P1 is extension closed in E . Assume K, C ∈ P1 . Since C ∈ P1 , there is deflation g : P  C in E , with P ∈ Obj(P). Consider the pullback p

P ×C M q



M

//P 

j

g

//C P0

f

By hypothesis (2), we have the commuting diagram



P ×C M



p

g0

//P

where P0 ∈ Obj(P), g0 : P0  P is a deflation, and f ∈ M or(P0 , P ×C M ). Then d0 := gg0 : P0  C is a deflation. Let P1 = ker(d0 ). By (1), P1 ∈ Obj(P). Also, let η = qf : P0 −→ M . So, jη = d0 . With this, we construct the following commutative diagram: 0

0

0

 / Q1

/N

 / P1

/0

0

 / Q0

 / Q0 ⊕ P0

 / P0

/0

∂0

0



/K 

0



i

δ=(i∂0 +β)

/M

j

d0



/C

/0



0

Here the first vertical line is a conflation, with Q0 , Q1 ∈ P, obtained by the hypothesis. The map β : P0 −→ M is obtained by hypothesis (2) of (3.5.3), with jβ = d. Recall, by (A.4.5), E can be embedded as a fully exact extension closed subcategory of an abelian category

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C . Let N = ker(δ) in C . By Snake Lemma A.3.7, Q1

/N

/ P1

/ coKer(∂)

/ coKer(δ)

/ coKer(d)

is exact. Since coKer(∂) = coKer(d) = 0, it follows coKer(δ) = 0. Further, /N

Q1 Also, 0

/ P1 / Q1

/0 /N

is exact in C .

is exact in C .

Therefore, 0

/ Q1

/N

/ P1

/0

is exact in C .

Since P → E → C are extension closed, N ∈ Obj(P). Now, the middle vertical line is a conflation in C . Since E → C is fully exact, the middle vertical line is conflation in E . Therefore, M ∈ Obj(P1 ). This completes the proof that P1 is an extension closed subcategory of E . Now, we prove (3) for n = 0. So, let M, C ∈ Obj(P1 ). Consider the pullback diagram



P0 ×M K  

K

/ / 0 _

P1 _

P1 _

 / P0 



 i

jd

 //C

d

/M

j

//C

Here the middle vertical line is a conflation, with P0 , P1 ∈ Obj(P). By (1), P0 ×M K ∈ P. Since the first vertical map is also a conflation, K ∈ P1 . So, (3) is established in this case. So, the lemma is established in the case n = 0. Now, the proof is completed by induction. Resolving Subcategories In the commutative algebra literature, some authors [AB] have introduced the following concept of resolving categories, in this context of resolutions theorem.

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Definition 3.5.5. Suppose E is an exact category (with a set of isomorphism classes). For subsequent reference, let /K

/M

/C

/0

be an exact sequence in E . (3.18) Now suppose P is an extension closed subcategory of E , (hence an exact subcategory). Then P is said to be a resolving subcategory of E if the following two conditions are satisfied: 0

1. Given an exact sequence (3.18) in E , if M, C are in P, then so is K. 2. For all objects M in E , there is an exact sequence: 0

/ Pn

/ Pn−1

/ ···

/ P1

/ P0

/M

/0

with Pi in P. In this case, we say P• is a P-resolution of M . Remark 3.5.6. Before we start commenting on the K-theory of rings, we make a decision that our rings A are not necessarily commutative unless stated so. The methods do not really simplify in the case of commutative rings. However, there are only a limited number of applications in [Q] that concern noncommutative situations. For this reason, some readers may choose to assume all rings are commutative. Example 3.5.7. Suppose A is a ring. By an A-module, we usually mean a left A-module. In analogy to the commutative situation, Coh(A) will denote the category of finitely generated left Amodules. The category of finitely generated left projective A-modules is denoted by P(A). 1. Note, P(A) is an exact subcategory of Coh(A). It is not a resolving subcategory of Coh(A). However, if A is a regular ring (see 5.2.3), then P(A) is a resolving subcategory of Coh(A). 2. Let H(A) be the full subcategory of Coh(A) whose objects are the A-modules M in Coh(A), with finite projective dimension. Then H(A) is an exact subcategory of Coh(A). Further, P(A) is a resolving subcategory of H(A).

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Proposition 3.5.8. Let E be an exact category and P ⊆ E be a resolving subcategory. Then QP −→ QE

is a homotopy equivalence. ∼

In particular, ∀ n ≥ 0, Kn (P) −→ Kn (E ) is an isomorphism. Proof. We apply (3.5.3). The first condition of (3.5.3) follows from the definition of resolving subcategory. Now suppose j : M  P in E is a deflation in E , with P ∈ Obj(P). Again by the definition of resolving category, there is a deflation ι : P0  M , with P0 ∈ Obj(P). Therefore, the diagram of deflations P0

ι





M

j

jι

commutes.

//P

So, the second condition of (3.5.3) is satisfied. By the second condition in the definition of resolving category, P∞ = E . So, the proof is complete by (3.5.3). The following is an extension of the classical Example 3.5.1. Corollary 3.5.9. Let A be a commutative Noetherian ring and P(A) ⊆ H(A) be as in (3.5.7). Then QP −→ QH(A) is a homotopy ∼ equivalence. Therefore, ∀ n ≥ 0, Kn (P(A)) −→ Kn (H(A)) is an iso∼ morphism. In particular, if A is a regular ring, then Kn (P(A)) −→ Kn (Coh(A)). More generally, suppose X is a Noetherian scheme. Let Coh(X) denote the category of coherent sheaves on X, and P(X) denote the category of locally free sheaves on X. Let H(X) denote the full subcategory of Coh(X), whose objects have finite P(X)-resolution. ∼ Then ∀ n ≥ 0, Kn (P(X)) −→ Kn (H(X)). In particular, if X is a regular scheme that is Noetherian, integral, and separated, then ∼

Kn (P(X)) −→ Kn (Coh(X))

∀n ≥ 0.

Proof. Follows from (3.5.3 or 3.5.8). Recall that if dim A = d, then H(A) = (P(A))d (see [Mh]). When X is Noetherian, integral, separated, and regular, then Coh(X) = P∞ (see [Hr, Ex. III.6.9]).

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A Transfer Map: Let f : A −→ B be a homomorphism of commutative rings. Use the notation H(A) → Coh(A) as above (3.5.7). Assume, as an A-module, B ∈ H(A). In this case, M ∈ H(B) =⇒ M ∈ H(A). (This can be seen by double complexes [Wc1, p. 99].) Therefore, f defines an exact functor H(B) −→ H(A). By (3.5.9), this functor induces the homomorphisms f∗ , as follows: Kn (B) 



_ _f∗_ _/ Kn (A)

Kn (H(B))



∀ n ≥ 0,



Transfer map.

to be called the

/ Kn (H(A))

Given another ring homomorphism g : B −→ C, with C ∈ H(B), we have (gf )∗ = f∗ g∗ : Kn (C) −→ Kn (A)

∀ n ≥ 0.

We have the natural functor P(A) × H(A) −→ H(A)

sending (P, M ) → P ⊗A M.

This is exact in both coordinates. By Additivity Theorem (3.4.4), we obtain the maps K0 (A) ⊗ Kn (A) _ _ _ _ _/ Kn (A) ∀n≥0





K0 (A) ⊗ Kn (H(A))





/ Kn (H(A))

where both horizontal maps are obtained, by sending [P ] ⊗ z → (P ⊗A −)∗ z. So, we have two products:  K0 (A) ⊗ Kn (A) −→ Kn (A) to be denoted by x ⊗ y → x · y K0 (B) ⊗ Kn (B) −→ Kn (B) Further, we have the pullback maps f ∗ : Kn (A) −→ Kn (B)

∀ n ≥ 0.

Now, we have the projection formula agreeable.

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Theorem 3.5.10. Under the above setup, we have f∗ (f ∗ x · y) = x · f∗ y

x ∈ K0 A, y ∈ Kn B.

In other words, the diagram K0 (A) ⊗ Kn (B) 1⊗f∗

f ∗ ⊗1

·

/ K0 (B) ⊗ Kn (B)

/ Kn (B)





K0 (A) ⊗ Kn (A)

commutes.

f∗

/ Kn (A)

·

Proof. For P ∈ P(A) and M ∈ H(B), we have the identity M → P ⊗A B ⊗ M = P ⊗A M. Now, the theorem follows from additivity theorem (3.4.4). Exact Connected Sequences Corollary 3.5.11. Suppose E is an exact category. Let T = {Ti : i ≥ 1} be an exact connected sequence of functors from an exact category E to an abelian category A . Meaning 1. Ti : E −→ A are functors (not necessarily exact), 2. For each conflation K



//C

/M

in E

(3.19)

and i ≥ 2, there is a morphism δi : Ti (C) −→ Ti−1 (K) such that the following: ···

/

T2 (K)

/

T2 (M)

/

T2 (C)

δ2

/

T1 (K)

/

T1 (M)

/

T1 (C)

is exact. Let P be the full subcategory of T -acyclic objects in E . Meaning Obj(P) = {M ∈ Obj(E ) : Ti (M ) = 0 ∀ i ≥ 1}

is the full subcategroy E .

Then P is an extension closed subcategory of E . Further assume that 1. For all M ∈ Obj(E ), there is a deflation P  M for some P ∈ Obj(P). 2. For all M ∈ Obj(E ), Tn (M ) = 0 ∀ n  0.

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106 ∼

Then Ki (P) −→ Ki (E ) ∀ i ≥ 0. Proof. It follows immediately that P is extension closed and hence Ki (P) is defined. For integers n ≥ 0, define the full subcategories Pn = {M ∈ Obj(E ) : Ti (M ) = 0 ∀ i ≥ n + 1}. It follows that Pn ⊆ Pn+1 is an extension closed subcategory, for all n ≥ 0. It follows that, given a conflation (3.19) in Pn+1 , if M, C ∈ Obj(Pn ), then K ∈ Obj(Pn ). Both the conditions of (3.5.3) ∼ are satisfied. Hence, for all integers i ≥ 0, Ki (Pn ) −→ Ki (Pn+1 ) are isomorphisms. By hypothesis, E = P∞ = ∪n≥0 Pn . Therefore, Ki (E ) = Ki (P0 ) for all i ≥ 0. The proof is complete. Remark 3.5.12. The connected sequence of functors defined above essentially comprises the δ-functors mentioned in [Hr, Section III.1, p. 205]. The δ-functors have much broader applications. Example 3.5.13. For a Noetherian ring A (not necessarily commutative), the category of finitely generated left A-modules is denoted by Coh(A). The K-groups would be denoted by Gn (A) = Kn (Coh(A)). Let f : A −→ B be a homomorphism of Noetherian rings. Define functors Tn : Coh(A) −→ Ab by Tn (M ) = T ornA (B, M ). Then {Tn : n ≥ 0} defines a connected sequence of functors. Assume ∀ M ∈ Coh(A), Tn (M ) = 0, ∀ n  0. Let P ⊆ Coh(A) be the full subcategory objects M , with T ornA (M, B) = 0 ∀ n = 0. Then Kn (P) ∼ = Gn (A) for all n ≥ 0, by (3.5.11). Note, the functor P −→ Coh(B), sending M → M ⊗A B, is exact. So, there is a well-defined homomorphism Gn (A) ∼ = Kn (P) −→ Gn (B) ∀ n ≥ 0. The conditions above are satisfied if B has finite Tor/Flat dimension (5.4.1) over A, as a right A-module. When f : A −→ B is flat, we have the same, without having to use (3.5.11). 3.6.

D´ evissage and Localization in Abelian Categories

To provide context, let A be a commutative Noetherian ring and √ A = √A0 be the reduced ring, where 0 denotes the nil radical of A.

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Among the goals of the D´evissage theorems is to prove Gn (A) ∼ = Gn (A) for all n ≥ 0 (3.6.3). We will be working on the following setup. Definition 3.6.1. Let A be an abelian category, with a set of isomorphism classes of objects. (Recall abelian categories are exact categories.) Let B ⊆ A be a subcategory such that (1) B ⊆ A is a full subcategory, (2) B is closed under subobjects, quotient objects, and finite products in A . In this case, 1. B is an abelian category, and the inclusion B → A is an exact functor. 2. Further, QB → QA is a full subcategory. Proof. Let M, N ∈ Obj(B). The map M orQB (M, N ) −→ M orQA (M, N ) is injective from the definition of morphisms. Now a map f ∈ M orQA (M, N ) can be written as f = p! i! where p : U  M is a deflation and i : U → N is an inflation. Since B is closed under subobjects, U ∈ Obj(B). Since, B is also closed under quotient, U → N is an inflation in B. Similarly, p is a deflation in B. So, the map is a bijection. The following is known as D´evissage Theorem. Theorem 3.6.2 (D´ evissage). Let B → A be as in the setup above (3.6.1). Assume that every object M ∈ Obj(A ) has a filtration: 0 = M0 → M1 → · · · → Mr =: M



∀ j = 1, . . . , r − 1

Mj ∈ Obj(B). Mj−1 ∼

Then QB → QA is a homotopy equivalence, and hence Kn (B) −→ Kn (A ) is an isomorphism for all integers n ≥ 0. Proof. Let f : QB → QA be the inclusion functor. Fix M ∈ Obj(A ). By Theorem A (2.2.2), it suffices to show that f /M is contractible. An object of f /M is given by a pair (N, u), where N ∈ Obj(B) and u ∈ M orQA (N, M ). Such a morphism u is given by an A -admissible layer (M0 , M1 ) in M , as in the diagram M 0_ N oo



p

M1

ι0





ι1

/M

with M0 = ker(p).

Algebraic K-Theory

108

Let L(M ) denote the category (partially ordered set) of all such A -admissible layers ((M0 , ι0 ), (M1 , ι1 )) in M . We abbreviate (M0 , M1 ) := ((M0 , ι0 ), (M1 , ι1 )). Let L(M, B) = {(M0 , M1 ) ∈ L(M ) : M1 /M0 ∈ Obj(B)} be the full subcategory of L(M ). Further, (M0 , M1 ) ≤ (M0 , M1 )

if M0 ≤ M0 ≤ M1 ≤ M1

(see 3.1.9).

∼

The bijection M orQE /M (N, N  ) −→ M orL(M ) ((M0 , M1 ), (M0 , M1 )) is clarified by the diagram M 0_ o

? _ M 0_



  / M  1

M1  q







N oo

M1  M0





/M

 with

M0 = ker(p), M0 = ker(q)

p

/ N

1 ∈ Obj(B). Since B is assumed to be closed under subobjects, M M0 Since L(0, B) is trivial and M has a filtration, with quotients in B, it is enough to prove that the inclusion ι : L(M  , B) → L(M, B) is a M homotopy equivalence, with M  ⊆ M and M  ∈ Obj(B). We define two functors:  r : L(M, B) −→ L(M  , B) (M0 , M1 ) → (M0 ∩ M  , M1 ∩ M  ) s : L(M, B) −→ L(M, B) (M0 , M1 ) → (M0 ∩ M  , M1 )

r, s are well defined because

M1 ∩ M  M1 M1 M1 ⊆ ⊆ × ∈ Obj(B). M0 ∩ M  M0 ∩ M  M0 M

Consider the following: 1. rι = IdL(M  ,B) . 2. There are natural transformations:  ιr −→ s (M0 ∩ M  , M1 ∩ M  ) ≤ (M0 ∩ M  , M1 ) (M0 , M1 ) ≤ (M0 ∩ M  , M1 ) IdJ (M ) −→ s Now, by (2.1.9), r is a homotopy inverse of ι. The proof is complete.

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109

The following is an important corollary of our primary interest. √ Lemma 3.6.3. Let A be a Noetherian ring (commutative) and 0 be the nil radical. Let A = √A0 . As before, we denote Gn (A) = Kn (Coh(A)), for all n ≥ 0. Then for all integers n ≥ 0, we have Gn (A) = Gn (A). Proof. Consider the inclusion functor Coh A → Coh(A). Write √ I = 0. Then I r = 0 for some integer r ≥ 0. So, for any object M in Coh(A), there is a filtration: 0 ⊆ I r−1 M ⊆ · · · ⊆ IM ⊆ M.

with

I kM ∈ Coh A . I k+1

The lemma follows from (3.6.2). The proof is complete. The following provides a contrast. Remark 3.6.4. The following are results of Richard Swan. Let A and A be as in (3.6.3). Let U (A) denote the groups of units of A, and as before, Kn (A) = Kn (P(A)), for all n ≥ 0. In general, K1 (A) = K1 (A). To see this, assume A is local. Matrices α ∈ GL(A) are diagonalizable, up to multiplication by some elementary matrices. So, K1 (A) = U (A) and K1 (A) = U (A). So, K1 (A) = K1 (A) unless √ 0 = 0. However, even when A is nonlocal, K0 (A) = K0 (A). Define the natural functor q : P(A) −→ P(A) by q(P ) := P = √P0P . Since q is ∼

an exact functor, it induces a homomorphism q0 : K0 (A) −→ K0 (A). By Nakayama’s lemma, P ∼ = Q =⇒ P ∼ = Q. It follows from this that q0 is an injective map. To prove q0 is surjective, let P ∈ Obj(P(A)). We will construct P ∈ Obj(P(A)), such that P ∼ = P. To see this, note that there is an idempotent matrix ϕ ∈ GLn (A) for some n ≥ 0 such that n P = ker(ϕ). Write F = An and F = A . Let Φ ∈ GLn (A) be any lift of ϕ. Write P = ker(Φ) ⊆ F , Q = Φ(F ) ⊆ F , and Q = ϕ(F ) ⊆ F . We have the following commutative diagram of exact sequences: 0

0

/P

ι

Φ





/P

/F

i

/F

/Q 

ϕ

/Q

/0 where the vertical maps are surjective.

/0

Algebraic K-Theory

110

In fact, the second line splits. With t = 1 − ϕ : F −→ P, we have ti = 1P . Let η : F −→ P be any lift of t. We have (ηι) ⊗ A = ti = 1. By Nakayama’s Lemma, co ker(ηι) = 0. So, ηι : P  P is surjective. So, ηι is an isomorphism, by the Noetherian property of P . Therefore, ι splits and P ∈ Obj(P(A)), with P = P. So, q0 is surjective. 3.6.1.

Semisimple Objects in abelian categories

Definition 3.6.5. Let A be an abelian category. 1. An object M in A is called a simple object if only subobjects of M are 0, M . 2. An object M in A is called a semisimple object if M is an image of a direct sum  of simple objects. In other words, if there is a surjective map i∈I Si  M , where {Si :∈ I} is a (possibly infinite) set of simple objects. 3. We say that an object M has finite length if M has a finite filtration 0 = M0 → M1 → M2 → · · · → Mn =: M

 ∀ j = 1, . . . , n

Mj is simple. Mj−1

(Define length (M ) := n. It is left as an exercise (3.8.4) that

(M ) is well defined.) Lemma 3.6.6. Let A be an abelian category and M be an object. Then the following are equivalent: 1. M is semisimple. 2. M ∼ = ⊕i∈I Si , where {Si :∈ I} is a set of simple objects. (So,

(M ) = Card(I).) Proof. (2) =⇒ (1) is obvious. To prove the converse, let f : ⊕i∈I Si  M be a surjective map, where {Si : i ∈ I} is a set of simple objects. Let I := {I0 ⊆ I : f|∪i∈I0 Si is injective}. By Zorn’s lemma, I has a maximal element J ⊆ I. Write N = ⊕i∈J Si ⊆ ⊕i∈I Si . Claim ∼ f0 := f|N : N −→ M is an isomorphism. If not, there is i1 ∈ I \ J such that f (Si1 ) ⊆ f0 (N ). Let f1 := f|Si1 : Si1 −→ M . We have f0 : N −→ M and f1 : S1 −→ M are injective. Write g = f|(N ⊕Si1 ) : N ⊕ Si1 −→ M . Then g = f0 ⊕ f1 . We prove g is injective. Let α : W −→ N ⊕ Si1 be a map such that gα = 0. Since N ⊕ Si1 ∼ = N × S i1 , it follows that f0 p0 α = 0 and f1 p1 α = 0, where p0 : N ⊕ Si1 −→

Quillen K-Theory

111

N and p1 : N ⊕ Si1 −→ Si1 are the projections. Since, f0 , f1 are injective, we have p1 α = 0, p0 α = 0. Hence, α = 0. So, g is injective. This is a contradiction to the maximality of J. Therefore, f0 is an isomorphism. The proof is complete. Corollary 3.6.7. Let A be an abelian category (with a set of isomorphism classes of objects) such that every object has finite length. Then  Kn (Dj ) ∀n≥0 Kn (A ) = j∈J

where {Sj : j ∈ J} is a set of representatives, of isomorphism classes of simple objects of A , and Dj = End(Sj )o . (Note that Dj is a division ring.) Proof. Let B be the full subcategory of semisimple objects in A . ∼ Then by D´evissage (3.6.2), Kn (B) −→ Kn (A ) is an isomorphism, ∀ n ≥ 0. For j ∈ J, let Bj be the full subcategory of B consisting of all objects which areisomorphic to a direct sum of Sj (3.6.6). So, B =  m B j = lim ( k=1 Bjk ) is the limit of the finite direct product. j∈J By Lemma 3.3.2(4), 3.3.3, it follows that    ∼ Kn (lim ( m Bjk )) −→ lim (Kn ( m k=1 k=1 Bjk )) m ∀ n ≥ 0 Kn (B) = ∼ ∼  −→ lim ( k=1 Kn (Bjk )) −→ j∈J Kn (Bj ) are isomorphisms. Now, the proof follows from the following Lemma (3.6.8). Lemma 3.6.8. Suppose B is an abelian category such that (1) each object is semisimple and has finite length, and (2) B has only a single simple object X, up to isomorphism. Let D := M or(X, X)o and P(D) denote the category of finite dimensional left D-vector spaces. Then the functor F : B −→ P(D)



F (M ) = M orB (X, M )

is an equivalence of categories. Proof. It follows from Lemma 3.6.6 that, each object M ∼ = X n for some integer n ≥ 0.

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112

Quillen’s localization theorem

3.6.2.

The following is the celebrated localization theorem of Quillen. Theorem 3.6.9 (Localization Theorem). Let A be an abelian category and B ⊆ A be a Serre subcategory (A.5.6), ι : B → A being the inclusion functor. Let A B be the quotient abelian category be the canonical functor. Then the sequence and q : A −→ A B /A

ι

B

q

/ A

is a homotopy fibration.

B

Consequently, there is an exact sequence ···

q∗

/ K1 A B

/ K0 (B)

ι∗

/ K0 (A )

q∗

/ K0 A B

/ 0.

Proof. Fix a zero object 0 in A and let 0 denote its image in A B, A which is also a zero object in B . From the definition of Serre subcategories (A.5.6) and the construction (A.5.7) of A B , it follows that B is the full subcategory of objects M in A such that qM  0. The diagram Qι

QB 0

/ QA '



Qq

commutes up to natural equivalence.

QA B

(Sometimes, we may ignore distinguishing between q and Qq. It will be clear from the context whether we mean q or Qq.) There is an obvious natural functor F : QB −→ 0/Qq, and a functor G, as follows:  ∼ F : QB −→ 0/Qq F (M ) = (M, 0 −→ qM ) G : 0/Qq −→ QA G(N, u) = N F

QB

/ 0/Qq

 the diagram Qι

'



G

QA

Quillen K-Theory

113

commutes. To establish the theorem, we check the hypotheses of Theorem B (2.2.3) to Qq, and establish that F is a homotopy equivalence. The following would settle the proof of the theorem: 1. To apply Theorem B (2.2.3) to Qq, we prove that for all u : ∗  V  −→ V in Q A B , the base change functor u : V /Qq −→ V /Qq ∗ is a homotopy equivalence. Recall the functor u sends (W, v) → ! (W, vu). Such morphisms u in Q A B factors u = i! p , as composition of surjective and injective maps. Therefore, it suffices to prove the same when u = i! is injective and when u = p! is surjective, in Q A B. We will prove this for injective maps u = i! only. The case when u = p! is surjective follows by similar arguments or by reversing the arrows, which changes surjective maps to injective maps. Further, let iV : 0 → V denote the unique arrow in A B . For  , we have i = ii . So, we only an inflation i : V  → V in A V V B A prove that for u = (iV )! : 0 −→ V in Q B , the base chance functor u∗ : V /Qq −→ 0/Qq is a homotopy equivalence. This will be completed in the following (3.6.19). 2. For objects V in Q A B , let FV be the full subcategory of V /Qq, ∼ consisting of objects (M, u) such that u : V −→ qM is an iso∼ morphism in Q A B . It is clear QB −→ F0 is an equivalence of categories. We prove in the following (3.6.10) that the inclusion FV → V /Qq is a homotopy equivalence. In particular, F : QB −→ 0/Qq is a homotopy equivalence. So, it remains to establish (3.6.10) and prove (iV )∗! is a homotopy equivalence (3.6.19). Thus, the proof of the theorem will be complete. Lemma 3.6.10. Use the notation form (3.6.9). For V in Q A B , the inclusion functor ιV : FV → V /Qq is a homotopy equivalence. Proof. Fix (N, v) ∈ Obj(V /Qq). We prove that ιV /(N, v) is contractible. By Theorem A (2.2.2), this will complete the proof. The map v : V −→ qN in Q A B is represented by an admissible layer ∼ A 1 (N0 , N1 ) in qN , and an isomorphism θ(v) : N N0 −→ V , in B (that is, N0 → N1 → qN ). Write C = ιV /(N, v). An object in C = ιV /(N, v) is given by ((M, μ), w) with (M, μ) ∈ Obj(FV ) and w ∈ M orC ((M, μ), (N, v)). Indeed, w ∈ M orQA (M, N ) with (qw)μ = v in Q A B . Further,

Algebraic K-Theory

114 ∼

μ : V −→ qM is an isomorphism. As an admissible layer, μ is given ∼ by (0, qM ) and isomorphism θ(μ) : qM −→ V in A B . As an admissible layer, w is represented by (M0 (w), M1 (w)) in N and an isomorphism ∼ 1 (w) θ(w) : M M0 (w) −→ M in A (i.e. M0 (w) → M1 (w) → N in A ). To translate the identity (qw)μ = v as admissible layers in N , consider the diagram qM1 (w) qθ(w) qM0 (w) ∼

/ qM  θ(μ)

∼

N1 N0

θ(v)

 /V

in

A . B

Here the composition (qw)μ represents the composition θ(μ)(qθ(w)), and θ(v) represents v. Since (qw)μ = v, we have (N0 , N1 ) = (qM0 (w), qM1 (w)). As before, let L(N ) denote the set of admissible layers ((M0 , ι0 ), (M1 , ι1 )) in N in the category A . We will abbreviate (M0 , M1 ) := ((M0 , ι0 ), (M1 , ι1 )). Write L(N, v) = {(M0 , M1 ) ∈ L(N ) : (qM0 , qM1 ) = (N0 , N1 )}. The partial order relation in L(N ) induces a partial order relation in L(N, v). It follows from the above discussions that the following functors: ⎧ ⎪ (M0 (w),  M1(w)) ⎨ ιV /(N, v) −→ L(N, v) ((M, μ), w) → ⎪ ⎩

L(N, v) −→ ιV /(N, v) where μ = v :

1 qM M0

=

N1 N0

(M0 , M1 ) → ∼

M1 M0 , μ

,w

−→ V, and w is defined by (M0 , M1 ).

are inverses of each other. So, ιV /(N, v) is equivalent to the partially ordered set of layers L(N, v). Now, for (M0 , M1 ), (M0 , M1 ) ∈ L(N, v), we have (M0 , M1 ) ≤ (M0 ∩ M0 , M1 + M1 ), ≥ (M0 , M1 ). Recall that M0 ∩ M0 , M1 + M1 are defined as follows:  N N  / , M0 ∩ M0 = ker N M0 ⊕ M0   /N . M1 + M1 = Image M1 ⊕ M1

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Since q is an exact functor, it follows that q(M0 ∩ M0 ) = N0 and q(M1 + M1 ) = N1 . So, (M0 ∩ M0 , M1 + M1 ) ∈ L(N, v). Therefore, L(N, v) is a directed set. So, by (2.1.19), L(N, v) is contractible. The proof is complete. Next, we introduce the following categories and notation. Definition 3.6.11. Use all the notation in (3.6.9). Let N ∈ Obj(A ). Define the category EN as follows: (1) Objects of EN are the pairs (M, h), where h ∈ M orA (M, N ) such that ker(h), co ker(h) ∈ B, or equivalently qh is an isomorphism in A B (such an arrow h is also called mod-B isomorphism). (2) For objects (M1 , h1 ), (M2 , h2 ), let M orEN ((M1 , h1 ), (M2 , h2 ))   = u ∈ M orQA (M1 , M2 ) : u = i! j ! , and h2 i = h1 j in A . So,  a morphism u ∈ M orQA (M1 , M2 ), represented by u =  i j / M2 , is a morphism u ∈ M orEN (M1 , h1 ), U M1 o o

(M2 , h2 ) , if the diagram U j



i



M1

/ M2 

h1

h2

commutes.

(3.20)

/N

One checks that commutativity of (3.20) is independent of the factorization u = i! j ! . Further, the diagram (3.20) induces the following morphism in A :

κ(u) :=

ker(h1 )

oo

ker(h1 j)



/ ker(h2 )



∈ M orQA (ker(h1 ), ker(h2 ))

Since B is closed under subobjects in A , ker(h1 j) ∈ Obj(B). So, κ(u) ∈ M orQB (ker(h1 ), ker(h2 )). Define the functor  κ((M, h)) = ker(h) ∀ (M, h) ∈ Obj (EN ) κN : EN −→ QB by κ(u) = as above So, κ((M, h)) = ker(h) is determined up to canonical isomorphisms, by the exact sequence 0

/ κ (M, h)

/M

/ N.

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Before we proceed, we provide a roadmap toward the completion of the proof of the Localization theorem (3.6.9). For any choice (and we would have some) of a pair (N, ϕ) with N ∈ Obj(A ) and an ∼ isomorphism ϕ : qN −→ V in A B , we develop the following diagram of functors: EN κN



QB

ψ(N,ϕ)

∼

/ FV   / F0  

∼

∼

/ V /Qq 

((iV )! )∗

(3.21)

/ 0/Qq

The functor ψ(N,ϕ) will be defined, and we will prove that it is a homotopy equivalence (3.6.18). All the other functors have been defined. ∼ We observed that QB −→ F0 is a homotopy equivalence. The last two inclusions are homotopy equivalences, by (3.6.10). We also prove that κN is a homotopy equivalence (3.6.13). The diagram is commutative up to a natural transformation. So, the diagram is homotopy commutative. It follows from all the above that (iV )! )∗ is a homotopy equivalence. This will complete the proof of (3.6.9). Now, we proceed, according to this plan. Lemma 3.6.12. Use notation as in (3.6.9, 3.6.11). Let EdN denote to full subcategory of EN consisting of objects (M, h) such that h is a deflation (i.e. epimorphism). Then the restriction map κdN : EdN −→ QB of κN is a homotopy equivalence. Proof. As usual, by (2.2.2), it is enough to prove that κdN /T is contractible ∀ T ∈ Obj(QB). So, N and T remain fixed for the rest of the proof, and we write C = κdN /T . Also, let C  ⊆ C be the full subcategory of objects ((M, h), u) ∈ C , with u ∈ M orQB (ker(h), T ) such that u is surjective. We will prove that the inclusion C  ⊆ C has a left adjoint. Let X = ((M, h), u) be an object in C . So, (M, h) ∈ Obj(C ), u ∈ M orQB (ker(h), T ).Factorize u = j ! i! , as in u = j ! i! =  

i / T0 o o j T , where i is an inflation and j is a deflaker(h)  tion in B. Further, h : M  N being a deflation in A , we have the

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pushforward diagram  ker(h)  _

i



/ M  _   i 

  T0  _ _ _ _/ i∗ M

h

//N

where i∗ M is the pushforward. h

//N

(3.22)

Write X = (i∗ M, h), j ! . Note X = (i∗ M, h), j ! is an object in C  . From diagram (3.20), it is clear that i! : (M, h) −→ (i∗ M, h) is an arrow in EN . It follows that i! : X −→ X is an arrow in C . As in (3.5.2), one checks that the arrow X −→ X is universal. Therefore, the inclusion C  → C has left adjoint (A.1.15). Hence, this inclusion is a homotopy equivalence (2.1.10). However, C  has an initial element, namely ((N, 1N ), jTj ). So, C  is contractible, and so is C . The proof is complete. Lemma 3.6.13. Use notation as in (3.6.9, 3.6.11, 3.6.12). The functor κN : EN −→ QB is a homotopy equivalence. Proof. Due to (3.6.12), it suffices to prove that the inclusion EdN → EN is a homotopy equivalence. Recall that S(N ) denotes the partially ordered set of subobjects ι : I → N (we abbreviate I ⊆ N ). Let S(N, B) = {I ∈ S(N ) : N/I ∈ B}. Let F : EN −→ S(N, B) be the functor F (M, h) = Im(h). For I ∈ S(N, B), it is clear that F −1 I = EdI . Further, I/F = {(M, h) ∈ EN : I ≤ h(M )}. There is a right adjoint functor R : I/F −→ F −1

R(M, h) = M ×N I I defined by (more simplistically, R(M, h) := h−1 (I)  I ). So, F is prefibered. For I ≤ J, the base change map F −1 J −→ F −1 I sends (M,

h) → (M ×I J  J) (more simplistically, (M, h) → h−1 (I)  I ). So, it is established that F is fibered, with fiber F −1 I = EdI . We have the commutative diagram EdI _



EdJ

κdI

/B G κdJ

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By (3.6.12), κdJ and κdI are homotopy equivalences, and hence so is the vertical inclusion. So, the fibered version of Theorem B (2.2.6) applies to F : EN −→ S(N, B). However, S(N, B) is contractible because it has a terminal object N . Therefore, EdI → EN is a homotopy equivalence for all I ∈ S(N, B). In particular, EdN → EN is a homotopy equivalence. The proof is complete. Lemma 3.6.14. Use notation as in (3.6.9, 3.6.11, 3.6.12). Let g : N −→ N1 be an arrow in A , which is a mod-B isomorphism (i.e. ker(g), co ker(g) ∈ Obj(B)). Then the functor g∗ : EN −→ EN1 sending (M, h) → (M, gh) is a homotopy equivalence. Proof. For (M, h) ∈ Obj(EN ), there is a natural inflation ι : κN (M, h) −→ κN  (g∗ (M, h)), defined as follows: ker(h) 

 _



/M

h

 ι 

/N g

ker(gh)



/M



gh

g∗

EN

/ N1

This induces a natural  transformation ι : κN −→ κN1

κN

/ EN1 ( 

κN1

QB

commutes via ι. Since κN , κN1 are homotopy equivalences (3.6.13), so is g∗ . We will prove that FV is homotopically equivalent to EN , when qN ∼ = V . For convenience of our discussions, recall (Ex. 3.8.1) that there is a bijection ∼

Iso A (M, N ) −→ IsoQ A (qM, qN ) B

B

sending ϕ → ϕ! .

For this reason, subsequently we may not distinguish these two sets of isomorphisms. Now, we further define a filtering category. Definition Use notation as in (3.6.9, 3.6.11, 3.6.12). Let

3.6.15. . Define the category IV as follows: (1) Objects of V ∈ Obj A B ∼ IV are pairs (N, ϕ) where N ∈ Obj(A ) and ϕ : qN −→ V is an isomorphism in A B , (2) and ∀ (N0 , ϕ0 ), (N1 , ϕ1 ) ∈ IV , define M orIV ((N0 , ϕ0 ), (N1 , ϕ1 )) = {g ∈ M orA (N0 , N1 ) : ϕ0 = (qg)ϕ1 } . Lemma 3.6.16. The category IV , as defined in (3.6.15), is a filtering category.

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Proof. The category IV is nonempty because (V, q1V ) is in it (considering V as an object in A ). Now, let (N0 , ϕ0 ), (N1 , ϕ1 ) be objects −1 in IV . From the construction of A B , we can write ϕ0 = s0 g0 =   g

0 / M0 o s0 V , where s0 is a mod − B isomorphism. Now, N0 −1 g0 : (N0 , ϕ0 ) −→ (M0 , (qs 0 ) ) is a morphism in IV . Replacing

(N0 , ϕ0 ) by M0 , (qs0 )−1 , we can assume ϕ0 = (qs0 )−1 for some mod-B isomorphism s0 : V −→ N0 . Similarly, we can assume ϕ1 = (qs1 )−1 for some mod-B isomorphism s1 : V −→ N1 . ModB-isomorphisms satisfy both right and left Ore conditions (A.5.2). So, we have a commutative diagram

V s1

s0



N1

/ N0 

t1

t0

where t0 , t1 are mod−B isomorphisms.

/N ∼

Let ϕ = q(t0 s0 )−1 = q(t1 s1 )−1 : N −→ V . Then (N, ϕ) is an object in IV . The arrows t0 , t1 define the arrows ti : (N0 , ϕi ) −→ (N, ϕ) in IV for i = 1, 2. Now suppose g1 , g2 : (N1 , ϕ1 ) −→ (N2 , ϕ2 ) are arrows in IV . So, g1 , g2 ∈ M orA (N1 , N2 ) such that ϕ2 qg1 = ϕ1 = ϕ2 qg2 . Therefore, q(g1 − g2 ) = 0. So, I := Image(g1 − g2 ) ∈ B. Let N = NI2 and −1 g : N2 −→ N be the quotient map, and ϕ := (qg) ϕ2 . Then g represents an arrow g : (N2 , ϕ2 ) −→ N , ϕ such that gg1 = gg2 . So, g equalizes g1 and g2 . The proof is complete. Lemma 3.6.17. Continue to use the notation in (3.6.9, 3.6.11, 3.6.12, 3.6.15). We define a functor Θ : IV −→ Cat by  ∀ (N, ϕ) ∈ Obj(IV ) Θ(N, ϕ) = EN Θ(g) = g∗ : EN0 −→ EN1 ∀ g ∈ M orIV ((N0 , ϕ0 ), (N1 , ϕ1 )) So, Θ is a filtering family of categories. Also, for (N, ϕ) in IV , define the functor ψ(N,ϕ) : EN −→ FV , as follows:

⎧ ∀ (M, h) ∈ Obj(EN ) ⎨ ψ(N,ϕ) (M, h) = M, (ϕqh)−1 (u) = u ∀ u = i! j ! ∈ ψ ⎩ (N,ϕ) M orEN ((M0 , h0 ), (M1 , h1 )) . (3.23)

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Note that ∀ g ∈ M orIV ((N0 , ϕ0 ), (N1 , ϕ1 )), we have ψ(N0 ,ϕ0 ) = Φ(N0 ,ϕ0 ) g∗ . By the universal property (2.1.15), there is a functor Ψ, as follows: Θ(N, ϕ) θ(N,ϕ)

ψ(N,ϕ)



!

limIV Θ(N, ϕ) _ _ _/ FV Ψ

Then Ψ is a homotopy isomorphism. Proof. Given (M, ϕ) ∈ Obj(FV ), we have ψ(M,ϕ−1 ) (M, 1M ) = (M, ϕ). So, Ψ is surjective on objects. By construction (2.1.15), objects in limIV Θ(N, ϕ) = limIV EN are represented by those in Θ(N, ϕ). We prove Ψ is injective. Suppose Ψ(θ(N,ϕ) (M, h)) = Ψ(θ(N1 ,ϕ1 ) (M1 , h1 )), for some (M, h) ∈ Obj (Θ(N, ϕ)), (M1 , h1 ) ∈ Obj (Θ(N1 , ϕ1 )). Since IV is directed, we can assume (N1 , ϕ1 ) = (N, ϕ). Therefore, ψ(N,ϕ) (M, h) = ψ(N,ϕ) (M1 , h1 ). So, M = M1 , and A N qh = qh1 in Q A B . So, qh = qh1 in B . Let N = Im(h−h1 ) and

g : N −→ N be the quotient map. Let ϕ = (qg)−1 ϕ. Then qg : (N, θ) −→ (N , ϕ) is an arrow in IV , which induces g∗ : EN −→ EN . Now, g∗ (M, h) = (M, gh) = (M, gh1 ) = g∗ (M, h1 ). So, it is established that Ψ is bijective on the objects. The morphisms in limIV Θ(N, ϕ) are represented by morphisms in Θ(N, ϕ) = EN . Suppose u1 , u2 ∈ M orΘ(N,ϕ) ((M1 , h1 ), (M2 , h2 )) = ψ(N,ϕ) (u2 ) in M orFV (M1 , are such that ψ(N,ϕ) (u1 ) −1 −1 (ϕqh1 ) ), (M2 , (ϕqh2 ) )). By definition (3.23) u1 = ψ(u1 ) = ψ(u2 ) = u2 . So, Ψ is injective on the morphisms. We already proved that Ψ is bijective on the objects. So, to

prove surjectivity, let u ∈ M orFV ψ(N,ϕ) (M1 , h1 )), ψ(N,ϕ) (M2 , h2 )) , where (M1 , h1 ), (M2 , h2 ) are objects in EN for some (N, ϕ) ∈ IV . Then u ∈ M orQA (M1 , M2 ) such that the diagram V

(ϕqh1 )−1

(ϕqh2 )−1

+

/ qM1 

qu

qM2

commutes in

A , B

 and

qu is isomorphism, qh2 qu = qh1

Quillen K-Theory

j!



We write u = i! = M1 o o have a commutative diagram qM0  qj

j

M0



i

121

 / M2 . It follows that we

 qi / qM2  qh2



qM1

∼ qh1

Let N =



So, q(h2 i−h1 j) = 0, hence Im(h2 i−h1 j) ∈ B,

/ qN

N Im(h2 i−h1 j)

and g : N  N be the quotient map. With ϕ =

ϕ(qg)−1 , we have an arrow qg : (N, ϕ) −→ (N , ϕ) in IV . Consider the diagram M0 j



i



M1

/ M2 

h1

h2

/N AA AA AA g A A ,N

⎧ ⎨ Outer square commutes, inner square may not. g∗ (M1 , h1 ) = (M1 , gh1 ) ⎩ g∗ (M2 , h2 ) = (M2 , gh2 )

(3.24) Further, u = i! j ! represents an arrow g∗ (M1 , h1 ) −→ g∗ (M2 , h2 ) in EN . And, ψ(N ,ϕ) (u) = u, as required. The proof is complete. ∼

Lemma 3.6.18. For any isomorphism ϕ : qN −→ V is functor ψ(N,ϕ) : EN −→ FV is a homotopy equivalence.

A B,

the

Proof. By (3.6.17) FV = limIV Θ(N, ϕ). Also, ∀ g ∈ M orIV ((N0 , ϕ0 ), (N1 , ϕ1 )), the functors Θ(g) : Θ(N0 , ϕ0 ) −→ Θ(N1 , ϕ1 ) are homotopy equivalences, by (3.6.14). Now the lemma follows, by an application of (2.1.18). The proof is complete. To finish the proof of the Localization Theorem (3.6.9), we prove the following Lemma. Lemma 3.6.19. For the inflation iV : 0 → V in A B , and (iV )! ∈ M orQ A (0, V ), the functor ((iV )! )∗ : V /Qq −→ 0/Qq is a homotopy B equivalence.

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122

Proof. Let (N, ϕ) ∈ IV , be any object (e.g. (N, ϕ) = (V, q1V )). So, ∼ ϕ : qN −→ V in A B . Recall the diagram (3.21). EN κN



QB

ψ(N,ϕ)

∼



/ FV  / F0 



∼

∼

/ V /Qq 

((iV )! )∗

/ 0/Qq

As promised, the functors ψ(N,ϕ) were defined and proven to be homotopy equivalences (3.6.18). The functor κN is also a homotopy equivalence (3.6.13). The lower left and upper right compositions are, respectively, given by ⎧   ∼ ⎨ (M, h) → ker(h), 0 −→ q(ker(h)   ∼ ⎩ (M, h) → M, (iqM )! : 0 −→ qM because qhϕ is an isomorphism. The inclusion ker(h) → M induces a natural transformation, from the first one to the second one. So, the diagram is homotopy commutative. Therefore, ((iV )! )∗ is a homotopy equivalence. The proof is complete. With this, the proof of the Localization Theorem (3.6.9) is complete. Remark 3.6.20. We comment on the conclusion of the proof of the Localization theorem in Lemma 3.6.19. In the proof of (3.6.19), the only tangible choice of (N, ϕ) in IV was (N, ϕ) = (V, q1V ). Then we asserted that ψ(V,q1V ) : EV −→ FV was a homotopy equivalence, using (3.6.18). One could ask why was it not possible to give a direct proof of this fact, by defining a homotopy inverse FV −→ EV ? We attempt to provide some clarity. 1. In the proof of (3.6.17), preimage of (M, ϕ) ∈ FV was first defined as (M, 1M ) ∈ Θ(M, ϕ−1 ) = EM . However, ϕ ∈ IsoQ A (V, M ) ∼ = B

Iso A (V, M ). So, we can write ϕ = (qt)−1 (qs), where s : V −→ N B and t : M −→ N are mod-B isomorphisms. By (3.6.14), these

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123

give zigzag homotopy equivalences EM

t∗

/ EN o s∗

EV

(3.25)

This is still discomforting, in that this depends on (M, ϕ). In any case, given an object (N, ϕ) in IV , we have zigzag homotopy equivalences, as in (3.25). 2. Now suppose u ∈ M orFV ((M1 , ϕ1 ), (M2 , ϕ2 )) and ϕi ∈ M or A (V, Mi ) for i = 1, 2. Then u ∈ M orQA (M1 , M2 ). For B

= (qti )−1 (qhi ) for some mod-B isomori = 1, 2, write ϕ−1 i phisms hi : M −→ Ni and ti : V −→ Ni . By the Ore condition, s1 t1 = s2 t2 =: t for some mod-B isomorphisms si : Ni −→ N . So, −1 ϕ−1 i = (q(ti si )) (q(si hi )). Adjusting hi := si hi and ti := si ti = t, we can assume N1 = N2 = N , and t = t1 = t2 . We reproduce the diagram (3.24), with same notation, for this new situation: M0  j



i

h2



M1

/ M2

h1

 /N o V AA t AA A g AAA  +N

⎧ ⎨ g∗ (M1 , h1 ) = (M1 , gh1 ) g∗ (M2 , h2 ) = (M2 , gh2 ) ⎩ And, u ∈ M or ((M , h ), (M , h )) EN 1 1 2 2

Now, (gt)∗ : EV −→ M orEN (g∗ (M1 , h1 ), g∗ (M2 , h2 )) is a homotopy

equivalence. The preimage of u ∈ M orFV (M1 , ϕ1 ), (M2 , ϕ2 ) in EV is the preimage of u, via (gt)∗ . For further insights for such a homotopy inverse of (gt)∗ , see the proof of (3.6.9). 3.7.

K-Theory Spaces and Reformulations

For an exact category E , the K-groups are defined as Kn (E ) := πn+1 (BQE , 0). Conversely, for CW complexes Y , the homotopy groups πn (Y, y) determine Y , up to homotopy equivalences (C.3.8). From this perspective, it may make more sense to define K-Theory as a topological space, as follows.

Algebraic K-Theory

124

Definition 3.7.1. Let E be an exact category. The K-theory space of E is defined to be the pointed topological space K(E ) := Ω (BQE )

the loop space of BQE

at 0

with the constant loop (see B.4.7) as the base point. Therefore, we have Kn (E ) = πn+1 (BQE , 0) = πn (K(E ), 0)

∀ n ≥ 0.

Comparing the homotopy groups, by the theorem of Whitehead (C.3.8), most of the above can be reformulated in terms of the KTheory spaces. Lemma 3.7.2. Let ι : P −→ E be an exact functor of exact categories. Then ι induces a map K(P) −→ K(E ) of K-theory spaces. The Resolution Theorem is reformulated, as follows. Theorem 3.7.3. Let E be an exact category and P ⊆ E be a resolving subcategory. Then the map K(P) −→ K(E )

is a homotopy equivalence.

Proof. For integers n ≥ 0, by (3.5.8), it follows that πn (K(P, 0) = πn+1 (K(QP, 0)) = πn+1 (K(QE , 0)) = πn (K(E ), 0)). The proof is complete, by (C.3.8). Theorem 3.7.4 (D´ evissage). Under the hypotheses of Theorem 3.6.2, the map K(B) −→ K(A )

is a homotopy equivalence.

Proof. Same as above. Theorem 3.7.5 (Localization). Under the hypotheses of the Localization Theorem (3.6.9) / K(A ) /K A K(B) B is a homotopy fibration.

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125

Proof. This follows from the Localization Theorem (3.6.9) in conjunction with (C.3.10). 3.8.

Exercises

Exercise 3.8.1. Let E be an exact category. Then ∼

1. Let θ : N −→ M be an isomorphism. Then θ ! = (θ −1 )! ∈ M orQE (M, N ) is an isomorphism. 2. There are bijections  ∼ IsoE (N, M ) −→ IsoQE (M, N ) θ → θ ! ∼ IsoE (M, N ) −→ IsoQE (M, N ) θ → θ! where IsoC (M, N ) denotes the set of isomorphisms in the respective category C . 3. Suppose u = i! p! ∈ M orQE (M, N ). If u is an isomorphism, then so are i and j. Exercise 3.8.2. Let F : P −→ E be an exact functor, of exact categories. Then the following conditions are equivalent: 1. The map BQP −→ BQE is a homotopy equivalence. 2. The map KP −→ KE is a homotopy equivalence. 3. The maps Kn (P) −→ Kn (E ) are isomorphisms ∀n ≥ 0. Exercise 3.8.3. Suppose E is an exact category and P ⊆ E is a resolving subcategory (3.5.5). Give an elementary proof that ∼ K0 (P) −→ K0 (E ). (Do not use the resolution theorem (3.5.3).) Exercise 3.8.4. Let A be an abelian category. Suppose M ∈ Obj(A ). Then M is said to have length n, and write (M ) = n if there is a filtration 0 = M0 → M1 → M2 → · · · → Mn =: M Mj is simple.  ∀ j = 1, . . . , n Mj−1 Prove that the length (M ) is well defined. (Hint: See [Wc2, p. 137].)

Algebraic K-Theory

126

Exercise 3.8.5. [Q, p.105] Suppose D is a Dedekind domain, with quotient field F . Then there is an exact sequence ···

/



m∈max(D) K0



m∈max(D) K0

D m

D m

/ K1 (D)

/ K1 (F )

/

/ K0 (D)

/ K0 (F )

/0

(Hint: Use Ex. A.6.9.) Exercise 3.8.6. Let (A, m) be a Noetherian local ring and κ = A m. Let M (A) denote the category of A-modules of finite length and V (κ) denote category of the κ-vector spaces of finite dimension. Prove that map K(M (A)) −→ K(V (κ)) is a homotopy equivalence. Problem 3.8.7. Let E be an exact category. Give an elementary description of K1 (E ). In particular, for a commutative Noetherian ring A, obtain an elementary description of K1 (Coh(A)). (Hint: Consult [GG, N].) Remark 3.8.8. A major part of our remaining interest concerns Ktheory schemes, some of what could be included as exercise in this section. However, we postpone this till Chapter 6.

Chapter 4

The Agreement with Classical K-Theory

Perhaps central to the study of algebraic K-theory has always been the K-theory of (commutative) rings A, that is the study of the K-theory of the exact category P(A) of finitely generated projective A-modules. As was mentioned at the end of Section 3.3, originally three groups, which we denoted (temporarily) by K0c (A), K1c (A), K2c (A), were defined. Some exact sequences of these three groups already existed in the early part of development of the theory [Mi, Sw3]. The desire to define higher K-groups and to extend these exact sequences is explicitly mentioned in the book of Swan [Sw3, p. 213]. It would not be far from the truth to say that higher algebraic K-theory, discussed in the previous chapters, evolved out of this same desire. The whole theory works for non-commutative rings, with unity. So, we will not assume that rings are necessarily commutative in this chapter. For such a ring A the category of finitely generated projective (right) A-modules will be denoted by P(A). We already saw that K0c (A) coincides with K0 (A) (3.3.6). The main goal of this chapter is, for i = 1, 2, to establish that Kic (A) coincides with Quillen Ki (A) := Ki (P(A)). However, the chapter is rich with various constructions, approaches to higher K-theory, and jargon. The original source for this chapter is the paper of Grayson [Gd1], and the results are mostly due to Quillen.

127

128

Algebraic K-Theory

4.1.

Symmetric Monoidal Categories

The direct sum operation on P(A) gives it the structure of a symmetric monoidal category in the following sense. Definition 4.1.1. A symmetric monoidal category is a triple (S, , e), where S is a category,  : S × S −→ S is a functor (to be referred to as “product”, “addition”, or “sum”), and e is a distinguished object (which would behave like an identity for the addition ). These three components satisfy the following conditions. 1. (Associativity) The addition  is associative, in the sense that there is a natural equivalence ∼

α : (1 × ) −→ ( × 1) More precisely, ∀ A, B, C ∈ Obj(S), there are natural isomorphisms ∼

αABC : A  (B  C) −→ (A  B)  C In other words, the diagram (S × S) × S ×1S

((a,b),c)→(a,(b,c))

/ S × (S × S)





S×S

1S ×

commutes, via α.

S×S 

,St



2. (Identity) There are natural equivalences 

S

∼

e : e  X −→ X ∼

e : X  e −→ X

(e,−)



commute via e, e.

1S

/S×S o 

(−,e)



+Ss

1S

S

The Agreement with Classical K-Theory

129

3. (Symmetry) There are natural equivalences S×S

∼

χ : A  B −→ B  A

(a,b)→(b,a)

 

/ S×S  ,S



commute via χ.

4. The following two diagrams commute up to natural equivalences: A  (B  (C  D)) 



/ (A  B)  (C  D)

α ∼

α ∼

/ ((A  B)  C)  D

Aα

αD

A  (e  B) 



AB



/ (A  (B  C))  D

∼ α

A  ((B  C)  D)



α ∼

/ (A  e)  B

∼

s

We only consider symmetric monoidal categories, as defined here, even when we drop the qualification “symmetric”. However, if the triple (S, , e) satisfies only the conditions (1, 2, 4), it should be referred to as a monoidal category. (While our definition already incorporates associativity (1) and symmetry (3) in it, the standard literature on monoidal categories may not always do so. So, all our (symmetric) monoidal categories are what the standard literature may call associative, symmetric monoidal categories.) Before we provide some examples, we make two observations. Definition 4.1.2. Let (M , ) be a symmetric monoidal category such that the isomorphism classes of objects in M form a set M Iso . Then M Iso is an abelian monoid. The Grothendieck group of this monoid is denoted by K0 M . It is a group given by the following universal property: M Iso f

/ K M 0 ϕ   )

G

Meaning that given a group G and a monoid homomorphism f , there is a unique group homomorphism ϕ, so that the above diagram

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Algebraic K-Theory

commutes. (Since M is assumed to be symmetric, it does not make any difference whether the diagram has the universal property in the category of groups Gr, or of abelian groups Ab.) Corollary 4.1.3. Suppose (S, , e) is a symmetric monoidal category. Then the addition  : S × S −→ S induces a product μ : k(BS × BS) −→ BS on the classifying space, where k(BS × BS) denotes the compactly generated topology on the product space (2.1.8). It also follows that, up to homotopy μ is commutative and associative and e ∈ BS acts as an identity. This endows BS structure of a Hc -space (see Section 4.8). Given a symmetric monoidal category S, the following is a standard way to construct a new symmetric monoidal category, which is of our primary interest. Definition 4.1.4. Let S be a symmetric monoidal category. Let Iso(S) be the subcategory of S, whose objects are the same as that of S, and morphisms are the isomorphisms of S. The following are some useful observations: 1. For objects P in S, let P denote the isomorphism classes of objects in S. ∀ P, Q ∈ Iso(S), P = Q =⇒ M orIso(S) (P, Q) = φ, M orIso(S) (P, P ) = AutS (P ). 2. Let {P i : i ∈ I} be the set of all isomorphism classes of objects in S, where I is an indexing set, and Pi is a choice  of representative of its isomorphism class P i . Then the inclusion i∈I AutS (Pi ) −→ Iso(S) is an equivalence of categories, where AutS (Pi ) denotes the one-object category. 3. Consequently, we have the homotopy equivalence of the classifying spaces  BAutS (Pi ) disjoint (disconnected) union in Top. BIso(S) i∈I

Example 4.1.5. The main example of our interest is the category P(R), of finitely generated (right) projective R-modules, where R is a ring, with  := ⊕, being the direct sum and e := R0 = {0} being the free R-module of rank zero. The following related examples are also among our primary interest.

The Agreement with Classical K-Theory

131

1. Let S := IsoP(R) ⊆ P(R) be the monoidal subcategory, as above. It follows  BAut(Pi ) disjoint (disconnected) union in Top BS 

i∈I

where runs through an indexing set I, with choices of representatives Pi ∈ P i , as in (4.1.4). 2. Let F (R) be the subcategory of S = IsoP(R), whose objects are based free R-modules {Rn : n ≥ 0}, and  φ if m = n n m M orF (R) (R , R ) = GLn (R) if m = n Then F (R) inherits the symmetric monoidal category structure from S = Iso(P(R)). Here GLn (R) denotes the group of automorphisms of Rn . Since Rn ∈ P(R) is treated as right module (of columns), matrices α ∈ GLn (R) act from the left side of Rn . We caution that in a non-commutative situation, rank of free modules may not make sense [Wc2, p. 2]. It is possible that Rm ∼ = Rn but n = m. However, such a possibility is ruled out in the category F (R). It follows that

B (F (R))



BGLn (R)

n≥0

disjoint (disconnected) union in Top. We also write

F (R) :=



GLn (R).

Note that, ignoring the monoidal structure on F (R), the righthand side can be viewed as a disjoint union of the one-element categories GLn (R). 3. Note that, since P(R) has an initial object, the classifying space BP(R) is contractible. 4.2.

S −1 S-Construction

For a symmetric monoidal category S, we define categories X with S-action. Then we define localization S −1 X in this section.

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Definition 4.2.1. Let (S, , e) be a symmetric monoidal category. An action of S on a category X (from left) is a functor  : S×X −→ X (we use the same notation ) such that 1. Action of e on X is a natural identity, meaning that ∀ E ∈ Obj(X ), there are natural equivalences e−

X

∼

ιE : e  E −→ E 

/X 

1X

ι

commute via ι.

+X

2. The action  is naturally associative. This means that ∀ P, Q ∈ Obj(S), ∀ E ∈ Obj(X ) there are natural equivalences αP QE : ∼ (P  Q)  E −→ P  (Q  E) such that ((P,Q),E)→((P,Q),E)

/ S × (S × X )

(S × S) × X ×1



S×X 



X

1×



S×X ∼ αP QE



commute via α.



/X

3. Further, ιE and αP QE are required to be coherent, as well with four factors associativity P1  P2  P3  E, where P1 , P2 , P3 ∈ Obj(S), and E ∈ Obj(X ). In this case, we say that X is an S-category, or X is a category with S-action. We would be mainly concerned with examples X of S-categories, which would have the flavor of the following. Example 4.2.2. Let R be a ring, and P(R) be the category of finitely generated (right) projective R-modules. Consider the symmetric monoidal category S = Iso(P(R)). Let X be the cat  / / C of finitely / egory of exact sequences E := K M generated (right) R-modules. The arrows in X are maps of such Then S has an action on X , defined by P  E :=  sequences.  /P ⊕M / / C. P ⊕ K

The Agreement with Classical K-Theory

133

Definition 4.2.3. Given (symmetric) monoidal categories (S, , eS ), (T, , eT ), a functor f : S −→ T is defined to be a monoidal functor if ∼

f (P Q) −→ f (P )f (Q),

∼

f (eS ) −→ eT

are naturally isomorphic.

In other words, the diagrams f ×f

S×S 



S

/T ×T 

f



eS 

S f

/ T _ _ _ _/ T ∼



T

/S 

eT 

eT f

/ T _ _ _/ T ∼

are naturally isomorphic. Further, f must respect the associativity in S and T , meaning the functors 

(P1 , P2 , P3 ) → f (P1 )  (f (P2 )  f (P3 )) must be naturally equivalent. (P1 , P2 , P3 ) → (f (P1 )  f (P2 ))  f (P3 )

Definition 4.2.4. Let (S, , e) be a symmetric monoidal category, and X , Y be categories with S-actions. A functor f : X −→ Y , of such S-categories, is said to preserve the S-action, if ∀ P ∈ ObjS, E ∈ ObjX

f (P  E) ∼ = P  f (E)

are naturally isomorphic. Further, f must also respect associativity, as in (4.2.3). We proceed to do a few basic and important constructions of categories. Definition 4.2.5. Let (S, , e) be a symmetric monoidal category, and X be an S-category. Define a new category S, X , as follows: 1. Objects of S, X  are the same as the objects of X . 2. Given E, F ∈ ObjX , an arrow f : E −→ F in S, X  is an equivalence class of pairs (P, f ), where P ∈ ObjS and f : P  E −→ F is an arrow in X . Two such pairs (P, f ), (Q, g) are

134

Algebraic K-Theory ∼

defined to be equivalent if there is an isomorphism τ : P −→ Q in S such that the diagram f

P E

/F D

commutes.

τ 1E 



g

QE

In this case, we write f := (P, f ) or f := [(P, f )], while P is understood to be the -summand of the domain of f . 3. Given two arrows, ϕ : E −→ F , γ : F −→ G in S, X , represented by (P, f ) and (Q, g), the composition γϕ is defined by the representation (Q  P, h), where h is defined by the composition, as in the commutative diagram 1Q f

QP E

O

Q

/QF

g S U W Y,  G

h

4. For E ∈ ObjS, X  = Obj X , the identity arrow 1E in S, X  is represented by (e, 1E ). Example 4.2.6. Suppose (S, , e) is a symmetric monoidal category and all morphisms of S are isomorphisms. Then ∀ P ∈ Obj(S), we have M or S,S (e, P ) = {(P, 1P )}. So, e is an initial object in S, S. The morphism (P, 1P ) is equivalent to all other of its representations (U, ϕ), as in the diagram U e 



P e

ϕ ∼ 1P

 /P

So, S, S is contractible. The following construction will be a basic object of study in the subsequent sections. Definition 4.2.7. Let (S, , e) be a symmetric monoidal category, and X be an S-category. Then S-acts on S × X diagonally (from

The Agreement with Classical K-Theory

135

left). So, S × X is also an S-category. Define S −1 X := S, S × X  We may write that F := (V, ϕ1 , ϕ2 ) : (Q1 , E1 ) −→ (Q2 , E2 ) is an arrow in S −1 X , to mean F is represented by (V, ϕ1 , ϕ2 ), with V ∈ Obj(S), ϕ1 ∈ M orS (V  Q1 , Q2 ), ϕ2 ∈ M orX (V  E1 , E2 ). We remark that S −1 X has two S-actions, one from left l and one from right r . For P, Q ∈ Obj(S) and E ∈ Obj(X ), the actions are defined by and P r (Q, E) := (Q, P  E). P l (Q, E) := (P  Q, E), Given an arrow f : P1 −→ P2 in S, and an arrow F := (V, ϕ1 , ϕ2 ) : (Q1 , E1 ) −→ (Q2 , E2 ) in S −1 X , define arrows 

f l F : (P1  Q1 , E1 ) −→ (P2  Q2 , E2 )

with f l F := (V, f  ϕ1 , ϕ2 )

f r F : (Q1 , P1  E1 ) −→ (Q2 , P2  E2 )

with f r F := (V, ϕ1 , f  ϕ2 )

See (4.3.8) for further information. Remark 4.2.8. In particular, S −1 S is defined in (4.2.7), for any symmetric monoidal category S. For a ring R, let S = F (R), T = Iso(P(R)), as in (4.1.5). Our main interest in this chapter is the study of the classifying spaces BS −1 S, BS −1 T , and BT −1 T , and their relationship with Quillen K-theory. All these will culminate in Sections 4.5 and 4.6.

4.3.

The Projection Functors

For a symmetric monoidal category S and an S-category X , in this section, we investigate the fibered and cofibered properties of the projection functors S −1 X −→ S, S and S −1 X −→ S, X . Definition 4.3.1. Let (S, , eS ) be a symmetric monoidal category, and X be an S-category. Then the projection p1 : S −1 X −→ S, S, defined by  ∀ (P, X) ∈ Obj S −1 X p1 (P, X) = P p1 (U, f, g) = (U, f ) ∀ (U, f, g) ∈ M orS −1 X ((P, X), (Q, Y )) (4.1)

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Algebraic K-Theory

defines a functor. For P ∈ Obj(S, S), the fiber category p−1 1 P was defined in (2.2.4). For future reference, we introduce the notation p1P , p1P for two obvious functors. Define p1P : X −→

p−1 1 P

 by

p1P (X) = (P, X) p1P (g) = (eS , 1P , g)

X ∈ Obj X g ∈ M orX (X, Y ) (4.2)

Further, define p1P by the commutative diagram / p−1 P 1 _

p1P : X p1P

(

(4.3) 

S −1 X

The vertical functor is a subcategory, but not necessarily a full subcategory. Lemma 4.3.2. Let (S, , e) be a symmetric monoidal category. Consider an arrow F = (U, f ) = (V, g) : P −→ Q in S, S, represented by f : U  P −→ Q, as well as g : V  P −→ Q. Then

∼

∃ isomorphism τ : U −→ V in S 

U P

f

/Q C

commutes.

τ 1P 



V P

g

(4.4)  Assume

U  P −→ Q

is monic, and the map

M orS (U, U ) −→ M orS (U  P, U  P )

is injective (faithful)

(4.5)

The latter property is read as that every translation is faithful. (Subsequently, it will be assumed that all morphisms in S are monic.) Under these hypotheses, it follows that there is only a unique choice of the isomorphism τ in (4.4), when U, V are given. In particular, if U = V, then τ = 1U and f = g.

The Agreement with Classical K-Theory

137

Lemma 4.3.3. Let (S, , e) be a symmetric monoidal category, and X be an S-category. Let p1 : S −1 X −→ S, S be the projection functor (4.1). Under the conditions of monicity and faithfulness (4.3.2), the functor p1P : X −→ p−1 1 P is an equivalence of categories, for all P ∈ Obj(S). Proof. We have Obj(p−1 1 P ) = {(P, X) : X ∈ Obj X }, and M orp−1 P ((P, X), (P, Y )) 1    f : U  P −→ P, = [(U, f, g)] :  f ∼ 1P g : U  X −→ Y = {[(eS , 1P , g)] : g : X −→ Y ∈ X } ∼ = M orX (X, Y ).

(4.6)

The proof is complete. In preparation for further discussions on S −1 X , we set up some notation and list some preliminary observations. Notation 4.3.4. Let S be a symmetric monoidal category, and X be an S-category. As indicated before, we make a notational convention that we will write v := (V, v) ∈ M or S,X (X, Y ), or v := (V, v) : X −→ Y in S, X , to mean that the arrow v : X −→ Y in S, X  is represented by the arrow v : V X −→ Y in X . By definition, V is unique up to isomorphism. Further, under the hypotheses of monicity and faithfulness (4.3.2), v : V X −→ Y in X is uniquely determined. We have the following lemma. Lemma 4.3.5. Let S be a symmetric monoidal category, and X be an S-category. Assume that every morphism in S is monic and translations in S are faithful (4.3.2). Then the projection functor p1 : S −1 X −→ S, X  is cofibered. Proof. For P ∈ Obj(S, S), we proceed to describe p1 /P . Obj (p1 /P ) =

((Q, X), v) : (Q, X) ∈ ObjS −1 X , v := (V, v) ∈ M or S,S ((Q, P ) . For i = 1, 2, and let ((Qi , Xi ), vi ) ∈ Obj(p1 /P ), where vi := (Vi , vi ) ∈ M or S,S (Qi , P ). A morphism, u := (W, f, g) ∈

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Algebraic K-Theory

M orp1 /P (((Q1 , X1 ), v1 ), ((Q2 , X2 ), v2 )) is given by f ∈ M orS (W  Q1 , Q2 ), g ∈ M orX (W  X1 , X2 ) such that the diagram Q1

v1

/P O v2

ρΓ=f

'

commutes in S, S.

Equivalently,

Q2 ∼

there is an isomorphism τ : V1 −→ V2 W in S such that the diagram V1  Q 1 τ 1Q1

v1



V2  W  Q1

1V2 f

/ V2  Q 2 v2

commutes in S.

)/

P (4.7)

∼

In fact, τ : V1 −→ V2  W is unique, under the monicity and faithfulness hypotheses (4.3.2). 1. Note that for P ∈ Obj S = Obj(S, S), the identity morphism 1P ∈ M or S,S (P, P ) is represented by 1P := (eS , 1P ). 2. For X1 , X2 ∈ Obj(X ), we describe M orp1 /P (((P, X1 ), 1P ), ((P, X2 ), 1P )). Let u := (W, f, g) ∈ M orρ/P (((P, X1 ), 1P ), ((P, X2 ), 1P )). With v1 = v2 = 1P := (eS , 1P ), we have V1 = ∼ ∼ V2 = eS . It follows τ : eS −→ W ∼ = eS  W −→ W . Reinterpreting (4.7), we have M orp1 /P (((P, X1 ), 1P ), ((P, X2 ), 1P )) = {(eS , 1P , g) ∈ Iso(S) × M orS (P, P ) × M orX (X1 , X2 )} ∼ M orX (X1 , X2 ) . (4.8) = 3. We describe M orp1 /P (((Q1 , X1 ), v1 ), ((P, X2 ), 1P )), for ((Q1 , X1 ), v1 ) ∈ Obj(p3 /P ), and X2 ∈ Obj(X ). So, v2 := (V2 , v2 ) = 1P := ∼ (eS , 1P ). Consequently, V2 = eS and τ : V1 −→ W . So, f = v1 in S, S. Reinterpreting (4.7), we have M orp1 /P (((Q1 , X1 ), v1 ), ((P, X2 ), 1P )) = {(V1 , v1 , g) ∈ Iso(S) × M orS (V1  P, P ) × M orX (V1  X1 , X2 )} ∼ = M orX (V1  X1 , X2 ) .

(4.9)

The Agreement with Classical K-Theory

139

4. Consider the natural (inclusion) functor Φ : p−1 1 P −→ p1 /P , defined by ⎧  −1  ⎪ ⎨ Φ(P, X) = ((P, X), 1P ) ∀(P, X) ∈ Obj p1 P Φ(u) = u ∀ u := (eS , 1P , g) ∈ ⎪ ⎩ M orp1 /P ((P, X2 ), (P, X2 )) . Note that ∼

Φ : M orp−1 P ((P, X2 ), (P, X2 )) −→ M orp1 /P (((P, X1 ), 1P ), ((P, X2 ), 1P )) 1

is a bijection (4.6), (4.8). 5. In the opposite directions, define the functor L : (p1 /P ) −→ p−1 1 P, with notation as in (4.3.4) ⎧ −1 ⎪ ⎨ L((Q1 , X1 ), v1 ) = (P, V1  X1 ) ∈ Obj(ρ P ), L(u) = L(W, f, g) := (eS , 1P , 1Q2  g) ∈ ⎪ ⎩ M or −1 ((P, Q1 , X1 ), (P, Q2 , X2 )) . p P 1

6. With notation as in (4.3.4), for ((Q1 , X1 ), v1 ) ∈ Obj(p1 /P ), with v1 : V1 Q1 −→ P and (P, X2 ) ∈ Obj(p−1 1 P ), we have two columns of isomorphisms M orp−1 P (L ((Q1 , X1 ), v1 ) , (P, X2 )) 1

M orp−1 P ((P, V1  X1 ), (P, X2 )) 1





M orX (V1  X1 , X2 )

M orp1 /P (((Q1 , X1 ), v1 ), Φ(P, X2 ))

_ _ _η _/ M orp1 /P (((Q1 , X1 ), u1 ), ((P, X2 ), 1P )) 



M orX (V1  X1 , X2 )

The first vertical bijection follows from (4.6) and the second vertical bijection follows from (4.9). This establishes that L is left adjoint to Φ, given by the following bijection: η(eS , 1P , G) = (U1 , u1 , G). The bijection η is natural with respect to either coordinates. This establishes that p1 is precofibered. We will prove that p1 is cofibered.

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Algebraic K-Theory

7. Let u = (U, u) : P −→ P  be an arrow in S, S, with u : V P −→ P  in S. The functor u∗ : p1 /P −→ p1 /P  given by  u∗ ((Q1 , X1 ), v1 ) = ((Q1 , X1 ), uv1 ) u∗ (W, f, g) = (W, f, g) for ((Q1 , X1 ), v1 ) ∈ Obj(p1 /P ), with v1 := (V1 , v1 ) : V1 Q1 −→ P in S. ⎧ Let (eS , 1P , g) ∈ M orp−1 P ((P, X1 ), (P, X2 )) ⎪ ⎪ 1 ⎪ ⎪ ⎪ Φ(eS , 1P , g) = (eS , 1P , G) ∈ ⎪ ⎪ ⎪ ⎪ ⎪ M orp1 /P (((P, X1 ), 1P ), ((P, X2 ), 1P )) ⎨ u∗ Φ(eS , 1P , G) = (eS , 1P , g) ∈ ⎪ ⎪ ⎪ M orp1 /P  (((P, X1 ), u), ((P, X2 ), u)) ⎪ ⎪ ⎪ ⎪ ⎪ Lu∗ Φ(eS , 1P , G) = (eS , 1P  , 1P  g) ∈ ⎪ ⎪ ⎩ M orp−1 P  ((P  , P  X1 ), (P  , P  X2 )). 1

−1 Using the same notation, for the functor u∗ : p−1 1 P −→ p1 P , we have

p−1 1 P 

u∗

Φ

/ (p1 /P )





 p−1 1 P l

Φ

-





u∗

(p1 /P  )

u∗ (P, X1 ) = (P  , V  X1 ) u∗ (eS , 1P , g) = (eS , 1P  , 1P  g)

L

(4.10) It follows that, for the composable arrows P0

u

wu

/ P1 ' 

w

in S, S,

w∗ u∗ = (wu)∗

P2

Therefore, p1 is cofibered. The proof is complete. We introduce the following definitions before we state some of the results.

The Agreement with Classical K-Theory

141

Definition 4.3.6. Let S be a symmetric monoidal category, and X be an S-category. For P ∈ Obj S, define the translation functor  TP : X −→ X

by

TP (X) = P  X TP (f ) = 1P  f

X ∈ Obj X f ∈ M orX (X1 , X2 )

We say that S acts invertibly, on X , if the translation maps TP are homotopy equivalences, ∀ P ∈ Obj S. The following is a direct application of the cofibered versions of Theorem B (2.2.6). Proposition 4.3.7. Let S be a symmetric monoidal category, and X be an S-category. Assume that every morphism in S is monic and translations in S are faithful (4.3.2). Consider the projection functor p1 : S −1 X −→ S, S. For P ∈ Obj(S), recall the notation (4.3.1), as in the commutative diagram X

p1P ∼ p1P

&

/ p−1 P 1

see (4.3.3). 

S −1 X

1. Suppose S acts invertibly on X . Then the sequence X

p1P

/ S −1 X

p1

/ S, S

is a homotopy fibration.

∀ P ∈ Obj(S, S) Consequently, if all maps in S are isomorphisms, then p1P : X −→ S −1 X is a homotopy equivalence. (There is no obvious functor, representing the homotopy inverse. The projection is not a functor.) 2. Conversely, if p1P is homotopy equivalence ∀ P ∈ Obj(S), then S acts invertibly on X . Proof. Assume S acts invertibly on X . By (4.3.5), p1 is precofibred (indeed, cofibered). For v := (V, v) ∈ M or S,S (P, P  ), we have the

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Algebraic K-Theory

commutative diagram (4.10) TV

X p1P



p−1 1 P

/X 

v∗

p1P 

/ p−1 P  1

The vertical functors are homotopy equivalences, by Lemma 4.3.3, and so is TV , by hypothesis. Therefore, v∗ is a homotopy equivalence. So, the precofibered version of Theorem B (2.2.6) applies. Therefore, the sequence p−1 1 P

p1

/ S −1 X

/ S, S

is a homotopy fibration.

∀ P ∈ Obj(S, S) ∼

Using the identification p1P : X −→ p−1 1 P , the first part of (1) is established. If all maps in S are isomorphisms, then S, S has an initial object, and S, S is contractible. Therefore, p1P is homotopy equivalence, for all P ∈ Obj(S, S). ∼ To see the converse, assume that p1P : X −→ S −1 X is a homotopy equivalence, ∀ P ∈ Obj(S, S). Now, for P ∈ Obj(Obj(S, S)), consider the commutative diagram of functors X TP

p1e

S





X

/ S −1 X

p1e

TrP

/ S −1 X S

where TrP is the translation, by the right action on S −1 X , defined in (4.3.8). In fact, TrP is a homotopy equivalence (4.3.8). Since the horizontal functors are homotopy equivalences, so is TP . So, the converse (2) is established. The following would be of our particular interest. Lemma 4.3.8. Let S be a symmetric monoidal category, and X be an S-category. There is a left S-action l and a right S-action r on S −1 X , as defined in (4.2.7). Then both actions act invertibly on S −1 X .

The Agreement with Classical K-Theory

143

Proof. For P ∈ Obj S, let TlP , TrP : S −1 X −→ S −1 X be the translations, with respect to l , r , defined as, TlP (Q, X) = (P  Q, X), and TrP = (Q, P  X). Now, TlP TrP (Q, X) = TrP TlP (Q, X) = (P  Q, P  X) = P  (Q, X). There is a natural arrow (Q, X) −→ TrP TlP (Q, X) in S −1 X , represented by (P, 1(P Q,P X) ). This defines a natural transformation 1S −1 X −→ TrP TlP = TlP TrP . Hence, by (2.1.9), both Tl (P ), Tr (P ) are homotopy equivalences. Proposition 4.3.7 has a useful application (4.3.10), by iterating the process of S −1 X -construction. This requires straightening some scope of confusions. Remark 4.3.9. Let (S, ) be a symmetric monoidal category, and (X , ) be an S-category. The availability of two S-actions l and r , on S −1 X , provides flexibility. It also creates confusion if we attempt to iterate the process. Namely, we obtain two localized categories Sl−1 S −1 X := (S, l )−1 (S −1 X )

and

Sr−1 S −1 X := (S, r )−1 (S −1 X ). Corollary 4.3.10. Let S be a symmetric monoidal category and X be an S-category. Assume that every morphism in S is an isomorphism (i.e., S = Iso(S)) and translations in S are faithful (4.3.2). As before, p1 : Sl−1 S −1 X −→ S, S denotes projection to the 1st coordinate. With other notation as in (4.3.1), the functors S −1 X

p1P

/ S −1 S −1 X l

are homotopy equivalences

∀ P ∈ Obj(S). Further, the homotopy inverse ΓP : Sl−1 S −1 X −→ S −1 X of p1P is given by ΓP (L, (Q, X)) = (P  Q, L  X), which is a well-defined functor.

∀ (L, (Q, X)) ∈ Obj(Sl−1 S −1 X )

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Algebraic K-Theory

Proof. By (4.3.8), the left S-action l acts invertibly on S −1 X . By Proposition 4.3.7 it is established that p1P is a homotopy equivalence. That ΓP is a well-defined functor, and is the homotopy inverse, follows from (4.3.11), which follows. The proof is complete. Now we proceed to define the homotopy inverse functor ΓP , mentioned above. Definition 4.3.11. Let S be a symmetric monoidal category and X be an S-category. Let (Pi , (Qi , Xi )) ∈ Obj(Sl−1 S −1 X ) for i = 1, 2. An arrow u ∈ M orS −1 S −1 X ((P1 , (Q1 , X1 )), (P2 , (Q2 , X2 ))), is first l represented by u = (U0 , u0 , f ) : (P1 , (Q1 , X1 )) −→ (P2 , (Q2 , X2 )), where U0 ∈ Obj(S), u0 : U0  P1 −→ P2 is in S and f : U0 l (Q1 , X1 ) = (U0  Q1 , X1 ) −→ (Q2 , X2 ) is in S −1 X . So, f is represented by f = (U1 , u1 , u2 ), with U1 ∈ Obj(S), u1 : U1  U0  Q1 −→ Q2 in S and u2 : U1  X1 −→ X2 in X . So, we write u = (U0 , U1 , u0 , u1 , u2 ). It follows, M orS −1 S −1 X ((P1 , (Q1 , X1 )), (P2 , (Q2 , X2 ))) = l ∪U0 ,U1 ∈ObjS M orS (U0  P1 , P2 ) × M orS (U1  U0  Q1 , Q2 ) ×M orX (U1  X1 , X2 ) where U0 , U1 runs through all the isomorphism classes of objects in S. Define  Γ(P, (Q, X)) = (Q, P  X) ∀ (P, (Q, X)) ∈ Obj(Sl−1 S −1 X ) Γ(u) = (U0  U1 , u1 , u0  u2 ) ∀ u = (U0 , U1 , u0 , u1 , u2 ). Note Γ(u) : (Q1 , P1  X1 ) −→ (Q2 , P2  X2 ) is an arrow in S −1 X . One checks that the composition rule works. It follows that Γ : Sl−1 S −1 X −→ S −1 X is a well-defined functor. The following provides a homotopy inverse ΓP to p1P : Sl−1 S −1 X −→ S −1 X in (4.3.10). Lemma 4.3.12. Let S be a symmetric monoidal category and X be an S-category. As before, p1 : Sl−1 S −1 X −→ S, S denotes the first projection map and other notation, as in (4.3.1). Let P ∈ Obj(S),

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145

and define the functor ΓP by the commutative diagram Γ

Sl−1 S −1 X F

I

/ S −1 X

TlP M O  R) S −1 X

where Γ is as in (4.3.11).

ΓP

Then ΓP is a homotopy left-inverse of p1P : S −1 X −→ Sl−1 S −1 X . Proof. For (Q, X) ∈ Obj(S −1 X ), we have ΓP p1P = TlP Γp1P (Q, X) = TlP Γ(P, (Q, X) = TlP (Q, P  X) = (P  Q, P  X) = P  (Q, X). The natural arrows (P, 1P (Q,X) ) : (Q, X) −→ TlP Γp1P (Q, X) in S −1 X lead to a natural transformation 1S −1 X −→ ΓP p1P . Therefore, B1S −1 X and BΓP p1P are homotopic. So, ΓP is a homotopy left-inverse of p1P . The proof is complete. 4.4.

The X -Coordinate Functor

Having dealt with the first coordinate map p1 : S −1 X −→ S, X  in the last section (Section 4.3), we consider the X -coordinate functor in this section. Lemma 4.4.1. Let S be a symmetric monoidal category and X be an S-category. Suppose every map in X is monic, and ∀ P ∈ Obj(S), ∀X ∈ Obj(E ) the translation map AutS (P ) −→ AutX (P  X) is an injection (faithfulness, analogous to (4.3.2)). Consider the 2ndcoordinate projection p : S −1 X −→ S, X . Then p is cofibered. Also, p−1 Y = S × Y ∼ = S, and for morphisms u = (U, u) : Y1 −→ Y2 in S, X , represented by u : U Y1 −→ Y2 in X , the induced functor u∗ : p−1 Y1 −→ p−1 Y2 (i.e., the cobase change map) is given by u∗ (P, Y1 ) = (U  P, Y2 ). S TU



S

So, the diagram,

P →(P,Y1 )

/ p−1 Y1 

u∗

commutes,

/ p−1 Y2

where TU : S −→ S is the translation functor P → U  P .

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Algebraic K-Theory

Proof. Given Y ∈ ObjS, X  = Obj X , we describe the categories p/Y and p−1 Y . To simplify notation, we may write (P, X; v) := ((P, X), v) for ((P, X), v) ∈ Obj(p/Y ). ⎧ ⎪ ⎪ Obj(p/Y )=

⎪ ⎪ (P, X; v) : P ∈ Obj(S), X ∈ Obj(X ), v ∈ M orS,X  (X, Y ) ⎪ ⎪ ⎪ ⎪ ⎪ M orp/Y ((P1 , X1 ; v1 ), (P2 , X2 ; v2 )) = ⎪ ⎪ ⎪ ⎪ ⎨ {(u1 , u2 ) ∈ M orS −1 X ((P1 , X1 ), (P2 , X2 ))  v2 u2 = v1 ∈ S, X }

Obj(p−1 Y ) = {(P, Y ) : P ∈ Obj(S)} ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ M orp−1 Y ((P1 , Y ), (P2 , Y )) = ⎪ ⎪ ⎪ ⎪ {(u1 , u2 ) ∈ M orS −1 X ((P1 , Y ), (P2 , Y ))  u2 = (eS , 1Y ) ∈ S, X } ⎪ ⎪ ⎪ ⎩ = M orS (P1 , P2 )

With v1 := (V1 , v1 ) : X1 −→ Y , in S, X , represented by v1 : V1  X1 −→ Y in S, we have ⎧ ⎪ ⎨ M orp/Y ((P1 , X1 ; v1 ), (P2 , Y ; 1Y )) = {(u1 , u2 ) ∈ M orS −1 X ((P1 , X1 ), (P2 , Y ))  u2 = v1 in S, X } ⎪ ⎩ ∼ Hom (V  P , P ). = S 1 1 2 An arrow (u1 , u2 ) ∈ M orp/Y ((P1 , X1 ; v1 ), (P2 , X2 ; v2 )) is represented by u1 : U  P1 −→ P2 in S and u2 : U ⊕ X1 −→ X2 in X . Also, v1 = (V1 , v1 ), v2 = (V2 , v2 ) are represented by v1 : V1  X1 −→ Y in S, v2 : V2  X2 −→ Y in X . And, v2 u2 = u1 means, there is ∼ an isomorphism τ : P1 −→ V2  U such that the following diagram commutes: V1  X1

v1

τ 1X1 



V2  U  X1

1V2 u2

V1  P1 W U S Q τ 1P  1



V2  U  P1

1V2 u1

/ V2  X2 v2

O

/) Y

We compose

& / V2  P2

By the monicity and faithfulness hypotheses, τ is uniquely determined. Define L(u1 , u2 ) = ((1V2  u1 )(τ  1P1 ), 1Y ) ∈ M orp−1 Y

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147

((V1  P1 , Y )), (V2  P2 , Y )). With such notation, define, L : p/Y −→ p−1 Y by ⎧ ∀ (P, X; v) ∈ Obj(p/Y ) ⎪ ⎨ L(P, X; v) = (V  P, Y ) L(u1 , u2 ) = ((1V2  u1 )(τ  1P1 ), 1Y ) ⎪ ⎩ ∀ (u1 , u2 ) ∈ M orp/Y ((P1 , X1 ; v1 ), (P2 , X2 ; v2 )) . Then L : p/Y −→ p−1 Y is a functor. We check that L is left adjoint to Φ : p−1 Y −→ p/Y , as follows. Let ((P1 , X1 ), v1 ) ∈ Obj (p/Y ), with v1 = (V1 , v1 ) : X1 −→ Y in S, X , represented by v1 : V1  X1 −→ Y in X and (P2 , Y ) ∈ Obj p−1 Y . Then M orp−1 Y (L ((P1 , X1 ), v1 ) , (P2 , Y ))

M orp−1 Y ((P1  V1 , Y ), (P2 , Y )) 



M orS (P1  V1 , P2 )

M orp/Y (((P1 , X1 ), v1 ), Φ(P1 , Y ))

_ _ _η _/ M orp/Y (((P1 , X1 ), v1 ), ((P2 , Y ), 1Y )) 



M orS (V1  P1 , P2 )

This establishes that L is left adjoint to Φ, by defining the following bijection: η(eS , f, 1Y ) = (V1 , f, v1 ) Therefore, p is precofibered. Further routine checking completes the proof that p is cofibered, and of the rest of the assertions. The proof is complete. Using essentially the same arguments in (4.4.1), we have the following. Lemma 4.4.2. Let S be a symmetric monoidal category, and X be an S-category. Assume that every map in X is monic, and ∀ P ∈ Obj S, ∀X ∈ Obj X the translation maps AutS (P ) −→ AutX (P  X) are injections (faithfulness). Treat S −1 X as an S-category, with its left action l . Then the X -coordinate map

148

Algebraic K-Theory

p3 : Sl−1 S −1 X −→ S, X  is cofibered, and the sequence S −1 S

ιY

/ S −1 S −1 X p3 l

/ S, X 

is a homotopy fibration, ∀Y ∈ Obj X , where ιY is the functor, defined by ⎧ ∀ (P, Q) ∈ Obj(S −1 S) ⎪ ⎨ ιY (P, Q) = (P, (Q, Y )) ιY (U0 , u0 , u1 ) = (U0 , eS , u0 , (u1 , 1Y )) ⎪ ⎩ ∀ (U0 , u0 , u1 ) ∈ M orS −1 S ((P1 , Q1 ), (P2 , Q2 )) . (4.11) Proof. To simplify notation, we write (P, Q, X) := (P, (Q, X)) ∈ Obj Sl−1 S −1 X . An arrow u : (P1 , Q1 , X1 ) −→ (P2 , Q2 , X2 ) in Sl−1 S −1 X is given by (U0 , (u0 , f )), with u0 : U0  P1 −→ P2 in S, and f : U0  (Q1 , X1 ) = (U0  Q1 , X1 ) −→ (Q2 , X2 ) in S −1 X . So, f = (U1 , u1 , u2 ), with u1 : U0  U1  Q1 −→ Q2 in S and u2 : U1  X1 −→ X2 in X . In summary, the arrow u : (P1 , Q1 , X1 ) −→ (P2 , Q2 , X2 ) in Sl−1 S −1 X is represented by u := (U0 , U1 , u0 , u1 , u2 ) with U0 , U1 ∈ Obj S ⎧ ⎪ ⎨ u0 ∈ M orS (U0  P1 , P2 ) u1 ∈ M orS (U0  U1  Q1 , Q2 ) ⎪ ⎩ u2 ∈ M orX (U1  X1 , X2 )

(4.12)

Let IsoCl(S) denote the set of all isomorphism classes of objects in S. Then we have M or −1 −1 ((P1 , Q1 , X1 ), (P2 , Q2 , X2 )) =  Sl S X [U0 ],[U1 ]∈IsoCl(S) M orS (U0  P1 , P2 ) × M orS (U0  U1  Q1 , Q2 ) ×M orX (U1  X1 , X2 ) So, the union runs over choices of one U0 , U1 from each isomorphism class in Obj(S). The third coordinate projection p3 (u) = u2 ∈ M orX (U1  X1 , X2 ) is to be considered as a morphism X1 −→ X2 in

The Agreement with Classical K-Theory

149

S, X . Let Y ∈ Obj(S, X ) = Obj(X ). We describe the categories p3 /Y and p−1 3 Y , as follows: ⎧ p3 /Y = ⎪ ⎪

⎪ ⎪ ⎪ (P, Q, X; v) : P, Q ∈ ObjS, X ∈ Obj X , v ∈ M or S,X (X, Y ) ⎪ ⎪ ⎪ ⎨M or  p3 /Y ((P1 , Q1 , X1 ; v1 ) , (P2 , Q2 , X2 ; v2 )) = ⎪ ⎪ [U0 ],[U1]∈IsoCl(S) ⎪ ⎪ ⎪ ⎪ (U0 , U1 , u0 , u1 , u2 ) ∈ M orS −1 S −1 X ((P1 , (Q1 , X1 )), (P2 , (Q2 , X2 ))) : ⎪ ⎪ l ⎩ v2 u2 = v1 ∈ S, X } Let vi := (Vi , vi ) be represented by the maps vi : Vi  Xi −→ Y in X for i = 1, 2. The equality, v2 u2 = v1 , above, is given by the commutative diagram V1  X1 τ 1X1

v1



V2  U1  X1

/ V2  X2 v2

1V2 u2

/) Y

under the monic and faithfulness hypotheses, for a unique

(4.13)

∼

isomorphism τ : V1 −→ V2  U1 in S. So, we have M orp/Y ((P1 , Q1 , X1 ; v1 ), (P2 , Q2 , X2 ; v2 ))  = [U0 ],[U1]∈IsoCl(S) {(u0 , u1 , u2 ) ∈ M orS (U0  P1 , P2 ) × M orS (U0  U1  Q1 , Q2 ) ×M orX (U1  X1 , X2 ) :

 ∼ ∃τ : V1 −→ V2  U1 , v1 = v2 (1V2  u2 )(τ  1X1 )

Further, if X2 = Y and v2 = (V2 , v2 ) = 1Y := (eS , 1Y ), then the above diagram reduces to V1  X1 τ 1X1

v1



U1  X1

# u2

/Y

V2 = eS , under the hypotheses, for a unique isomorphism ∼ τ : V1 −→ U1 in S.

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Algebraic K-Theory

(Not surprisingly, it precisely means v1 = u2 in S, X .) Therefore, M orp/Y ((P1 , Q1 , X1 ; v1 ), (P2 , Q2 , Y ; 1Y ))  = [U0],[U1 ]∈IsoCl(S) {(u0 , u1 , u2 ) ∈ M orS (U0  P1 , P2 ) × M orS (U0  U1  Q1 , Q2 ) ×M orX (U1  X1 , Y ) :  ∼ ∃τ : V1 −→ U1 , v1 = u2 (τ  1X1 ) i.e., u2 = v1 = M orS (U0  P1 , P2 ) × M orS (U0  V1  Q1 , Q2 ) × v1 up to isomorphisms of U0 ∼ = M orS −1 S ((P1 , Q1  V1 ), (P2 , Q2 )) If X1 = Y and v1 = (V1 , v1 ) = (eS , 1Y ), then V1 = eS . From this, the description of p−1 Y follows: ⎧ −1 p Y = {(P, Q, Y ) : P, Q ∈ ObjS} ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ M orp−1 Y ((P1 , Q1 , Y ), (P2 , Q2 , Y )) = M orS (U0  P1 , P2 ) × M orS (U0  Q1 , Q2 ) × 1Y ⎪ ⎪ ⎪ up to isomorphisms of U0 ⎪ ⎪ ⎪ ⎩= ∼ M orS −1 S ((P1 , Q1 ), (P2 , Q2 ))

(4.14)

Let Φ : p−1 3 Y −→ p3 /Y be the natural functor. We proceed to define a left adjoint L : p3 /Y −→ p−1 3 Y , of Φ, as follows. With notation, as in (4.12), (4.13), let u = (u0 , u1 , u2 ) : ((P1 , (Q1 , X1 )), v1 ) −→ ((P2 , (Q2 , X2 )), v2 ) be an arrow in p3 /Y . From the description above, we have arrows u0 : U0  P1 −→ P2 and u1 : U0  U1  Q1 −→ Q2 , ∼ and τ : V1 −→ V2  U1 , in S, commuting the respective diagram. So, we obtain a commutative diagram U0  V1  Q1 τ 1Q1 U0

V

Uu

S



V2  U0  U1  Q1

1V2 u1

Q

' / V2  Q2

where u is the composition in S.

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151

Then Lu := (U0 , eS , u0 , u, 1Y ) : (P1 , V1  Q1 , Y ) −→ (P2 , V2  Q2 , Y ) is a map in p−1 3 Y . Accordingly, define  L((P, Q, X), v) = (P, (V  Q, Y )), v := (V, v) : V  X −→ Y L(u) = (U0 , eS , u0 , u, 1Y ) : (P1 , V1  Q1 , Y ) −→ (P2 , V2  Q2 , Y ) Let (P1 , (Q1 , X1 ), v1 ) ∈ Obj (p3 /Y ), (P2 , (Q2 , Y )) ∈ Obj p−1 Y , with v1 : V1  X1 −→ Y in S. Then we have two columns of bijections:

According to this diagram, a natural bijection η (the dotted arrow) is defined, as follows: η(U0 , eS , u0 , u1 , 1Y ) = (U0 , V1 , u0 , u1 , v1 ) Therefore, p3 is precofibered. Let v := (V, v) : Y −→ Y  , w = (W, w) : Y  −→ Y ” be two composable arrows in S, X . The cobase change functor v∗ is given by v∗ (P, (Q, Y )) := Lv∗ Φ(P, (Q, Y )) = Lv∗ ((P, (Q, Y )), 1Y ) = L(((P, (Q, Y )), v) = (P, (V  Q, Y  ). It follows (wv)∗ (P, (Q, Y )) = w∗ v∗ (P, (Q, Y )). An arrow u ∈ M orp−1 Y ((P1 , (Q1 , Y )), (P2 , (Q2 , Y ))), is given by u = (U0 , eS , u0 , 3 u1 , 1Y ), with u0 : U0  P1 −→ P2 in S, and u1 : U0  Q1 −→ Q2 in X . ⎧ Φ(u) = v∗ ((U0 , eS , u0 , u1 , 1Y )) : ⎪ ⎪ ⎪ ⎪ ⎪ ((P1 , (Q1 , Y )), 1Y )) −→ ((P2 , (Q2 , Y )), 1Y )) ⎪ ⎪ ⎪ ⎨ v∗ Φ(u) = (U0 , eS , u0 , u1 , 1Y ) : ((P1 , (Q1 , Y )), v)) −→ ((P2 , (Q2 , Y )), v) ⎪ ⎪ ⎪ ⎪ ⎪ v∗ (u) := Lv∗ Φ(u) = (U0 , eS , u0 , 1V  u1 , 1Y  ) ⎪ ⎪ ⎪ ⎩ ∈ M orp−1 Y  ((P1 , (V  Q1 , Y  )), (P2 , (V  Q2 , Y  ))) 3

152

Algebraic K-Theory

From this it follows that (wv)∗ (u) = w∗ v∗ (u). So, p3 is cofibered. We have the commutative diagram of functors / p−1 Y 3

S −1 S TrV





v∗

where

/ p−1 Y  3

S −1 S

TrV (P, Q) = V r (P, Q) = (P, V  Q). By (4.14), the horizontal functors are homotopy equivalences, and so is the translation Tr (V ) (4.3.10). Therefore, the cobase change functors v∗ are homotopy equivalences. Hence, the cofibered version of Theorem B (2.2.6) is applicable, and the lemma is established. The proof is complete. The following is the final theorem in this section. Theorem 4.4.3. Let S be a symmetric monoidal category, and X be an S-category. 1. Assume that every morphism in S is an isomorphism (i.e., S = Iso(S)), and translations are faithful (4.3.2). 2. Also assume that all maps in X are monic, and the translation maps AutS (P ) −→ AutX (P  X) are injective (i.e., faithful), ∀ P ∈ Obj S, ∀X ∈ Obj X . Let p2 : S −1 X −→ S, X  be the X -coordinate projection. Then the sequence S −1 S

νY

p2

/ S −1 X

/ S, X 

is a homotopy fibration, ∀Y ∈ Obj X ,

where νY : S −1 S −→ S −1 X denotes the functor (P, Q) → (Q, P  Y ). Proof. Consider the diagram p2

S −1 X S −1 S

ιY

/



p1e

/ S, X 

S

Sl−1 S −1 X

p3

/ S, X 

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153

The vertical arrow p1eS is the functor p1eS (Q, X) = (eS , (Q, X)), which is a homotopy equivalence (4.3.10). The homotopy inverse of p1eS is given by ΓeS (P, (Q, X)) = (Q, P  Q). With ιY , as in (4.4.2), we have νY = ΓeS ιY . The sequence in the 2nd-line is a homotopy fibration, by Lemma 4.4.2. Therefore, the proof is complete by identifying S −1 X with Sl−1 S −1 X , via p1eS . 4.5.

Split Exact Categories

Given an exact category E , we define a new category EE , whose objects are conflations. This is not to be confused with the category of extensions ε(E ) defined before (Section 3.4). Definition 4.5.1. Let E be an exact category. We define the category EE as follows: 1. The objects of EE are the admissible exact sequences (conflasions), in E  .  2. Let Ei := Ki   ιi / Mi ζi / / Ci ∈ Obj EE , for i = 1, 2. A morphism f := (α, β) : E1 −→ E2 is given by an equivalence class of diagrams E1

 ι1 / MO 1

KO 1  α

?

 

K2



f



E2

1K2

1M1 ι1 α

 

K2 

ζ1

ι2

/ M 1  _ β   / M2

/ / C1 OO / / C _

ζ2

(4.15)

 / / C2

Two such diagrams are defined to be equivalent if there is an isomorphism, with all the horizontal maps equal to identity, except C. Indeed, up to a choice of C, the diagram (4.15) is determined by the commutative diagram  ι1 / M 1  _ β  α  .  ?    / M2 K2

KO 1 

ι2

We also write f = (α, β).

(4.16)

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Algebraic K-Theory

Having defined the category EE , we have the following observations: 1. We have not said the EE is an exact category. 2. With the above notation in (4.15), define C

t(f ) = ( C1 o o



/ C2 ) ∈ M orQE (C1 , C2 ).

Define the target functor, t : EE −→ QE by  E1 ∈ Obj EE as in (4.15) t(E1 ) = C1 t(f ) = as above f ∈ M orEE (E1 , E2 ) as in (4.15) 3. For C ∈ Obj(QE ), we sometimes denote the fiber subcategory EEC := t−1 C. Recall that the identity map 1C ∈ M orQE (C, C) is given by C o o 1C C   1C / C . It follows from diagram (4.15), the morphisms in t−1 C = EEC are given by commutative diagrams: KO 1 

 ι1 / M1

α 

 β

K2



α, β



ι2

being isomorphisms in E .

/ M2 ∼

4. Further, t−1 0 = EE0 −→ Iso(E ), where Iso(E ) denotes the category whose objects are the same as the objects in E , and morphisms are isomorphisms (4.1.4). More explicitly, the equivalence is defined as follows: (a) For M ∈ Obj E , let F (M ) denote the conflation ( M   1M / M / / 0 ). ∼

(b) For an isomorphism f : M1 −→ M2 , define the morphism F (M1 ) −→ F (M2 ), by the diagram  1M1 / M1

MO 1  f −1 

M2

 f



1M2

 / M2

Lemma 4.5.2. Let E be an exact category, S = Iso(E ) and C ∈ Obj E . Note that S is a symmetric monoidal category under the direct sum ⊕. Then the fiber category EEC = t−1 C has a natural symmetric monoidal category structure  (given in the proof ). Further,

The Agreement with Classical K-Theory

155

1. Every arrow in EEC = t−1 C is monic. 2. There is a natural monoidal functor ηC : S −→ EEC . Further, for P ∈ Obj(S), E ∈ Obj (EEC ) the map AutS (P ) −→ AutEEC (P  E) is injective (faithfulness). (As is customary, for P ∈ Obj(S), E ∈ Obj(EEC ), we denote P  E := ηC (P )  E.) Proof. An arrow f := E1 −→ E2 in EEC is given by f = (α, β), as in (4.16). As both α and β are monic (in fact, isomorphisms), so is f .  Now, we define the  monoidal structure on EEC . Let Ei = ζi   ιi / / / C ∈ Obj EEC , for i = 1, 2. Consider the pullKi Mi back diagram

K1 K1

K2 _

K2 _



 /M

 / / M2



 / M1

 //C

ι1

Define,

ι2

where M := M1 ×C M2

is the pullback.

ζ2

ζ1

E1  E2 :=



K1 ⊕ K2 

 ι

/M

ζ



//C

∈ Obj EEC .

monoidal This addition operation  on EE   C defines a symmetric  1 structure on EEC , where eC := 0  / C C / / C would be the zero-object. We proceed to define the monoidal functor ηC : S −→ EE  C.  //C /P ⊕C ∈ For P ∈ Obj(E ), define ηC (P ) := P  ∼ Obj (EEC ). For an isomorphism f : P −→ Q in E , define ηC (f ) = (f −1 , f ⊕ 1C ) ∈ M orEE (ηC (P ), ηC (Q)). We display the same by the diagram, ηC (P )



PO 

/P ⊕C O

f −1 

Q

ηC (f )



ηC (Q)

1Q



  

Q

//C OO 1C

1

/P ⊕C  f ⊕1C



/Q⊕C

/ / C _ 

1C

//C

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Algebraic K-Theory

These associations P → ηC (P ), f → ηC (f ) define the functor ηC : S −→ EEC . It is also clear that ηC (P ⊕ Q) = ηC (P )  ηC (Q). Hence, ηC is a monoidal functor. In an additive category, the map Aut(P ) −→ Aut(P ⊕K) is injective. It follows from this that the map AutS (P ) −→ AutEEC (P  E) is injective. The proof is complete. In this chapter, our primary interest has been the category of projective modules, in which every exact sequence splits. We define split exact category in the obvious way. Definition 4.5.3. An exact category E is said to be a split exact category if every exact sequence splits. Remark  4.5.4. Let E be a split exact category and C ∈ Obj E . Let X := K   ι / P ζ / / C ∈ Obj(EEC ). Then X ∼ = ηC (K) in EEC . Further, S, EEC  is connected. Proof. Consider the following commutative diagram: w

  

K

~ ι

r P 1P −sζ

 u _s U /P i //C

where ζs = 1C .

ζ

Since ι = ker(ζ), there is an arrow w so that the triangle commutes. Since ιwι = ι, it follows that wι = 1K . Define two arrows in EEC , as follows:

One checks that gf = 1 and f g = 1. So, ηC (K) ∼ = X in EEC , hence also in S, EEC . Further, in S, EEC , we have an obvious arrow eC −→ ηC (K) in S, EEC , represented by a map 1ηC (K) : eC  ηC (K) −→ ηC (K) in EEC . Therefore, S, EEC  is connected. The proof is complete.

The Agreement with Classical K-Theory

157

The following construction will be of some use to us in the proof of the next theorem. Definition 4.5.5. Suppose η : (S, S , eS ) −→ (E, E , eE ) is a functor of symmetric monoidal categories. Then S, E has a natural symmetric monoidal category structure, as follows. Write T = S, E. For, X1 , X2 ∈ Obj(T ) = Obj(E), define X1 T X2 := X1 E X2 . For the morphisms f1 = (P1 , f1 ), f2 = (P2 , f2 ) ∈ M orT (X1 , Y1 ), with P1 , P2 ∈ Obj(S), define f1 T f2 := (P1 S P2 , f1 E f2 ). This defines the monoidal structure in T . Theorem 4.5.6. Let E  be a split exact category,  and S = Iso(E ).  ζ E Fix C ∈ Obj(E ) and E = K  / / C ∈ Obj(EEC ). Then /M the functor νE : S −1 S −→ S −1 EEC

is a homotopy equivalence,

where νE (P, Q) = (Q, P  E) with P  E :=  0⊕ζE / / C (see 4.4.3). P ⊕M

  P ⊕ K /

Proof. Because of (4.5.2), Theorem 4.4.3 is applicable. So, the sequence S −1 S

νE

/ S −1 EEC

/ S, EEC 

is a homotopy fibration.

So, we need to prove S, EEC  is contractible. Write T = S, EEC . By (4.5.5), T has the monoidal product T : T × T −→ T . This induces a map μ : B(T × T ) −→ BT , of the classifying spaces. However, ∼ B(T ×T ) −→ k(BT ×BT ) is a homeomorphism (2.1.8). This provides an Hc -space structure on BT (refer to Section 4.8 for Hc -spaces). Further, BT is connected (4.5.4). By Theorem 4.8.6, BT is group like. In particular, there is a map j : 1BT −→ 1BT such that j · 1BT and 1BT · j are null homotopic.

158

Algebraic K-Theory

 Suppose E = K   notation, consider the

ι

ζ

/M



//C

∈ Obj (EEC ). To establish

MJ J

K _ JJ Δ JJ ι2 JJ J$   / K ι1 M × C M q 2

pullback diagram

q1

K



ι



//M



/M

ι

K _



ζ

//C

ζ

Δ being the diagonal arrow. EE =

We have

 K ⊕K

  (ι1 ,ι2 )/

M ×C M

ζq1

//C



Define the functor 2T : T −→ T by the composition, as follows: T N

Δ

P

/ T × T 1T ×1T/ T × T R

So, 2T (E) = E T E.

T T V W Y -  2T T

Recall that B1T · B1T := B2T (consult Section 4.8). (Ideally, in multiplicative notation, 2T would be denoted by (1T )2 .) Consider the following arrow in EEC : K E

 (1K ,ι) /

K ⊕O K  K ⊕K

τE



E T E

1



K ⊕K



(0,ζ)

/K ⊕M

α

//C O 1C

1

β



K ⊕O M

(0,ζ)

 (ι1 ,Δ)



//C 

1C

/ M ×C M //C ζq1 =ζq2

(ι1 ,ι2 )

⎧ Δ is the diagonal map, ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎨ 1K −1K , α= 0 ι ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ β = 1K −1K ⎩ 0 1K

where

The Agreement with Classical K-Theory

159

Then (K, τE ) represents a natural arrow τE : E −→ ET E =: 2T (E) in T . We established that there is a natural transformation τ : 1T −→ 2T . So, the two maps 1BT B2T = B1T · B1T are homotopic. So, 1BT = B1T · B1T in [BT, BT ]. Therefore, j · 1BT = j · B1T · B1T in [BT, BT ]. Since j · 1BT is null homotopic (i.e., the identity element in [BT, BT ]), it follows B1T is null homotopic. Therefore, T is contractible. The proof is complete. 4.5.1.

S −1 S and QE

For a split exact category E , and S = Iso(E ), in this section, we / S −1 EE / QE is a homotopy prove that the sequence S −1 S fibration. To prepare for this, we prove the following. Lemma 4.5.7. As in (4.4.3), suppose S is a symmetric monoidal category, with S = Iso(S). Assume that translations in S are faithful. Let X be a S-category. Let F : X −→ Y be any functor, such that F (P  X) = F (X) (in other words, with trivial S-action on Y , F is a functor of S-categories.). Let S −1 F : S −1 X −→ Y be the functor induced by F, sending (P, X) → F (X). Then 1. For Y ∈ Obj(Y ), F −1 Y is S-subcategory of X . Further, (S −1 F )−1 Y ∼ = S −1 (F −1 Y ) are equivalent. 2. If F is fibered, then S −1 F : S −1 X −→ Y is fibered. Proof. The proof of (1) is routine. So, we prove (2). Let Y ∈ Y , ΦY : F −1 Y −→ Y /F be the natural functor and RY : Y /F −→ F −1 Y be its right adjoint. So, for X1 ∈ Obj(X ) and (X2 , v2 ), we have a natural isomorphism ∼

ηY : M orF −1 Y (X1 , R(X2 , v2 )) −→ M orY /F ((X1 , 1X1 ), (X2 , v2 )) . −1

Write F = S −1 F . Then F Y = {(P, X) : F (X) = Y } = S × F −1 Y . Note f ∈ M orS −1 X ((P1 , X1 ), (P2 , X2 )) is given by f = (U, f1 , f2 ) such that f1 ∈ M orS (U P1 , P2 ), f2 ∈ M orX (U X1 , X2 ). So, M orS −1 X ((P1 , X1 ), (P2 , X2 ))  M orS (U  P1 , P2 ) × M orX (U  X1 , X2 ) = U ∈S

160

Algebraic K-Theory

the union runs through choices of representatives U from each isomorphism class. So, ⎧ M or −1 ((P1 , X1 ), (P2 , X2 )) = ⎪ ⎪ ⎪ F Y ⎪ ⎪ ⎪ U ∈S {(f1 , f2 ) ∈ M orS (U  P1 , P2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ×M orX (U  X1 , X2 ) : F (f2 ) = 1Y } ⎪ ⎨ So, the 2nd-coordinate f2 ∈ M orF −1 Y (U  X1 , X2 ) ⎪ ⎪ ⎪ ⎪ M orY /F (((P1 , X1 ), v1 ), ((P2 , X2 ), v2 )) = ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ U ∈S {(f1 , f2 ) ∈ M orS (U  P1 , P2 ) ⎪ ⎪ ⎩ ×M orX (U  X1 , X2 ) : F (f2 )v1 = v2 } . Define, the functor −1

RY : Y /F −→ F Y = S × F −1 Y by  RY ((P, X), v) = (P, RY (X, v)) RY (U, f1 , f2 ) = (U, f1 , RY (f2 )) Define, η Y : M orF −1 Y ((P1 , X1 ), (P2 , RY (X2 , v2 ))) −→ M orY /F (((P1 , X1 ), 1X1 ), ((P2 , X2 ), v2 )) by η Y (U, f1 , f2 ) = (U, f1 , η(f2 )). Since ηY is a natural bijection, so is η Y . This establishes that F is prefibred. For v : Y −→ Z −1 −1 in Y , the functor v ∗ : F Z −→ F Y is given by v ∗ (P, X) = (P, RY (X, v)). Since F is fibered, it follows that F is fibered. The proof is complete. Lemma 4.5.8. Let E be an exact category and S = Iso(E ). Then natural functors 

t : EE −→ QE and q : S −1 EE −→ QE

are fibered.

The Agreement with Classical K-Theory

161

Proof. By Lemma 4.5.7, it is enough to prove that t : EE −→ QE is fibered. For C ∈ Obj(QE ), we have ⎧ ⎪ Obj(C/t) = {(E, v) : E ∈ Obj(EE ), v ∈ M orQE (C, t(E))} ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ M orC/t ((E1 , v1 ), (E2 , v2 )) = {f ∈ M orQE (E1 , E2 ) : t(f )v1 = v2 } ∀ (E1 , v1 ), (E2 , v2 ) ∈ Obj(C/t) ⎪ ⎪ ⎪ In particular, ⎪ ⎪ ⎪ ⎩ M or ((E , 1 ), (E , v )) = {f ∈ M or (E , E ) : t(f ) = v } 1 C 2 2 1 2 2 QE C/t Such an arrow f ∈ M orC/t ((E1 , v1 ), (E2 , v2 )) is presented by the following diagrams: 

∀ i = 1, 2 vi :=

 ι1 / MO 1

KO 1 

E1 =

α

?

 



1K2

E2 =

ζ1

  

K2

ι2

fs

/ / D _

/ M 1  _ β  / M2

,

/ / C1 OO

1M1



K2 

f



 qi / Ci Vi 

pi Coo



,

 t(f )v1 = v2

(4.17)

fm

/ / C2

ζ2

The equality t(f )v1 = v2 is given by the commutative diagram {

V2  I

I

Iτ ∼ I

q2

I$

V1 ×C1 D 

p2

k



Coo

p1



V1



 j

/ D   fm )/ C2 

q1

fs

/ C1

where τ is a unique isomorphism. We define a right adjoint R : C/t −→ t−1 C of Φ. For (E, v) ∈ Obj(C/t), consider the pullback

162

Algebraic K-Theory

diagram ˜ K O

R(E, v) :=

α

?

ι /





 

K

E=

pγ

//C OO

1



K 1K

M ×O B V

p

_γ _ _/ V  / M ×B _ _ V β q    /M //B ι

(4.18)

ζ

In this diagram, (E, v) ∈ Obj(C/t), where E is represented in the third horizontal line and v by the last vertical line, and M ×B V is ˜ := ker(pγ). Then the top line is defined to the pullback. Further, K −1 be R(E, v) ∈ t C. Let f ∈ M orC/t ((E1 , v1 ), (E2 , v2 )), as displayed in what follows, fixing the notation: R(E1 , v1 ) :=

 K˜O 1 

ι /



α1

 

KO 1  ϕ

?

 



K2 

f



E2 =

1K2

 

K2 

//C OO

1

/C O \\

p1

p2 / M1 ×C V1 _ _ _/ V1 p  _ 1 _ `A` A β q1 Ak  1 A   ζ1 ι1 τ / M1 / / C1 V 2 o ∼ Vq 2 O OO Q } Nn 1M1 fs } } ~} j / M 1 / / D _  _ q2 ψ fm    z / M2 / / C2 γ1

K1 E1 =

O

1

?  

1K1

p1 γ1

M1 ×C1 V1

ι2

with V 2 := D ×C1 V1

ζ2

(4.19)

In this diagram, the right-hand side displays the equation t(f )v1 = v2 . We develop the arrow R(f ) by a number of diagrams. We extract

The Agreement with Classical K-Theory

163

the following diagram from (4.19) as follows:  ι1 / M1 O

KO 1  

ϕ

?



K2  1K2



K2 

ι

 ι2

ζ1

/ / C 1 o q 1 ? _ V1 p 1 / / C O OO OO  1M1 fs 1 k / M 1 / / D o_ _ _? _ V 2 //C _ q p O  _ ζ ψ fm  τ 1    / M2 / / C2 o //C ? _ V2 q2

ζ2

(4.20)

p2

Fold the diagram along the middle line, with the left half horizontally, and the right half vertically. We raise three pullback diagrams (three vertical walls) as follows: ˜2  K O

1

α2

K2 1



/ M2 ×C V2 γ2 _2 β  2 / M2

/ / V2 _

K2

ι2

˜1  K O



1

q2

  

K1

1



 

K2 

1

K1

pγ

//C OO p

/ M1 × D V _ _ _/ / V 2 γ _  _  β q   / M1 //D ι

ι1

ζ

/ M1 ×C V1 p1 γ1 / / C OO O 1

α1



/ M1 ×D V O

K2

/ / C2

ζ2



α

p2

ι2

  

1

KO 

/ M2 ×C V2 p2 γ2 / / C OO O 2

p1

/ M1 ×C V1 γ1 _1 β  1 / M1

ι1

ι1

ζ1

/ / V1 _ 

(4.21)

q1

/ / C1

A cubical diagram is formed from (4.21), and (4.20) (think of them as three vertical walls, one behind another). For typographical reasons, display this cube in two cubical diagrams (4.22, 4.23) (sharing the middle wall). Note M1 ×D V = M1 ×C1 V1 . And, from the first diagram

164

Algebraic K-Theory

˜ 2 = K. (4.22), it would also be clear that K  O @@ @@ @@ @@ 1

˜2  K

α

/ M 1 ×C1 V O  Nt 1 N 

˜2  K

O

K2

 α2 AA AA AA 1 AA

ι1

K2

1

1



  1 ι CC CC CC CC 1 !   K

K2

2

pγ

NψN

NN & / M2 ×C2 V2 O

/ / CO @ O @@ @@1 @@ @@  p 2 γ2 / / CO p O

p2 / M1 ×C 1 V t 1 _ _ _ _ _ _1 _ _ _ _/ / V 2 γ _ `@@  _ NN @  @@τ NψN  ∼ @@ NN  @ &  β q / M2 ×C 2 V2 / / V2 γ2 _  ι2  _       / M1 O / / D Bp q2  β2 OOO ζ B B OOOψ  BBfm OOO BB OOO  B!  '/ / / C2 M2 ι2

ζ2

(4.22) ˜ 2 . This follows, In this diagram, we used the identification K = K  ˜ 2 = ker(p2 γ2 ) ∼ because ψ is an isomorphism, and p2 γ2 = τ pγ. So, K = ker(pγ) = K.  O `@@ @@ @@ @@ ϕ /O

˜1  K

α1

K1

1

/ ˜2  K

O

2

  1 ι1 aCC CC CC ϕ CC 0 P K   2

O



 α `AA ι1 AA AA ϕ AA 0 PK  

K1

M1 ×C1 V1

1

/

p 1 γ1

fNNN NNN1 NNN NN / M1 ×C1 V1 O

/ / CO _@ O @@ @@1 @@ @@ pγ p1 / / CO O

/ / V1 p 1 γ1 _ `@` @  _ fNNN @ NNN1 @@k  NNN @@  @ NN  β1 / M1 ×C 1 V1 _ _ _ _ q1_ _ _/ / V 2 γ _  ι1  _      q / M1 gO / / C1  β aBa B ζ1 OOO BB fm OOO1  BB OOO BB OOO   / M1 //D

M1 ×C 1 V1

ι

ζ

(4.23)

The Agreement with Classical K-Theory

165

The ceiling of the cube (4.22, 4.23) is defined to be the morphism R(f ), as follows: R(E1 , v1 ) =

˜1  K O ϕ

˜2  K 

R(E2 , v2 ) =

1

//C O

1

?

R(f )

p1 γ1

/ M1 ×C V1 O 1



˜2  K

1 pγ

/ M1 ×C V1 1

//C

ψ





/ M2 ×C V2 2

1

//C

p2 γ2

(4.24) This completes the definition of the functor R : C/t −→ t−1 C. We check that R is a right adjoint to the natural functor Φ : t−1 C −→ C/t.      Let E1 = K1 ι1 / M1 ζ1 / / C ∈ Obj t−1 C ,    (E2 , v2 ) ∈ Obj (C/t) with E2 = K2 ι2 / M2 ζ2 / C2    and v2 = C o o p2 V2 q2 / C2 . We have ⎧ M ort−1 C (E1 , R(E2 , v2 )) = ⎪ ⎪ ⎪ ⎨ {f ∈ M orEE (E1 , R(E2 , v2 )) : t(f ) = 1C } ⎪ M orC/t (Φ(E1 ), (E2 , v2 )) = M orC/t ((E1 , 1C ), (E2 , v2 )) ⎪ ⎪ ⎩ = {f ∈ M orEE (E1 , E2 ) : t(f ) = v2 } From construction, there is a natural arrow η : R(E2 , v2 ) −→ E2 in EE . By composition, there is a map η : M ort−1 C (E1 , R(E2 , v2 )) E1 −→ M orC/t ((E1 , 1C ), (E2 , v2 ))

f

/ R(E2 , v2 ) η

η(f )

) 

E2

166

Algebraic K-Theory

More explicitly, η(f ) is the composition, as follows: 

KO 1 

E1 =

ι1

/ M1 O

ϕ 

˜2  K

R(E2 , v2 ) =

ϕ



˜2 K O K2 1

E2 =

//C

/ M2 ×C V2 O 2

p2 γ2

1



//C OO

1

?



/ M 1 _ 



α2 η

1

 ψ

1



//C O

1

?

f

ζ1



K2



p2

/ M2 ×C V2 γ2 _2 β  2 / M2

ι2



ι2

ζ2

/ / V2 _ 

q2

/ / C2

First, by diagram chasing, it follows that η is injective. Further, chasing the pullback properties, it follows that η is surjective. So, it is established that R is a right adjoint of Φ. So, t is prefibered. For v ∈ M orQE (C, D), we consider the diagram t−1 D v∗

ΦD

/ D/t



R





t−1 C

r

ΦC

v∗

where the same notation v ∗

/ C/t

is used for two functors.   and w = B o o s2 W   m2 / C    //D K ι /M ∈ be two morphisms in QE . Let E := ζ  −1  Obj t D . To establish w∗ v ∗ (E) = (wv)∗ (E), consider the Let v =



Coo

s1

V

 m1 / D



The Agreement with Classical K-Theory

167

commutative diagram ˜ ˜ K O

w∗ v ∗ (E) =

1

?

˜ K 1



˜ K O

v ∗ (E) =

/ (M ×D V ) × W _ _ _/ / W h h _ NK _ G  m2 =   sγ //C / M ×D V V ×q C W OO O Q v

γ

 

K

s1

o / M ×D _ _ _ _ _/ / V  lu _ _ V  m1    /M //D



K E=

s2 γ

1

?

1



/ (M ×D V ) ×C W s_2 γ _ _/ / B OO O

ι

ζ

Since M ×D (V ×C W ) ∼ = (M ×D V )×C W , it follows that w∗ v ∗ (E) = ∗ (vw) (E). Therefore, t is fibered. Corollary 4.5.9. Let E be an exact category. Consider the functor t : EE −→ QE . Let  v ∈ M orQE (C, B) be represented by v =    C o o p M q / B . Then the base change functor v ∗ : t−1 C −→ t−1 B is given by the diagram v ∗ (E) = R(E, v), where the right adjoint R was defined in diagram (4.18). As is obvious in (4.18), there is a natural arrow ηE ∈ M orEE (v ∗ (E), E). Theorem 4.5.10. Let E be a split exact category, and S = Iso(E ). Then the sequence S −1 S

νE

/ S −1 EE

q

/ QE

is a homotopy fibration ∀ E ∈ Obj(EE )

where νE (P, Q) = (Q, P  E). Proof. By (4.5.8), q is fibered. Let C ∈ Obj(QE ). By Lemma 4.5.7, q−1 C = S −1 EEC . Let  v ∈ M orQE (C, B) be represented by  q p  / B . We will prove that v ∗ : q−1 B −→ q−1 C v := C o o M

168

Algebraic K-Theory

is a homotopy equivalence. Consider the following diagram of compositions of arrows in QE : 7 BO

v

C

7 MO

0!M

p!

q! p!

/M

0

BO

0!C

0B!

q!

/C

M

0M !

 /0

Therefore, to prove v ∗ is a homotopy equivalence, it is enough to prove the same for v = 0!C and v = 0C! , where 0!C =      0oo C  1 / C , and 0C! = 0 o o /C . 0 1. Write v = 0!C . Now, q−1 0 = S −1 t−1 0 = S −1 S and q−1 0 = S −1 t−1 0 = S −1 EEC . The equivalence    /Q /0 sends (P, Q) → P, Q . S −1 S −→ q−1 0 By definition, R(E, v) :=

˜ = M  ι K O α



/M O



/ M _ γ_ _/ C _

K E=

  

K

//0 OO

1

?  

1K

pγ



1

/M

ι



ζ

1

//C

So, v ∗ (P, E) = (P, M )   /C / C ∈ Obj(EE ), consider the Now, with 1 = 0 C 1 commutative diagram of functors S −1 S

(P,Q)→(Q,P 1) −1

(P,Q)→(Q,P ⊕C)

/S +

EEC 

v∗

S −1 S

The horizontal functor is a homotopy equivalence by (4.5.6). Up to a switch, the diagonal functor is a translation, and hence a homotopy equivalence (4.3.8). So, v ∗ is a homotopy equivalence.

The Agreement with Classical K-Theory



2. Now, let v = 0C! =

0o

˜ = K K O

R(E, v) :=

ι

/K O



E=

//0 OO

1

?  K

1K



/ C . By definition,

0 

 

K

ι

169

p

/ K _ _ _/ 0 _  _ ι q   /M //C

So,

v ∗ (P, E) = (P, K)

ζ

Again, consider the commutative diagram of functors S −1 S

(P,Q)→(Q,P 1) −1

/S

(P,Q)→(Q,P )

+

EEC 

v∗

S −1 S

Since the horizontal and the diagonal functors are homotopy equivalences, so is v ∗ . Therefore, the proof is complete by an application of the fibered version of Theorem B (2.2.6). The following exercise would be useful for proof of the next theorem. Exercise 4.5.11. Let X be a category. Define a new category Ar(X ), whose objects are the arrows in X . To define arrows in Ar(X ), consider the following commutative diagram of arrows in X , MO 1 u

f

 

M2

g

/ N1  v 

which is

g = uf v

/ N2

Then M orAr(X ) (f, g) is defined to be the set of all such pairs (u, v) as in the diagram. Then X and Ar(X ) are homotopically equivalent, by the functors defined in the proof.

170

Algebraic K-Theory

Proof. Define F : Ar(X ) −→ X , by F (f : M −→ N ) = N . And, define G : X −→ Ar(X ), by G(N ) = 1N . Clearly, F G = 1X . Also, GF (f : M −→ N ) = 1N . A natural transformation 1Ar(X ) −→ GF is given by the diagram 1

NO f

/N 1



M

f

/N

So, GF is homotopic to 1Ar(X ) −→ GF , by (2.1.9). The proof is complete. Theorem 4.5.12. Let E be a split exact category, and S = Iso(E ). Then S −1 EE is contractible. Proof. Let X be the subcategory of QE whose objects are the := same as that of QE , and M orX (P, Q) {ι! ∈ M orBQ (P, Q) : ι : N → M is an inflation in E }. Then 0 ∈ X is an initial object, and hence X is contractible. Let Ar(X ) be the category arrows of X , as in (4.5.11). There is a natural functor F : Ar(X ) −→ EE

sending ι! → ( K 



ι

/M

/ / co ker(ι) )

∀ inflations (ι : K → M ) Suppose (ιi! : Ki −→ Mi ) ∈ Obj (Ar(X )) for i = 1, 2. An arrow f := (α! , β! ) : ι1! −→ ι2! is given by the commutative diagram of inflations:  ι1 / M 1 _

KO 1  α

?

K2



 ι2

β

/ M2

This corresponds to an arrow F (ι1! ) −→ F (ι2 ) in EE (see Definition 4.5.1). Define F (f ) : F (ι1! ) −→ F (ι2 ) to be this arrow. So, the functor F : Ar(X ) −→ EE is defined. It is clear from (4.5.1) that F is an equivalence of categories. The inverse of F is given by the conflation E sent to its own inflation. Since X is contractible, so is EE . Since all

The Agreement with Classical K-Theory

171 ∼

maps in S are isomorphisms, by Proposition 4.3.7, EE −→ S −1 EE is a homotopy equivalence. Therefore, S −1 EE is contractible. The proof is complete. The following theorem is one of our main goals in this chapter. Theorem 4.5.13. Let E be a split exact category and S = Iso(E ). ∼ where Then there is a homotopy equivalence B(S −1 S) −→ ΩB(QE   ), −1 ∼ ΩB(QE ) is the loop space. In particular, Kn (E ) = πn BS S,  , for all n ≥ 0. Proof. Refer to Section B.5. By Theorem 4.5.10, we have homotopy fibration BS −1 S

/ B(S −1 EE )

|q|

/ BQE

where |q| := Bq.

∼

In other words, the map ϕ : BS −1 S −→ F(|q|, 0) is a homotopy equivalence, where the latter term denotes the homotopy fiber. By ∼ (4.5.12), B(S −1 EE ) is contractible. So, ΩB(QE ) −→ F (|q|, 0) is a homotopy equivalence. The proof is complete. 4.6.

Cofinality

First we define Cofinality. Definition 4.6.1. Suppose f : (S, S ) −→ (T, T ) is a functor of symmetric monoidal categories. We say that f is cofinal if ∀ P ∈ Obj(T ), there is an object F ∈ Obj(S) such that F ∼ = P T Q for some Q ∈ Obj(T ). Theorem 4.6.2. Suppose f : (S, S ) −→ (T, T ) is a functor of symmetric monoidal categories, and f is cofinal. Assume S = Iso(S) and T = Iso(T ), and every translation in S, T are faithful. 1. Suppose X is a T -category. Then S −1 X T −1 X is a homotopy equivalence. In particular, S −1 T ∼ = T −1 T . 2. Assume AutS (F ) ∼ = AutT (F ) for all F ∈ Obj(S). Then the component of (0, 0) in BS −1 S is homotopically equivalent to the component of (0, 0) in BT −1 T ∼ = BS −1 T . Proof. It is easy to see that Sl−1 T −1 X ∼ = Tl−1 S −1 X . Now S acts −1 −1 invertibly on T X (from left). So, T X −→ Sl−1 T −1 X is a

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homotopy equivalence (4.3.7, 4.3.10). In fact, T also acts invertibly on S −1 X . To see this, let F ∼ = P  Q for F ∈ Obj(S) and P, Q ∈ Obj(T ). Now consider the diagram TP

S −1 X

/ S −1 X +

TF



TQ

So, TQ TP is a homotopy equivalence.

S −1 X

So, TP has left homotopy inverse, and TQ has right homotopy inverse. Hence, TP is a homotopy equivalence. This establishes that T acts invertibly on S −1 X . Therefore, S −1 X −→ Tl−1 S −1 X is a homotopy equivalence (4.3.7, 4.3.10). In the following commutative diagram, S −1 X 

T −1 X

∼

/ S −1 T −1 X l 

∼

 / T −1 S −1 X l

three functors are homotopy equivalences.

Hence, the fourth one S −1 X T −1 X is also a homotopy equivalence. This completes the proof of (1). Now, we prove (2). 1. For (F1 , P1 ), (F2 , P2 ) ∈ Obj(S −1 T ), define (F1 , P1 ) ∼ (F2 , P2 ) to be equivalent if there are two arrows, as follows / (F, P ) o (F2 , P2 ) . So, there are U, V ∈ Obj(S), (F1 , P1 ) and the diagram of isomorphisms: ∼

U  (F1 , P1 )  U  F1 ⇐⇒ U  P1

/ (F, P ) o

∼ =F ∼ =P

∼

∼ = V  F2 ∼ = V  P2

V  (F2 , P2 ) (4.25)

It follows that this is an equivalence relation on S −1 T . For (F, P ) ∈ S −1 T , its equivalence class is denoted by [(F, P )]T . This equivalence relation restricts to S −1 S. For (F, G) ∈ S −1 S, its equivalence class in S −1 S is denoted by [(F, P )]S . Let IT be an indexing set, and {[(Fi , Pi )]T : i ∈ IT } be the set of all equivalence classes of the subcategories of S −1 T . Likewise, let IS be an indexing set, and {[(Ki , Gi )]S : i ∈ IS } be the set of all equivalence classes of the subcategories of S −1 S.

The Agreement with Classical K-Theory

173

 2. It follows that S −1 T = i∈IT [(Fi , Pi )]T is a disjoint union of sub categories. So, BS −1 T = i∈IT B[(Fi , Pi )]T . Since B[(Fi , Pi )]T are −1 (path) connected, these  are the connected components of BS T . −1 Likewise, BS S = i∈IS B[(Ki , Gi )]S , and B[(Ki , Gi )]S are its connected components of BS −1 S. 3. The inclusion [(0, 0)]S −→ [(0, 0)]T , induces the map ι : B[(0, 0)]S −→ B[(0, 0)]T . It remains to prove that this map is a homotopy equivalence. We write 0 = (0, 0) ∈ [(0, 0)]S , [(0, 0)]T . We use Whitehead’s theorem (C.3.8) to prove that ι is a homotopy equivalence. By (4.25), π0 (B[(0, 0)]S = π0 (B[(0, 0)]T = 0 are trivial. So, we prove that the maps πn ([(0, 0)]S , 0) −→ πn ([(0, 0)]T , 0) are isomorphisms ∀n ≥ 1. To prepare for the proof, we make a few observations. By (4.25), for (F, P ) ∈ [(0, 0)]T there are U, V ∈ Obj(S) such that U  F ∼ =V and U  P ∼ = V . Observe that, given finitely many objects (Fi , Pi ) ∈ [(0, 0)]T , for i = 0, 1, . . . , n, there are Ui , Vi ∈ Obj(S) such that Ui  Fi ∼ = Vi ∼ = Ui  Pi . Replacing Ui by U := U0  · · ·  Un , we ∼ have U  Fi = Vi ∼ = U  Pi , with U, Vi ∈ Obj(S). Now for U ∈ Obj(S), consider the functor τUT : [(0, 0)]T −→ [(0, 0)]T sending (F, P ) → (U  F, U  P ). The arrows (U, 1U F , 1U P ) : (F, P ) −→ (U  F, U  P ) in S −1 T define a natural transformation 1[(0,0)]T → τUT . So, τUT is a homotopy equivalence. Likewise, its restriction τUS : [(0, 0)]S −→ [(0, 0)]S is also a homotopy equivalence. The following diagram of functors: / [(0, 0)]T

[(0, 0]S S τU





commutes.

T τU

/ [(0, 0)]T

[(0, 0]S

It induces the commutative diagram πn ([(0, 0]S , 0)

πn (ι)

S πn (|τU |) 

T  πn (|τU |)



πn [(0, 0]S , (U, U ))

/ πn [(0, 0)]T , 0) 

πn (ι)

∀n ≥ 1.

/ πn [(0, 0)]T , (U, U ))

(4.26) where the vertical maps are isomorphisms.

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Algebraic K-Theory

Now fix, n ≥ 1. We prove πn (ι) : πn (B[(0, 0)]S ) −→ πn (B[(0, 0)]T ) is an isomorphism. Let γ : (Sn , ) −→ (B[(0, 0)]T , v0 ) be a map. Since γ(Sn ) is compact, by (C.1.8) and (1.2.11), there is a finite simplicial subcomplex K• ⊆ N• ([0, 0]]T ) such that K• is generated by finitely many nondegenerate simplexes and γ(Sn ) ⊆ |K• |. Consider such a nondegenerate set of generators {σ j } of K• . Such a generator can be written as . By the above observation, there are objects U and Vij in S, such that j j U ⊕ Fij ∼ = Vi ∼ = U ⊕ Pi for all i, j. Write

where Fij := U  fij . Now τUT (K• ) is generated by {U  σ j }. Let L• ⊆ N• [(0, 0)]S be the simplicial set generated by {U  σ j }. Then L• −→ K• is an isomorphism of simplicial sets and  the restriction ∼ ι||L• | : |L• | −→ |K• | is a homeomorphism. So, τUT  γ lifts to  Ta map n β : (S , ) −→ (|L| , (U, U )). So, it follows that πn ([β]) = [τU  γ] ∈ πn [(0, 0)]T , (U, U )). Now it follows from diagram (4.26) that [γ] has a preimage in πn [(0, 0]S , (U, U )). So, the map πn [(0, 0]S , (U, U )) −→ πn [(0, 0)]T , (U, U )) is surjective. To prove injectivity, let γ : (Sn , ) −→ [(0, 0)]S be such that πn (ι)([γ]) = 0. So, there is a homotopy H : (Sn , ) × I −→ [(0, 0)]T such that H(−, 0) = c(0,0) the constant map and H(−, 1) = ιγ. Again, H(Sn × I) is compact. So, using the same argument as above, we can find a simplicial subset L• ⊆ N• [(0, 0)]S , K• ⊆ N• [(0, 0)]T ∼ such that L• −→ K• is an isomorphism. And hence, H lifts to a ˜ : (Sn , ) × I −→ ([(0, 0)]S , (U, U )), from the conhomotopy in H   stant path to a lift of τUT  γ. Since the first vertical map in (4.26) is an isomorphism, it follows [γ] = 0. This establishes that the maps πn ([(0, 0)S , 0) −→ πn ([(0, 0)T , 0) are isomorphisms, ∀n ≥ 0. Therefore, by Whitehead’s theorem (C.3.8) the map ι is a homotopy equivalence. This completes the proof of (2). Remark 4.6.3. The proof of part (2) of Theorem 4.6.2 is more transparent for the case of our main interest. That is, when A is a ring.

The Agreement with Classical K-Theory

175

Fix such a ring A. S = {An : n ≥ 0}, where An is treated as a based free (right) module, as defined in (4.1.5 (2)) and was denoted by F (A). And T = Iso(P(A)), where P(A) is the category of finitely generated (right) projective A-modules. The main advantage is that it is easier to work with modules. While in the commutative case the rank(P ) provides a handle, it does not make much sense in the non-commutative situation [Wc2, p. 2]. However, from the structure of S = F (A), the ranks of the free modules An make sense, because M orS (An , Am ) = φ, unless n = m. 1. For P, Q ∈ Obj(T ), define P and Q to be equivalent, and write P ∼ Q, if F1 ⊕ P ∼ = F2 ⊕ Q, for some F1 , F2 ∈ Obj(S). This is an equivalence relation, and let [P ] denote the equivalence class of P . Indeed, P ∼ Q if and only if there is a sequence of arrows in S, T  in either direction, starting from P , ending at Q. With 0 := A0 , [0] is the class of all stably free modules. We fix an indexing set I so that {[Pi ] : i ∈ I} is the set of all equivalence classes of projective modules in P(A), and Pi ∈ [Pi ]. 2. As in (4.25), for (F1 , P1 ), (F2 , P2 ) ∈ Obj(S −1 T ) (F1 , P1 ) ∼ (F2 , P2 ) defined to be equivalent, if there are two arrows, as follows / (F, P ) o (F2 , P2 ) . So, there are U, V ∈ Obj(S), (F1 , P1 ) and the diagram of isomorphisms: ∼

U ⊕ (F1 , P1 )  U ⊕ F1 ⇐⇒ U ⊕ P1

/ (F, P ) o

∼ =F ∼ =P

∼

V ⊕ (F2 , P2 )

∼ = V ⊕ F2 ∼ = V ⊕ P2

As above, this is an equivalence relation. The second line means P1 ∼ P2 in the sense of (1). Since the first isomorphism is in S −1 S, U, V, F1 , F2 ∈ S = {An : n ≥ 0} and they have well-defined rank. In particular, rank(U ) + rank(F1 ) = rank(V ) + rank(F2 ). 3. Now consider the category S = {An : n ≥ 0}, as defined in (4.1.5 (2)), and the category S −1 S. Write Fn = An . In S −1 S there is an arrow (Fm1 , Fn1 ) −→ (Fm2 , Fn2 ) if and only if m2 − m1 = n2 − n1 ≥ 0. So, ⎛ ⎞ ⎛ ⎞    ⎝ [(0, Fn )]S ⎠ (4.27) S −1 S = ⎝ [(Fn , 0)]S ⎠ n≥0

n≥1

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Algebraic K-Theory

is a disjoint union, and they correspond to the connected components of BS −1 S. Therefore, [(0, 0)]S = {(Fn , Fn ) : n ≥ 0}. We denote B[(0, 0)]S = B{(Fn , Fn ) : n ≥ 0}. 4. For clarity, assume that rank(P ) is well defined (as in the case when A is commutative and connected) and is well behaved. With r1 = rank(P1 ), r2 = rank(P2 ), the first line means 

F1 = F2 ⊕ Ar1 −r2 if r1 ≥ r2 equivalently, if rank(F1 ) ≥ rank(F2 ) F2 = F1 ⊕ Ar2 −r1 if r2 ≥ r1 equivalently, if rank(F2 ) ≥ rank(F1 )

Let {[Pi ] : i ∈ I} be the set of all equivalence classes (1) of projective modules. Choose Pi ∈ [Pi ] such that ri := rank(Pi ) = min{rank(P ) : P ∈ [Pi ]}. Such a choice is unique up to isomorphism. With notation as above,  [(F, Pi )]T = {(F ⊕ Arank(P )−ri , P ) : P ∈ [Pi ]} In particular,

[(0, 0)] T = {(Arank(P ) , P ) : P ∈ [0]}

So, the correspondence (n, i) → [(An , Pi )]T is a bijective correspondence from Z × I to the set of equivalence classes in S −1 T . Theorem 4.6.4. Let A be a ring, and P(A) be the category of finitely generated (right) projective A-modules. Let S = {An : n ≥ 0} = F (A) be as defined in (4.1.5(2)), and T = Iso(P(A)). We use some of the notation in the proof of (4.6.2). Then 1. All the connected components of T −1 T are homotopically equivalent. 2. There are homotopy equivalences: BT −1 T ∼ = K0 (A)×B [(0, 0)]T ∼ = K0 (A) × B [(0, 0)]S . In particular, Kn (A) ∼ = πn (B [(0, 0)] T , 0)) ∼ = πn (B [(0, 0)] S , 0)) for all n ≥ 1. Proof. By (4.6.2), S −1 T ∼ = T −1 T are homotopically equivalent. So, −1 we work with S T . We use some of the notation in (4.6.3) and in the proof of (4.6.2). Let Pi ∈ [0]T be as in (4.6.3(1)) and F ∈ Obj(S). The map τ(F,Pi ) : S −1 T −→ S −1 T sending (P, Q) → (F ⊕ P, Pi ⊕ Q) is the composition of a left translation and a right translation. Since both are homotopy equivalences (4.3.8), so is τ(F,Pi ) . Now, Bτ(F,Pi ) maps the connected component B [(0, 0)]T into B [(F, Pi )]T ,

The Agreement with Classical K-Theory

177

and hence they are homotopically equivalent. So, (1) is established. Now, we prove (2). The second equivalence follows from (4.6.2). To prove the first we only need to proveK0 (A) is in   equivalence, α : K0 (A) −→ π0 BT −1 T by bijection to π0 BT −1 T . Define  −1 α([P ] − [Q]) = [(P,Q)] ∈ π0 BT T , which is well defined. The map β : π0 BT −1 T −→ K0 (A) given by β([(P, Q)]) = [P ] − [Q] is also well defined. So, α is an isomorphism. The descriptions of Kn (A) follow from Theorem 4.5.13. The proof is complete. Remark 4.6.5. We avoided the plus construction of Quillen. Given a pointed (path) connected space (X, 0) and a perfect normal subgroup E ⊆ π1 (X, x), another pointed space (X + , 0+ ) is constructed, satisfying certain universal properties. Given a commutative ring A, this construction is applied to the space (BGL(A), ), and (BGL(A)+ , + ) is obtained. Plus construction predates the S −1 S construction, and higher K-groups of A were defined as the homotopy groups πn (BGL(A)+ , + ). Refer to, for example, [Ha] , for plus construction. It turns out (BGL(A)+ , + ) is homotopically equivalent to B [(0, 0)]S , as above (see [Wc2, Thm. 4.9]).

4.7.

Agreement of Modern and Classical K-Theory

The goal of this important section is to prove that the Classical and Modern (Quillen) K-Theory coincide. Let A be a ring. Other than K0 (A), classically two other groups were defined, which we denote (temporarily) by K1c (A) and K2c (A). Among these two, more commonly discussed (and taught) was K1c (A), which is also known as the Whitehead group. In this section, we prove that Kic (A) coincides with   Ki (A), for i = 1, 2. In fact, we will prove Kic (A) ∼ = πi BS −1 S, 0 , for i = 1, 2, where S is as above (4.6.4). First, we recall the classical definition. 4.7.1.

The Whitehead Group

Let A be a ring. As usual, GLn (A) denotes the group of all invertible matrices of ordern (to be viewed as automorphisms of An ). α0 The association α → defines an inclusion map GLn (A) → 01

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Algebraic K-Theory

GLn+1 (A). The (infinite) General Linear group is defined to be GL(A) = n≥0 GLn (A). Given λ ∈ A, and 1 ≤ i, j ≤ n, with i = j, let eij (λ) be the square matrix (of order n) whose only nonzero entry is λ, at (i, j) position. Let ELn (A) ⊆ GLn (A) be defined as the subgroup generated by the  set {1 + eij (λ) : λ ∈ A, 1 ≤ i, j ≤ n, i = j}. Define EL(A) := n≥0 ELn (A) ⊆ GL(A) as the (infinite) elementary group. Before we define Whitehead groups, we insert the following basic lemma. Lemma 4.7.1. Let A be a ring. Let α, β ∈ GLn (A). Then   β 0 and ∈ EL2n (A) 0 β −1       βα 0 α 0 αβ 0 ≡ mod EL2n (A). ≡ 0 1n 0 β 0 1n Proof. Write   1n 1n β − 1n ε= 0 1n 1n   β 0 . = 0 β −1

0 1n



1n 0

β −1 − 1n 1n



1n −β

0 1n



All the matrices on the left-hand side are in EL2n (A), and hence so is the one on the right-hand side. This settles the first assertion. It follows that         α 0 αβ 0 α 0 βα 0 , = .  = 0 β 0 1n 0 β 0 1n The proof is complete. Recall, for a group G, and elements f, g ∈ G , we write [f, g] := f gf −1 g−1 , to be called the commutator. The commutator subgroup [G, G] is defined as the subgroup generated by all the commutators [f, g]. The following would be used to define the Whitehead group (4.7.3).

The Agreement with Classical K-Theory

179

Lemma 4.7.2. Let A be a ring. Then [GL(A), GL(A)] = [EL(A), EL(A)] = EL(A).

Proof. For λ ∈ A and 1 ≤ i = j ≤ n. We use the notation ij (λ) = 1 + eij (λ) ∈ ELn (A). For distinct 1 ≤ i, j, k ≤ n, it is easy to check [ij (λ), jk (μ)] = ik (λμ), which can be checked with i = 1, j = 2, k = 3. It follows from this that ELn (A) = [ELn (A), ELn (A)] for all n ≥ 3, and hence [EL(A), EL(A)] = E(A). Given α, β ∈ GLn (A), by (4.7.2), 

αβα−1 β −1 0

0 1n



 =

αβ 0

0 −1 β α−1



α−1 0

0 α



β −1 0

0 β



∈ EL2n (A). Therefore, [GL(A), GL(A)] = E(A). The proof is complete. Definition 4.7.3. With notation as above, the Whitehead group is defined as follows: K1c (A) =

GL(A) . EL(A)

We use superscript c (temporarily), to distinguish K1c (A) from the Quillen group K1 (A) := K1 (P(A)). However, most readers know that classically K1 -group of A was defined to be the Whitehead group. And, the notation K1 (A) rightfully belongs to the Whitehead group. 4.7.2.

The Agreement of K1c (A) and K1 (A)

In this section, we prove that Quillen K1 (A) agrees with the classical group K1c (A). Definition 4.7.4. Let A be a ring. Let S = {An : n ≥ 0} = F (A) ⊆ Iso(P(A)) be the symmetric monoidal category, as defined in (4.1.5(2)). Denote vn := [(An , An ))] ∈ BS −1 S and

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Algebraic K-Theory

1n := 1An . As before, B [(0, 0)] S denotes the connected component of (0, 0) in BS −1 S. Given α ∈ GLn (A) ⊆ GL(A), there is a loop n (α) in B [(0, 0)]S , at the vertex vn , corresponding to the   ∼ arrow A0 ; α, 1n : (An , An ) −→ (An , An ) in S −1 S. The arrow (An ; 1n , 1n ) : (0, 0) −→ (An , An ) in S −1 S defines a path from 0 −→ vn in BS −1 S. So, we have a commutative diagram of isomorphisms: π1 (B [(0, 0)] S , (0, 0))

ιn ∼

/ π1 (B [(0, 0))] , vn ) S

π1 (B [(0, 0)] S , (0, 0))

∼ ιn+1

 / π1 (B [(0, 0)] , vn+1 ) . S

The second vertical arrow is obtained by using the path vn −→ vn+1 corresponding to the arrow (A; 1n+1 , 1n+1 ) : (An , An ) −→ (An+1 , An+1 ) in S −1 S. Define, 0 (α) = ι−1 n (n (α)), which is a loop at (0, 0).       α0 α0 = 0 . Therefore, ιn ([0 (α)]) = [n (α)] = n+1 01 01 Define, η : GL(A) −→ π1 (B [(0, 0)] S , (0, 0)) by η(α) = ι0 (α)

∀ α ∈ GLn (A).

(4.28)

It follows from the above discussions that η is a well-defined group homomorphism. Lemma 4.7.5. surjective.

With the notation as in (4.7.4), the map η is

k the Proof. As before, let B[(0,  k-skeleton of B[(0, 0)]S .  0)]S denote By (C.3.9), the map π1 B[(0, 0)]1S , (0, 0)  π1 (B[(0, 0)]S , (0, 0)) is surjective. The space B[(0, 0)]1S consists of lines (F ) : (An , An ) −→ (An+r , An+r ), one for each morphism F : (Ar , f, g) : (An , An ) −→ (An+r , An+r ). However, for two composable arrows F, G, the concatenation (G)(F ) cannot be treated as the same (homotopic) path (GF ), because there is no nondegenerate 2-simplex in B[(0, 0)]1S . So, (F ), (G), (GF ) form a loop, which is not null homotopic in B[(0, 0)]1S . We argue to mitigate this situation.

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181

A loop α at (0, 0) in B [(0, 0)]S is given by finitely many (use (C.1.8)) composable arrows (Ar , f, g), as in the following diagram, in either direction, and they can be joined to (0, 0), back and forth, as follows: (An , An ) O

(Ar ,f,g)

 

_ _ _ _ _/ (0, 0) _ _ n+r (A ,1n+r ,1n+r )

f, g ∈ GLn+r (A) '

(An+r , An+r )

This triangle corresponds to a loop at (0, 0) and the loop α is homotopic to concatenation of such loops, given by a single arrow (Ar , f, g). Therefore, π1 (B [(0, 0))] S , (0, 0)) is generated by such loops given by such single arrows. Note that the arrow (Ar , f, g) in S −1 S is a composition as in the following diagram, and which extends as follows: (An , An ) O

(Ar ,1n+r ,1n+r ) n+r / (A , An+r )

SSS SSS SSS (A0 ,f,g) SS  (Ar ,f,g) SSS)  _ _ _ _ _/ (An+r , An+r ) (0, 0) _ _ n+r

(An ,1n ,1n )



(A

,1n+r ,1n+r )

Since the upper triangle corresponds to a 2-simplex in B [(0, 0)]S , the second vertical edge can be homotoped to coincide with the diagonal edge. Writing N := n + r, it follows from this that π1 (B [(0, 0)]S , (0, 0)) is generated by loops, given by single arrows, as follows: (AN , AN ) O

(

AN ,1N ,1N

)



(A0 ,f,g) &

f, g ∈ GL(AN )

(0, 0) _ _ _N_ _ _ _/ (AN , AN ) (A ,1N ,1N ) We denote this loop by (f, g). With notation in  0  (4.7.4), note that the arrow A , f, g is a composi(f, 1N ) = η(f ). Also note   0 0 tion of A , f, 1N and A , 1N , g . Note that the vertical canonical arrow (AN , 1N , 1N ) : (0, 0) −→ N (A , AN ) is identified with the (AN , α, α) : (0, 0) −→ (AN , AN )

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Algebraic K-Theory

for any α ∈ GLN (A). So, (f, g)) is equivalent to (f α, gα)). In particular, (f, g) = (f g−1 , 1N ). This establishes that η is surjective. The proof is complete. Theorem 4.7.6. Let A be a ring. With the notation as in (4.7.4), ∼ the map η induces an isomorphism η : K1c (R) −→ π1 (B[(0, 0)]S , (0, 0)). GL(A) , and K1c (A) denote the one Proof. As before, K1c (A) := EL(A) object () category, corresponding to the group K1c (A). We also have EL(A) = [GL(A), GL(A)] = [EL(A), EL(A)], the commutator subgroup (4.7.2). Recall B[(0, 0)]S is an Hc -space (see Section 4.8). Therefore, by (4.9.1) π1 (B[(0, 0)]S , (0, 0)) is an abelian group. Therefore, η factors through a surjective map η : K1c (A)  π1 (B[(0, 0)]S , (0, 0)), (4.7.5). To prove that η is injective, we define a left inverse of η. There is a well-defined functor, defined as follows:

Φ : [(0, 0)]S −→ K1c (A) where,  Φ(An , An ) =  ∀ n Φ((Ak , f, g)) = EL(A)(f g−1 ) ∀ f, g ∈ GLk+n (A) where (Ak , f, g) denotes the arrow (An , An ) −→ (Ak+n , Ak+n ), represented by (Ak , f, g), and EL(A)(f g−1 ) denotes the coset. We prove that Φ((Ak , F, G)) is well defined. Suppose (Ak , F, G) : (An , An ) −→ (Ak+n , Ak+n ) represents the same map. Then there is α ∈ GLr (A), and with αn := α ⊕ 1n , such that F = αn f and G = αn g. So, −1 −1 −1 −1 −1 F G−1 = αn f (αn g)−1 = αn (f g−1 )α−1 n = αn (f g )αn (f g ) f g

= [αn , f g−1 ]f g−1 where [αn , f g−1 ] denotes the commutator. Therefore, EL(A)(f g −1 ) = EL(A)(F G−1 ), and it follows that Φ((Ak , F, G)) is well defined. So, Φ defines the map of the classifying spaces BΦ : c B[(0,  0)]S −→ BK1 (R), and a map π1 (Φ) : π1 (B[(0, 0)]S , (0, 0)) −→ π1 BK1c (A),  = K1c (A). Now,

BΦρ(EL(A)α) = BΦ(A0 , α, 1n ) = EL(A)α. The proof is complete.

The Agreement with Classical K-Theory

4.7.3.

183

The Agreement of K2c (A) and K2 (A)

As before, let A denote a ring, and let S = {An : n ≥ 0} = F (A) and other notation be as in (4.7.4). The proof of coincidence of classical K2c (A) and Quillen K2 (A), at this stage, is more algebraic than topological. The proof is due to Gersten [G2]. As usual, given a group G, let G denote the category that has exactly one object  and M or(, ) = G. We also use the notation BG := BG to denote the classifying space. Before we define K2c (A), we define a natural map BGL(A) −→ B(S −1 S), as follows. Lemma 4.7.7. There is a natural map ρ : BGL(A) −→ BS −1 S, such that ρ induces the following familiar exact sequence: / EL(A)

1

 with

/ GL(A)

π1 (ρ)

/ K1 (A)

/1

π1 (BGL(A)) = GL(A) π1 (BS −1 S) = K1 (A).

(4.29)

Proof. For completeness, we include the proof (see [Wc2, p. 353]). We have a commutative diagram βn

GLn (A) ιn

 GLn+1 (A)

/ AutS −1 S (An , An ) A

βn+1

 / AutS −1 S (An+1 , An+1 )

 where

β(g) = (g, 1n ) ιn (g) = g ⊕ 1A .

We treat βn as functors GLn (A) −→ S −1 S, sending  → (An , An ). This commutative diagram yields a natural equivalence βn → βn+1 ιn . So, it defines a homotopy Hn : BGLn (A) × I −→ BS −1 S. Now, by mapping telescope, we have a description (refer to Exercise 4.9.2), BGL(A) =

∞

n=1 BGLn (A) × [n, n + 1] (xn , n + 1) ∼ (ιn (xn ), n + 1) ∀xn ∈ BGLn (A), n = 1, 2 . . .

(4.30)

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(Note, mapping telescope construction is an iteration of mapping cylinder construction (C.4.11).) So, the sequence βn defines a map ρ : BGL(A) −→ BS −1 S. The proof is complete. The following is an immediate consequence of (4.7.7). Lemma 4.7.8. Let ρ : BGL(A) −→ BS −1 S be the map as in (4.7.7) and F := F (ρ, [(0, 0)]) be the homotopy fiber. Then the sequence (4.29) extends to an exact sequence 1

/ π2 BS −1 S, [(0, 0)]

/ π1 (F )

/ GL(A)

/ K1 (A)

/1

(4.31) Proof. It follows from the fact that π2 (BGL(A)) = 0. The proof is complete. For the convenience of our discussions, we sketch the definitions of the Steinberg group, K2 (A), and some other results from [Mi]. We refer to [Mi] for details. Definition 4.7.9. Let A be a ring. Fix an integer n ≥ 3. The Steinberg group Stn (A) is defined to be the group generated by the set of generators   xλij : 1 ≤ i, j ≤ n, i = j, λ ∈ A subject to the relations, ⎧ λ μ xij xij = xλ+μ ⎪ ij ⎪ ⎪ ⎪ 

⎪ ⎨ λ μ xij , xjk = xλμ ik 

⎪ μ λ ⎪ xij , xkp = 1n ⎪ ⎪ ⎪ ⎩

∀1 ≤ i, j ≤ n, i = j, and λ, μ ∈ A ∀1 ≤ i, j, k ≤ n, i = j, j = k, i = k, and λ, μ ∈ A ∀1 ≤ i, j, k, p ≤ n, i = j, k = p, j = k, i = p, and λ, μ ∈ A

(4.32) So, Stn (A) = F R where ⎧ F = Free Group generated by ⎪ ⎪ ⎪ λ ⎨ {xij : 1 ≤ i, j ≤ n, i = j, and λ ∈ A} ⎪ R ⊆ F is the normal subgroup generated ⎪ ⎪ ⎩ by the relations (4.32)

The Agreement with Classical K-Theory

185

There is an obvious map Stn (A) −→ Stn+1 (A). Define the infinite Steinberg group by St(A) := limn Stn (A). As before, for 1 ≤ i, j ≤ n, i = j and λ ∈ A, let eλij denote the n × n matrix with λ in (i, j) position and zero elsewhere. So, 1n + eλij ∈ ELn (A) is a generator of the group ELn (A) of elementary matrices. The matrices ελij := 1n + eλij satisfy the relations (4.32). So, there is a surjective map ϕn : Stn (A)  ELn (A)

sending

xλij → .ελij

(4.33)

Taking the limit, we have a surjective map ϕ : St(A)  EL(A).

(4.34)

The classical group K2c (A) was defined in [Mi] as K2c (A) := ker(ϕ).

(4.35)

In what follows, we state some results and definitions from [Mi]. Theorem 4.7.10. With notation as in (4.7.9), we have K2c (A) = ker(ϕ) = center(St(A)). Proof. See [Mi, p. 40]. Remark 4.7.11. Consequently, K2c (A) is an abelian group. Further, the sequence (4.29) extends to an exact sequence 1

/ K c (A) 2

/ St(A)

/ GL(A)

/ K1 (a)

/ 1.

(4.36) This is yet another extension of the sequence (4.29), along with (4.31). Proof. Obvious from definition (4.35). Now we define central extensions and universal central extensions of groups. Definition 4.7.12. Consider an exact sequence 1

/ ker(f )

/X

f

/G

/1

of groups.

(4.37)

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Algebraic K-Theory

Such a sequence is denoted by (X, f ). We say that (4.37) splits if there is a homomorphism,  : G −→ X such that f  = 1G . Also recall that a group G is said to be perfect if the commutator [G, G] = G. In our context, by (4.7.2), the Elementary subgroup EL(A) is a perfect subgroup of GL(A). The sequence (4.37) is defined to be a Central extension of G if ker(f ) ⊆ Center(f ). We also say that X is a central extension of G. A central extension (U, v) of G is called a Universal central extension of G if it has the universal property, among the central extensions of G. This means, given any other central extension (X, f ), as in (4.37), there is a unique homomorphism h : U −→ X such that v = f h. In other words, there is a unique h such that the diagram 1

1

/ ker(g)

/ ker(f )

/U v    h

/X

f

/G

/1

commutes. /G

(4.38)

/1

As usual, if G has a universal central extension (U, v), then it is unique up to natural isomorphisms. The following is a characterization of universal extensions. Theorem 4.7.13. Let G be a group. A central extension (U, v) of G is universal if and only if (1) U is perfect and (2) every central extension of U splits. Proof. See [Mi, p. 43]. Theorem 4.7.14. A group G admits a universal central extension if and only if G is perfect. Proof. See [Mi, p. 45]. Corollary 4.7.15. Let A be a ring. Then the infinite elementary group EL(A) admits a universal central extension. Proof. In fact, E(A) is perfect, by (4.7.2). Now the corollary follows from (4.7.14). The following is the final theorem from [Mi].

The Agreement with Classical K-Theory

Theorem 4.7.16. extension. 1

187

Let A be a ring. Consider the following

/ K c (A) 2

/ St(A)

/ EL(A)

/1

(4.39) which is the left-hand side of (4.36). Then (4.39) is a universal central extension of EL(A). Proof. See [Mi, p. 47], and also [Sw3, p. 208]. The tail-end of the argument comes from [G2], as follows. Theorem 4.7.17. Let A be a ring. Consider the following extension:   / π2 BS −1 S, [(0, 0)] / π1 (F ) / EL(A) /1 1 (4.40) which is the left-hand side of (4.31). Then (4.40) is a universal central extension of EL(R). Proof. Recall π2 (X, x) is commutative, for any (X, x) ∈ Top• . Further, it is not difficult to see that for a homotopy fibration /X / B , the map π2 (B) −→ π1 (X) is central. So, (4.40) F is central. It is a result of Kervaire [Km] that (4.40) is a universal extension. Refer to Exercise 4.9.3, [Wc2, p. 289], [Sv, Prop. 2.5]) for more complete proofs. Finally, we state the coincidence theorem. Theorem 4.7.18. Let A be a ring. Then K2c (A) K2 (A) are isomorphic.   Proof. Recall, Quillen K2 (A) = π2 BS −1 S, [(0, 0)] . Now the theorem follows by comparing two universal central extensions (4.39) and (4.40). The proof is complete. 4.8.

Hc -Spaces

For completeness, we include this section on H-spaces. First, we recall the definition of H-spaces [S, p. 36], [W, pp. 116–120].

188

Algebraic K-Theory

Definition 4.8.1. Let (Y, ) ∈ Top• . Let p1 , p2 : Y × Y −→ Y denote the projections, and ι1 , ι2 : Y −→ Y × Y be the maps defined by ι1 (y) = (y, ), ι2 (y) = (, y). Then Y is defined to be a H-space if there is a map μ : k(Y ×Y ) −→ Y in Top• such that μι1 = μι2 = 1Y . There is a subtle difference between the literature on H-spaces [S,W] and the way it is used in homotopy theory, in particular in KTheory, literature (e.g. 4.7.6). To iron out this difference, we define Hc -Spaces. Definition 4.8.2. Let (Y, ) ∈ Top• . Let p1 , p2 , ι1 , ι2 , be as in (4.8.1), while we consider them as maps p1 , p2 : k(Y × Y ) −→ Y and ι1 , ι2 : Y −→ k(Y × Y ). Then p1 , p2 , ι1 , and ι2 are continuous. Further, p1 ι1 = p2 ι2 = 1Y . A topological space Y is said to be an Hc -space, if there is a map μ : k(Y × Y ) −→ Y , such that μι1 = μι2 = 1Y : i1

.

0 k(Y × Y )

Y

i2 1Y



μ

)Y

Fix an Hc -Space (Y, μ) as above, we further define the following and add comments: 1. Given X, Y ∈ Top• , we denote [X, Y ] := HomHomTop• (X, Y ), as in Section B.2. 2. For α, β ∈ HomTop• (X, Y ), define the product α · β by the commutative diagram X Q

Δ

/ k(X × X) α×β / k(Y × Y ) S T μ V W  Y α·β Z Y

For such α, β ∈ HomTop• (X, Y ), further define [α]·[β] := [α·β] ∈ [X, Y ].

The Agreement with Classical K-Theory

189

3. Let eX : X −→ Y denote the constant map in Top• . Then α·eX = α = eX · α, which follows from the commutative diagram Δ

X

/ X × X α×eX / Y × Y μ

α

)

Y



1

/Y ×Y

ι1

/Y C μ

Consequently, [α] · [eX ] = [α] = [eX ] · [α]. In other words, under this product · in [X, Y ], [eX ] acts as the identity. 4. In some literature, the conditions above that μι1 = μι2 = 1Y are loosened and it is assumed only that μι1 ∼ μι2 ∼ 1Y are homotopic, and the product above on [X, Y ] makes sense. One further defines (homotopy) associative and symmetric properties of Hc -spaces. (See [W, p. 118].) Under these conditions, [X, Y ] also has a structure of an abelian monoid, with identity, for all X ∈ Top• . 5. In particular, ∀ n ≥ 0, the multiplication μ induces a structure of an abelian monoid, with identity on πn (Y, ) = [Sn , Y ] (which is not to be mixed up with the homotopy group structures, for n ≥ 1). Definition 4.8.3. Let (Y, μ) be a homotopy associative Hc -space. We say (Y, μ) is group-like if there is a map j : Y −→ Y of pointed spaces, such that j · 1Y 1Y · j eY . This means, if [1Y ] has an inverse in the monoid [Y, Y ]. In this case, [X, Y ] also has a group structure for any X ∈ Top• . This follows from the following commutative diagram, with α : X −→ Y a map of pointed spaces, X α

Δ/

k(X × X)

joα×α

/ k(Y × Y ) μ