Extended abstracts Spring 2015 : interactions between representation theory, algebraic topology and commutative algebra 978-3-319-45440-5, 3319454404, 978-3-319-45441-2, 3319454412

Table of contents :
Front Matter ....Pages i-ix
Homological Algebra on an Adams Algebraic Stack (Leovigildo Alonso Tarrío)....Pages 1-5
Lyubeznik Numbers of Local Rings and Linear Strands of Graded Ideals (Josep Àlvarez Montaner)....Pages 7-13
The Heart of a t-Structure Induced by a n-Tilting Module (Silvana Bazzoni)....Pages 15-19
On Some Local Cohomology Spectral Sequences (Alberto F. Boix)....Pages 21-26
Homotopy Representations of Classifying Spaces (Natàlia Castellana)....Pages 27-33
Decomposing Gorenstein Rings as Connected Sums (Hariharan Ananthnarayan, Ela Celikbas, Zheng Yang)....Pages 35-39
Rigid and Test Modules (Olgur Celikbas)....Pages 41-46
Tate Homology Beyond Gorenstein Rings (Lars Winther Christensen)....Pages 47-51
On the Classification of Artin Algebras and the Inverse System of Macaulay (Joan Elias)....Pages 53-57
Purity in Categories of Sheaves (Sergio Estrada)....Pages 59-62
Idempotent Functors and Nilpotent Spaces (Wojciech Chachólski, Emmanuel Dror Farjoun, Ramón Flores, Jérôme Scherer)....Pages 63-67
Decomposition Spaces and Incidence (Co)Algebras (Imma Gálvez-Carrillo)....Pages 69-74
Cellular Approximations for Fusion Systems (Alberto Gavira-Romero)....Pages 75-80
Homological Epimorphisms and the Lie Bracket in Hochschild Cohomology (Reiner Hermann)....Pages 81-85
Gorenstein Projective Precovers (Alina Iacob)....Pages 87-90
Hochschild Homology on Schemes and Fundamental Class (Ana Jeremías López)....Pages 91-95
A Remark on Leclerc’s Frobenius Categories (Martin Kalck)....Pages 97-101
Atom-Molecule Correspondence in Grothendieck Categories (Ryo Kanda)....Pages 103-107
Proalgebraic Crossed Modules of Quasirational Presentations (Andrey Mikhovich)....Pages 109-114
Some F-Invariants for Quotient Singularities (Yusuke Nakajima)....Pages 115-119
Strong Generation of Some Derived Categories of Schemes (Amnon Neeman)....Pages 121-127
Regularity of Products over Quadratic Hypersurfaces (Hop D. Nguyen, Thanh Vu)....Pages 129-132
Phantom Maps and Representability (Oriol Raventós)....Pages 133-136
Six Operations on dg Enhancements of Derived Categories of Sheaves and Applications (Olaf M. Schnürer)....Pages 137-141
Tensor Product of Dualizing Complexes over a Field (Liran Shaul)....Pages 143-147
Strong Generators in Tensor Triangulated Categories (Johan Steen)....Pages 149-153
Abelian Model Structures and Applications (Jan Šťovíček)....Pages 155-159
Singularity Categories of Stable Resolving Subcategories and Applications to Gorenstein Rings (Ryo Takahashi)....Pages 161-165
Classes of Flat Modules Arising in Algebraic Geometry and Approximations (Jan Trlifaj)....Pages 167-171
The Dual Graph of an Arithmetically Gorenstein Scheme (Matteo Varbaro)....Pages 173-174
Baez–Dolan Stabilization via (Semi-)Model Categories of Operads (David White, Michael Batanin)....Pages 175-179
Vanishing of Tor (Olgur Celikbas, Roger Wiegand)....Pages 181-185
Prime Ideals in Noetherian Rings (Sylvia Wiegand)....Pages 187-192

Citation preview

Trends in Mathematics Research Perspectives CRM Barcelona Vol.5

Dolors Herbera Wolfgang Pitsch Santiago Zarzuela Editors

Extended Abstracts Spring 2015 Interactions between Representation Theory, Algebraic Topology and Commutative Algebra

Trends in Mathematics Research Perspectives CRM Barcelona Volume 5

Series editors Enric Ventura Antoni Guillamon

Since 1984 the Centre de Recerca Matemàtica (CRM) has been organizing scientific events such as conferences or workshops which span a wide range of cutting-edge topics in mathematics and present outstanding new results. In the fall of 2012, the CRM decided to publish extended conference abstracts originating from scientific events hosted at the center. The aim of this initiative is to quickly communicate new achievements, contribute to a fluent update of the state of the art, and enhance the scientific benefit of the CRM meetings. The extended abstracts are published in the subseries Research Perspectives CRM Barcelona within the Trends in Mathematics series. Volumes in the subseries will include a collection of revised written versions of the communications, grouped by events.

More information about this series at http://www.springer.com/series/13332

Dolors Herbera Wolfgang Pitsch Santiago Zarzuela •

Editors

Extended Abstracts Spring 2015 Interactions between Representation Theory, Algebraic Topology and Commutative Algebra

Editors Dolors Herbera Departament de Matemàtiques Universitat Autònoma de Barcelona Bellaterra, Barcelona Spain

Santiago Zarzuela Departament de Matemàtiques i Informàtica Universitat de Barcelona Barcelona Spain

Wolfgang Pitsch Departament de Matemàtiques Universitat Autònoma de Barcelona Bellaterra, Barcelona Spain

ISSN 2297-0215 ISSN 2297-024X (electronic) Trends in Mathematics ISSN 2509-7407 ISSN 2509-7415 (electronic) Research Perspectives CRM Barcelona ISBN 978-3-319-45440-5 ISBN 978-3-319-45441-2 (eBook) DOI 10.1007/978-3-319-45441-2 Library of Congress Control Number: 2016950386 Mathematics Subject Classification (2010): 13-06, 16-06, 18-06, 55-06 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This book is published under the trade name Birkhäuser, www.birkhauser-science.com The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

In the first semester of 2015, from January 7 to June 30, over a hundred mathematicians visited the Centre de Recerca Matemàtica (CRM) in Bellaterra (Catalonia, Spain) to participate in the thematic semester “Interactions Between Representation Theory, Algebraic Topology and Commutative Algebra”. The local organizers of this research semester were Dolors Herbera (Universitat Autònoma de Barcelona, Spain), Wolfgang Pitsch (Universitat Autònoma de Barcelona, Spain) and Santiago Zarzuela (Universitat de Barcelona, Spain), with the scientific advise of Bill Dwyer (University of Notre Dame, USA), Srikanth B. Iyengar (University of Nebraska, USA), Henning Krause (Universität Bielefeld, Germany) and Bernard Leclerc (Univeristé de Caen, France). The core of the program consisted of the following activities: The first advanced course: “(Re)Emerging Methods in Commutative Algebra and Representation Theory”, from February 9 to 13, 2015, followed by a more specialized workshop: “Homological Bonds Between Commutative Algebra and Representation Theory” (HobCART), from February 16 to 20, 2015. Then, the second advanced course: “Building Bridges Between Algebra and Topology”, from April 13 to 17, 2015 followed by its workshop: “Brave New Algebra: Opening Perspectives” (BnaOP), from April 20 to 24, 2015. Then finally, a conference: “Opening Perspectives in Algebra, Representations, and Topology” (OP-ART), from May 25 to 29, 2015. This resulted in a very intensive period of six months, during which mathematicians ranging from the young and enthusiastic PhD students to the respected and very active Emeritus Professor and whose research interests covered a variety of areas (singularity theory, representation theory, commutative algebra, algebraic topology) although with an overlapping interest in homological methods, gathered, exchanged ideas, got acquainted with each others research and results and in some cases started what we hope to be fruitful collaborations. The very dynamic and productive atmosphere we enjoyed during this semester translated into an equally active weekly seminar and a couple of special seminars on specialized topics: Grothendieck Derivators, and Differential Graded Categories.

v

vi

Preface

In this volume of the subseries Research Perspectives CRM-Barcelona (published by Birkhäuser inside the series Trends in Mathematics), we present 33 Extended Abstracts corresponding to selected talks given by participants in the several events of the research program. Seventeen come from the first workshop HobCART, eight from the second one BnaOP and eight more from the conference OP-ART. The variety of topics presented bears the testimony of the rich activity that made the success of the IRTATCA semester. We hope that this volume will give to the authors the opportunity to quickly communicate their recent research: most of the short articles here are brief and preliminary presentations of new results not yet published in regular research journals. We would like to express our gratitude to the CRM for hosting and supporting our research program. Also, our warm thanks to the CRM staff, its director, Joaquim Bruna, and all the secretaries, and especially Neus Portet, for providing great facilities and a very pleasant working environment. Last but not least, thanks are due to all those who attended the talks, for their interest, their active participation, and their enthusiasm towards mathematics. The program was possible thanks to the generous support of several organizations: Universität Bielefeld via the Collaborative Research Centre 701 “Spectral Structures and Topological Methods in Mathematics”, the Institut de Matemàtiques de la Universitat de Barcelona (IMUB) that hosted the workshop HobCART, the National Science Foundation, the Simons Foundation, and the research projects “Estructura y clasificación de anillos, módulos y C*-álgebras” (DGI MICIIN MTM2011-28992-C02-01, Spain) and “Análisis local en grupos y espacios topológicos” (DGI MICIIN MTM2010-20692, Spain) both hosted by the Universitat Autònoma de Barcelona. Finally, we also axknowledge the financial support of the Societat Catalana de Matemàtiques and the Intercountry Program of the Fulbright Foundation. Bellaterra, Spain Bellaterra, Spain Barcelona, Spain

Dolors Herbera Wolfgang Pitsch Santiago Zarzuela

Contents

Homological Algebra on an Adams Algebraic Stack . . . . . . . . . . . . . . . . Leovigildo Alonso Tarrío

1

Lyubeznik Numbers of Local Rings and Linear Strands of Graded Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Josep Àlvarez Montaner

7

The Heart of a t-Structure Induced by a n-Tilting Module . . . . . . . . . . . Silvana Bazzoni

15

On Some Local Cohomology Spectral Sequences . . . . . . . . . . . . . . . . . . . Alberto F. Boix

21

Homotopy Representations of Classifying Spaces . . . . . . . . . . . . . . . . . . . Natàlia Castellana

27

Decomposing Gorenstein Rings as Connected Sums . . . . . . . . . . . . . . . . . Hariharan Ananthnarayan, Ela Celikbas and Zheng Yang

35

Rigid and Test Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Olgur Celikbas

41

Tate Homology Beyond Gorenstein Rings . . . . . . . . . . . . . . . . . . . . . . . . . Lars Winther Christensen

47

On the Classification of Artin Algebras and the Inverse System of Macaulay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joan Elias

53

Purity in Categories of Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sergio Estrada

59

Idempotent Functors and Nilpotent Spaces . . . . . . . . . . . . . . . . . . . . . . . . Wojciech Chachólski, Emmanuel Dror Farjoun, Ramón Flores and Jérôme Scherer

63

vii

viii

Contents

Decomposition Spaces and Incidence (Co)Algebras . . . . . . . . . . . . . . . . . Imma Gálvez-Carrillo

69

Cellular Approximations for Fusion Systems . . . . . . . . . . . . . . . . . . . . . . Alberto Gavira-Romero

75

Homological Epimorphisms and the Lie Bracket in Hochschild Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reiner Hermann

81

Gorenstein Projective Precovers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alina Iacob

87

Hochschild Homology on Schemes and Fundamental Class . . . . . . . . . . . Ana Jeremías López

91

A Remark on Leclerc’s Frobenius Categories. . . . . . . . . . . . . . . . . . . . . . Martin Kalck

97

Atom-Molecule Correspondence in Grothendieck Categories . . . . . . . . . 103 Ryo Kanda Proalgebraic Crossed Modules of Quasirational Presentations . . . . . . . . 109 Andrey Mikhovich Some F-Invariants for Quotient Singularities . . . . . . . . . . . . . . . . . . . . . . 115 Yusuke Nakajima Strong Generation of Some Derived Categories of Schemes . . . . . . . . . . 121 Amnon Neeman Regularity of Products over Quadratic Hypersurfaces . . . . . . . . . . . . . . . 129 Hop D. Nguyen and Thanh Vu Phantom Maps and Representability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Oriol Raventós Six Operations on dg Enhancements of Derived Categories of Sheaves and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Olaf M. Schnürer Tensor Product of Dualizing Complexes over a Field . . . . . . . . . . . . . . . 143 Liran Shaul Strong Generators in Tensor Triangulated Categories . . . . . . . . . . . . . . . 149 Johan Steen Abelian Model Structures and Applications . . . . . . . . . . . . . . . . . . . . . . . 155 Jan Štovíček

Contents

ix

Singularity Categories of Stable Resolving Subcategories and Applications to Gorenstein Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Ryo Takahashi Classes of Flat Modules Arising in Algebraic Geometry and Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Jan Trlifaj The Dual Graph of an Arithmetically Gorenstein Scheme . . . . . . . . . . . . 173 Matteo Varbaro Baez–Dolan Stabilization via (Semi-)Model Categories of Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 David White and Michael Batanin Vanishing of Tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Olgur Celikbas and Roger Wiegand Prime Ideals in Noetherian Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Sylvia Wiegand

Homological Algebra on an Adams Algebraic Stack Leovigildo Alonso Tarrío

Abstract We will discuss joint work with Alonso Tarrío et al. (Expo Math 33:452– 501, 2015 [1]). We attach to a geometric stack a small non-trivial ringed site that has good functoriality properties when considering quasi-coherent sheaves. Our approach differs from (Various authors, Stacks Project [5]) in that they use big sites for which functoriality is obvious but not the existence of generators. MSC 2000: Primary 14A20, 14A20 · Secondary 14F05, 18F20

1 Generalities on Stacks and Quasi-coherent Sheaves Let S denote the big site of schemes over a base S (that we will assume affine), together with the étale topology. We will work inside the 2-category of stacks, see [6]. An S-scheme X defines a stack over S that we will not distinguish by its notation. An algebraic stack (see [2]) is a stack X satisfying two conditions: (i) the diagonal δ : X −→ X × S X is representable by algebraic spaces; (ii) it is locally a scheme, i.e., there exists a scheme U and a 1-morphism of S-stacks p : U −→ X smooth and surjective. A stack X that possesses a presentation p : U → X with U quasi-compact will be called quasi-compact. Thus, one may choose for U an affine scheme. A quasicompact algebraic stack is called geometric (see [3]) if the diagonal δ : X −→ X × S X is an affine morphism (and, therefore, representable). A scheme is a geometric stack if and only if it is quasi-compact and semi-separated. A (small) site C is a ringed site if it possesses a ring object O in its sheaf category (i.e., its associated topos). A sheaf of O-modules M is quasi-coherent if for every V ∈ C there exists a covering { f i : Vi → V }i∈L such that for every Vi , i ∈ L, there is a presentation L.A. Tarrío (B) Facultade de Matemáticas, Departamento de Álxebra, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_1

1

2

L.A. Tarrío ) (J ) O|(I Vi −→ O|Vi −→ M|Vi −→ 0.

There are several candidates for a Grothendieck topology on a category of schemes over X but they present certain drawbacks. The main ones are: (i) the étale topology may be empty; (ii) the lisse-étale topology, formed by smooth schemes over the stack with étale coverings, is not functorial for 1-morphisms of stacks.

2 The Small Flat Site: Faithfully Flat and Finite Presentation For a geometric stack X we define the category Aff/X as follows. (i) Objects: The couples (V, v) with V an affine scheme and v : V → X a flat and finitely presented 1-morphism of stacks. (ii) Morphisms: The couples ( f, α) : (V  , v  ) → (V, v) inducing a 2-commutative diagram of 1-morphisms of stacks: f

V

V

α v

v

X The category Aff/X is essentially small. Furthermore, Aff/X has a ring presheaf OX : Aff/X → Ring, given by OX (V, v) = B with V = Spec(B). We define a Grothendieck topology on Aff/X by taking as coverings flat and finitely presented morphisms such that their joint image covers V . The corresponding site is denoted Afffppf /X, and its associated topos Xfppf . The presheaf OX is a sheaf for this topology. We denote the category of quasi-coherent sheaves of OX -modules as Qco(X). Theorem 1 The category of quasi-coherent sheaves Qco(X) is Abelian. Let f : X → Y be a 1-morphism of geometric stacks. As on schemes, f does not induce a (continuous) functor from Afffppf /Y to Afffppf /X, unless f is an affine morphism. In general, topologies finer than the étale topology seldom possess exact inverse images. Nonetheless, there is a way to reconstruct functoriality, for quasi-coherent sheaves of modules. For (V, v) ∈ Afffppf /Y, we denote by J(v, f ) the category with objects 2-commutative squares

Homological Algebra on an Adams Algebraic Stack h

W

3

V

γ

w

X

v f

Y

(1)

with (W, w) ∈ Afffppf /X. Given G over X we define the presheaf f ∗ G by: ( f ∗ G)(V, v) = lim G(W, w). ←− J(v, f )

Let now (W, w) ∈ Afffppf /X, we denote by I(w, f ) the category with objects 2-commutative squares (1) with (V, v) ∈ Afffppf /Y. We define for F over Y the presheaf f p F by: ( f p F)(W, w) = lim F(V, v). −→ I(w, f )

Proposition 2 If G is a sheaf on Afffppf /X, then f ∗ G is also a sheaf on Afffppf /Y. Corollary 3 Let F ∈ Yfppf . Denote by f −1 F the sheaf associated to the presheaf f p F. One gets a pair of adjoint functors f −1

Xfppf ←− −→ Yfppf . f∗

Caution: in general f −1 is not exact, that is why the pair ( f −1 , f ∗ ) does not constitute a morphism of topos. However the following holds. Proposition 4 The category I(w, f ) possesses finite products. As a consequence, the functor f −1 preserves algebraic structures. Theorem 5 Let f : X → Y be a 1-morphism of geometric stacks. There is a pair of adjoint functors f∗

←− Qco(Y). Qco(X) −→ f∗

Proposition 6 Let f : X → Y and g : Y → Z be 1-morphisms of geometric stacks, it holds (g f )∗ = g∗ f ∗ . As a consequence, (g f )∗ ∼ = f ∗ g∗. Theorem 7 Let f 1 , f 2 : X → Y be 1-morphisms of geometric stacks, and let ˜ f 2∗ ζ : f 1 ⇒ f 2 be a 2-morphism. Then, there are natural isomorphisms ζ∗ : f 1∗ → and ζ ∗ : f 2∗ → ˜ f 1∗ .

4

L.A. Tarrío

3 An Algebraic Approach Let X be a geometric stack and p : U → X a presentation with U an affine scheme. The fiber product U ×X U is an affine scheme and the couple (U, U ×X U ) is a scheme in groupoids whose associated stack is X. Let A0 and A1 be the rings defined by = Spec(A0 ) and U ×X U = Spec(A1 ). Denote the couple A• := (A0 , A1 ). It has the structure of a Hopf algebroid; see [4]. An A• -comodule (M, ψ M ) (on the left) is an A0 -module M together with an A0 -linear map (the structure map) ψ M : M → A1 η R ⊗ A0 M, such that (∇ ⊗ id M )ψ M = (id A1 ⊗ψ M )ψ M and ( ⊗ id M )ψ M = id M . Theorem 8 If A1 is a flat A0 -module, A• , is a Grothendieck category. Theorem 9 Let X = Stck(A• ). The categories Qco(X) and A• , are equivalent, therefore Qco(X) is a Grothendieck category.

4 The Derived Category Let A(X) denote the category of all OX -modules. The inclusion functor ι : Qco(X) → A(X) possesses a right adjoint, the coherator functor, Q : A(X) → Qco(X). By derivation they induce an adjunction ι

←− D(Qco(X)). D(A(X)) −→ RQ

Theorem 10 The category D(Qco(X)) satisfies: (i) it is a triangulated category; (ii) it possesses arbitrary coproducts; (iii) a cohomological functor H taking values in abelian groups is representable, i.e., H = HomD(Qco(X)) (−, F) with F ∈ D(Qco(X)). Proposition 11 Let X be a geometric stack and p : U → X a presentation with U an affine scheme. We say that X satisfies the Adams condition if it satisfies one of the following equivalent conditions: (a) p∗ OU is a direct limit of locally free modules; (b) X has the strong resolution property. Proposition 12 Every quasi-coherent sheaf on an Adams geometric stack X is the target of an epimorphism with a flat quasi-coherent source. Theorem 13 Let X be an Adams geometric stack. The category D(Qco(X)) has a natural structure of a symmetric closed category:

Homological Algebra on an Adams Algebraic Stack

5

(i) the derived functor of tensor product − ⊗LOX− is computed by homotopically flat resolutions (as in the case of schemes); (ii) the internal hom is defined, for F ∈ D(Qco(X)), through the formula Hom•X (F, −) := Rqc QXHom•X (F, −). Acknowledgements This work has been partially supported by Spain’s MINECO and E.U.’s FEDER research projects MTM2011-26088, MTM2014-59456 and Xunta de Galicia’s GRC2013045.

References 1. L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, M.J. Vale Gonsalves, A functorial formalism for quasi-coherent sheaves on a geometric stack. Expo. Math. 33, 452–501 (2015) 2. M. Artin, Versal deformations and algebraic stacks. Invent. Math. 27, 165–189 (1974) 3. J. Lurie, Tannaka duality for geometric stacks. http://www.math.harvard.edu/~lurie/papers/ Tannaka.pdf 4. D.C. Ravenel, Complex cobordism and stable homotopy groups of spheres, in Pure and Applied Mathematics, vol. 121. Academic Press, Inc., Orlando, FL (1986) 5. Various authors, Stacks Project. http://stacks.math.columbia.edu 6. A. Vistoli, Grothendieck topologies, fibered categories and descent theory, in Fundamental Algebraic Geometry. Mathematical Surveys and Monographs Series, vol. 123, pp. 1–104. American Mathematical Society, Providence, RI (2005)

Lyubeznik Numbers of Local Rings and Linear Strands of Graded Ideals Josep Àlvarez Montaner

Abstract We report recent work on the study of Lyubeznik numbers and their relation to invariants coming from the study of linear strands of free resolutions.

The aim of this paper is to put in the same spotlight two different sets of invariants that apparently come from completely different approaches. On one hand we have the Lyubeznik numbers that are a set of invariants of local rings coming from the study of injective resolutions of local cohomology modules. On the other hand the ν-numbers that are defined in terms of the acyclycity of the linear strands of free resolutions of Z-graded ideals. It turns out that both sets of invariants satisfy analogous properties and, in the particular framework of Stanley–Reisner theory, they are equivalent. This note will survey some recent work done by the author with his collaborators Vahidi and Yanagawa in [1–3].

1 Lyubeznik Numbers of Local Rings Let (R, m) be a regular local ring of dimension n containing a field K, and I an ideal of R. Consider a minimal injective resolution of a local cohomology module HIn−i (R) E• (HIn−i (R)) : 0

/ H n−i (R) I

/ E0

/ E1

d0

d1

/ ··· ,

(1)

where the p-th term is of the form Ep =



E R (R/p)μ p (p,HI

n−i

(R))

,

p∈SpecR

J. Àlvarez Montaner (B) Departament de Matemàtiques, Universitat Politècnica de Catalunya, Av. Diagonal 647, 08028 Barcelona, Spain e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_2

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J. Àlvarez Montaner

with the invariants μ p (p, HIn−i (R)) being the so-called Bass numbers. Since local cohomology modules are not finitely generated R-modules, it shouldn’t be clear that this injective resolution is finite or that the Bass numbers are finite. These properties were proved to be true in the seminal works of Huneke–Sharp [6] and Lyubeznik [7] for the cases where the characteristic of the field K is positive and zero, respectively. Relying on these facts, Lyubeznik [7] proved that the Bass numbers λ p,i (R/I ) := μ p (m, HIn−i (R)) = μ0 (m, Hmp (HIn−i (R))) are numerical invariants of the local ring R/I . These invariants are nowadays known as Lyubeznik numbers and, despite its algebraic nature, encode interesting geometrical and topological information.

1.1 Properties of Lyubeznik Numbers It was already proved in [7] that these invariants satisfy the following properties: (i) Vanishing: λ p,i (R/I ) = 0 implies 0 ≤ p ≤ i ≤ d, where d = dim R/I ; (ii) Highest Lyubeznik number: λd,d (R/I ) = 0. Therefore, we can collect them in the so-called Lyubeznik table: ⎛

λ0,0 · · · ⎜ .. (R/I ) = ⎝ .

⎞ λ0,d .. ⎟ . ⎠

λd,d

and we say that the Lyubeznik table is trivial if λd,d = 1 and the rest of these invariants vanish. In [2, 3] we used the Grothendieck’s spectral sequence p,n−i

E2

= Hmp (HIn−i (R)) =⇒ Hmp+n−i (R)

to extract some further constraints for the shape of the Lyubeznik table: (iii) Euler characteristic: 0≤ p,i≤d (−1) p−i λ p,i (A) = 1. (iv) Consecutiveness of trivial diagonals: Let ρ j (R/I ) =

d− j

i=0

λi,i+ j (R/I )

Lyubeznik Numbers of Local Rings and Linear Strands of Graded Ideals

9

denote the sum of the entries of the diagonals of the Lyubeznik table. Then, (1) if ρ1 (R/I ) = 0, then ρ0 (R/I ) = 1; (2) if ρ0 (R/I ) = 1 and ρ2 (R/I ) = 0, then ρ1 (R/I ) = 0; (3) if ρ j−1 (R/I ) = 0 and ρ j+1 (R/I ) = 0, then ρ j (R/I ) = 0 for 2 ≤ j ≤ d − 1.

1.2 Some Examples Very little is known about the possible configurations of Lyubeznik tables. The first example one may think is when there is only one local cohomology module different from zero. Then, using Grothendieck’s spectral sequence, we obtain a trivial Lyubeznik table. This situation is achieved, among others, in the following cases: • R/I is Cohen–Macaulay and contains a field of positive characteristic. • R/I is Cohen–Macaulay and I is a square-free monomial ideal in any characteristic. In characteristic zero we may find examples of Cohen–Macaulay rings with nontrivial Lyubeznik table, e.g., the ideals generated by the 2 × 2 minors of a 2 × 3 matrix. We also point out that replacing the Cohen–Macaulay condition by sequentially Cohen–Macaulay in the cases considered above, the result still holds true; see [2]. Another case that has received a lot of attention is when all local cohomology modules HIn−i (R) have dimension zero for i = d. In this case the Lyubeznik numbers of R/I satisfy λ p,i (R/I ) = 0 if p = 0 or i = d. Moreover, λ0,i (R/I ) = λd−i+1,d (R/I ) for 2 ≤ i < d, and λ0,1 (R/I ) = λd,d (R/I ) − 1. On top of that, these Lyubeznik numbers can be described in terms of certain singular cohomology groups in characteristic zero or étale cohomology groups in positive characteristic; see [4, 5].

2 Linear Strands of Graded Ideals Now we turn our attention to Z-graded ideals I in the polynomial ring R = K[x1 , . . . , xn ]. Consider a minimal Z-graded free resolution of I , L• (I ) : 0

/ Ln

dn

/ ···

/ L1

d1

where the i-th term is of the form Li =

 j∈Z

R(− j)βi, j (I ) ,

/ L0

/I

/ 0,

(2)

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and the invariants βi, j (I ) are the so-called Betti numbers. Given r ∈ N, we also consider the r -linear strand of L • (I ), L• (I ) : 0

/ L n

dn

/ ···

/ L 1

d1

/ L 0

/ 0,

where L i = R(−i − r )βi,i+r (I ) , and the differential di is the corresponding component of di . In Àlvarez-Montaner–Yanagawa [3] we introduced a new set of invariants measuring the acyclicity of the linear strands. Namely, given the field of fractions Q(R) of R, we introduce the ν-numbers: j−i> (I ) ⊗ R Q(R))]. νi, j (I ) := dim Q(R) [Hi (L< •

2.1 Properties of ν-Numbers Quite nicely, these invariants satisfy analogous properties to those satisfied by Lyubeznik numbers. However, mimicking the construction of the Betti table, we will consider the following table for ν-numbers: νi,i+r 0 1 2 · · · 0 ν0,0 ν1,1 ν2,2 · · · 1 ν0,1 ν1,2 ν2,3 · · · .. .. .. .. . . . . Given a Z-graded ideal I , we denote Ii = { f ∈ I | deg( f ) = i}. Then we have: (i) highest ν-number: ν0,l (I ) = 0 where l := min{i | Ii = 0}; in particular, we will say that I has a trivial ν-table when ν0,l (I ) = 1 and the rest of these invariants are zero; (ii) Euler characteristic: i, j∈N (−1)i νi, j (I ) = 1; (iii) consecutiveness of trivial columns: with νi (I ) = j∈N νi, j (I ) denoting the sum of entries of the columns of the ν-table, we have (1) if ν1 (I ) = 0, then ν0 (I ) = 1; (2) if ν0 (I ) = 1 and ν2 (I ) = 0, then ν1 (I ) = 0; (3) if ν j−1 (I ) = 0 and ν j+1 (I ) = 0, then ν j (I ) = 0 for 2 ≤ j ≤ d − 1. We also obtain the following property which, in general, is not known for the case of Lyubeznik numbers: (iv) Thom–Sebastiani type formula: let I , J be Z-graded ideals in two disjoint sets of variables, say I ⊆ R = K[x1 , . . . , xm ] and J ⊆ S = K[y1 , . . . , yn ]. The νnumbers of I T + J T , where T = R ⊗K S = K[x1 , . . . , xm , y1 , . . . , yn ] satisfy:

Lyubeznik Numbers of Local Rings and Linear Strands of Graded Ideals

11

• if I1 = 0 or J1 = 0 then I T + J T has trivial ν-table; • if I1 = 0 and J1 = 0 then we have the equality νi, j (I T + J T ) = νi, j (I T ) + νi, j (J T ) +



νk,l (I T )νk  ,l  (J T ).



k+k =i−1 l+l  = j

2.2 Some Examples So far we cannot present too many examples of possible configurations of ν-tables, except for the case of monomial ideals via the correspondence given in Theorem 1. However, using similar arguments to those considered in [2] to prove that sequentially Cohen–Macaulay rings have trivial Lyubeznik numbers, one can prove that componentwise linear ideals have trivial ν-table; see [3] for details.

3 The Case of Stanley–Reisner Rings Let I be a square-free monomial ideal in the polynomial ring R = K[x1 , . . . , xn ]. In this case, I coincides with the Stanley–Reisner ideal I of a simplicial complex  in n vertices, and it is known that R/I reflects topological properties of the geometric realization || of  in several ways. The jewel of the paper [1] is the following correspondence between the Lyubeznik numbers of a Stanley–Reisner ring R/I and the ν-numbers of the Alexander dual ideal I∨ of I . Theorem 1 (Àlvarez-Montaner–Vahidi, [1, Corollary 4.2]) Consider a square-free monomial ideal I ⊆ R = K[x1 , . . . , xn ]. We have λ p,i (R/I ) = νi− p,n− p (I∨ ). In this setting, another interesting result is that Lyubeznik numbers of Stanley– Reisner rings are not only algebraic invariants but also topological invariants. Namely, Theorem 2 (Àlvarez-Montaner–Yanagawa, [3, Theorem 5.3]) Consider a squarefree monomial ideal I ⊆ R = K[x1 , . . . , xn ]. Then, λ p,i (R/I ) depends only on the homeomorphism class of || and char(K). Of course, we may find examples of simplicial complexes  and  with homeomorphic geometric realizations but R/I being non-isomorphic to R/I . It is also well know that local cohomology modules as well as free resolutions depend on the characteristic of the base field, the most recurrent example being the Stanley–Reisner ideal associated to a minimal triangulation of P2R . Namely, for the ideal I = (x1 x2 x3 , x1 x2 x4 , x1 x3 x5 , x2 x4 x5 , x3 x4 x5 , x2 x3 x6 , x1 x4 x6 , x3 x4 x6 , x1 x5 x6 , x2 x5 x6 )

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in R = K[x1 , . . . , x6 ], the Lyubeznik table in characteristics zero and two are ⎛ ⎛ ⎞ ⎞ 0000 0010 ⎜ 0 0 0⎟ ⎜ ⎟ ⎟ and Z/2Z (R/I ) = ⎜ 0 0 0⎟ , Q (R/I ) = ⎜ ⎝ ⎝ 0 0⎠ 0 1⎠ 1 1 respectively.

4 Open Questions There are still a lot of open questions concerning Lyubeznik numbers and, by analogy, one can also formulate similar questions for ν-numbers. To name a few: (i) What can we say about the vanishing of Lyubeznik numbers and possible configuration of Lyubeznik tables? In particular, is there an homological characterization of rings having trivial Lyubeznik table? (ii) Recent developments in the study of local cohomology of determinantal ideals suggest that this would be a good set of examples where one can try to compute their Lyubeznik table; in general, it would be interesting to have other families of examples. (iii) What kind of topological information is provided by Lyubeznik numbers? Is it possible to extend the results of [4, 5] to other situations? (iv) It follows from Theorem 1 that Lyubeznik numbers of Stanley–Reisner rings satisfy a Thom–Sebastiani type formula. Does this property hold in general? (v) Consider the local ring at the vertex of the affine cone for some embedding of a projective variety in a projective space. Zhang [8] proved that its Lyubeznik numbers depend only on the projective variety but not on the embedding when the base field has positive characteristic. However, it is still open whether this result is true in characteristic zero.

References 1. J. Àlvarez-Montaner, A. Vahidi, Lyubeznik numbers of monomial ideals. Trans. Am. Math. Soc. 366, 1829–1855 (2014) 2. J. Àlvarez-Montaner, Lyubeznik table of sequentially Cohen-Macaulay rings. Commun. Algebra 43, 3695–3704 (2015) 3. J. Àlvarez-Montaner, K. Yanagawa, Lyubeznik numbers of local rings and linear strands of graded ideals. arXiv:1409.6486 4. M. Blickle, R. Bondu, Local cohomology multiplicities in terms of étale cohomology. Ann. Inst. Fourier 55, 2239–2256 (2005) 5. R. García, C. Sabbah, Topological computation of local cohomology multiplicities. Collect. Math. 49, 317–324 (1998)

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6. C. Huneke, R.Y. Sharp, Bass numbers of local cohomology modules. Trans. Am. Math. Soc. 339(2), 765–779 (1993) 7. G. Lyubeznik, Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra). Invent. Math. 113, 41–55 (1993) 8. W. Zhang, Lyubeznik numbers of projective schemes. Adv. Math. 228, 575–616 (2011)

The Heart of a t-Structure Induced by a n-Tilting Module Silvana Bazzoni

Abstract We describe the t-structure induced by an infinitely generated n-tilting module, and we ask when its heart is a Grothendieck category.

1 Introduction The notion of a t-structure in a triangulated category was introduced in Beilinson– Bernstein–Deligne [1]. A t-structure is a pair of full subcategories satisfying suitable axioms which guarantee that their intersection is an abelian category H, called the heart of the t-structure. According to Happel–Reiten–Smalø [2], every torsion pair in a Grothendieck category A induces a t-structure in the unbounded derived category D(A) of A. In particular, every (infinitely generated) 1-tilting module T over a ring R induces a torsion pair, and hence a t-structure in D(R) such that D(H) is triangle equivalent to D(R). Moreover, if T is finitely generated, then the heart H is equivalent to the module category over the endomorphism ring of T . We describe the t-structure induced by an infinitely generated n-tilting module and we ask when its heart is a Grothendieck category. In the recent paper Parra– Saorín [5], the authors proved that if T is a 1-tilting module, then the heart of the induced t-structure is a Grothendieck category if and only if the torsion free class associated to T is closed under direct limits. We prove that the latter condition holds if and only if T is a pure projective 1-tilting module. Moreover, we show that if R is a commutative ring then T is projective so, equivalent to a finitely generated tilting module. We also exhibit a class of rings for which there exist pure projective 1-tilting modules which are not equivalent to finitely generated tilting modules.

S. Bazzoni (B) Università di Padova, Padua, Italy e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_3

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2 Definitions and Notations Definition 1 (Beilinson–Bernstein–Deligne, [1]) A t-structure in a triangulated category (D, [−]) is a pair (U, V) of subcategories such that (i) U[1] ⊆ U; (ii) V = U ⊥ [1], where U ⊥ = {Y ∈ D | HomD (U, Y ) = 0}; (iii) for every object D ∈ D there is a triangle U → D → Y → U [1] with U ∈ U and Y ∈ U ⊥ . A t-structure plays the rôle of a torsion pair in a triangulated category. Theorem 2 (Beilinson–Bernstein–Deligne, [1]) The heart H = U ∩ V of a t-structure (U, V ) is an abelian category. We are interested in the t-structures in the unbounded derived category of a ring R induced by n-tilting modules. To this aim we recall the notion of tilting modules. Definition 3 An R-module T is n-tilting if (T1) p.dim.T ≤ n; (T2) ExtiR (T, T (I ) ) = 0, for all i ≥ 1 and all sets I ; (T3) there exists an exact sequence 0 → R → T0 → · · · → Tr → 0, where Ti ∈ Add(T ), i.e., they are direct summands of direct sums of copies of T . If T is finitely generated then it is called classical. The class T ⊥ = {M ∈ Mod-R | Exti (T, M) = 0, ∀i ≥ 1} is called the tilting class. If T is a 1-tilting module, then T ⊥ coincides with the class of modules generated by T and it is a torsion class.

3 The t-Structure Induced by a n-Tilting Module Every n-tilting R-module T gives rise to a t-structure (U, U ⊥ [1]) in D(R), which can be better described using the tool of model structures; see, e.g., [3]. The starting point is to note that if T is a n-tilting module, then (⊥ T , T ) is a complete cotorsion pair, that is a pair of mutually orthogonal classes with respect to the Ext-functor providing for approximations. By Hoevey’s Theorems [4] there is a model structure on Ch(R) corresponding to the tilting cotorsion pair. The fibrant objects of this model structure can be explicitly described. They are exactly the complexes quasi isomorphic to complexes with terms in T and, if HomD(R) (T [i], X ) = 0 for all i < 0, then X is quasi isomorphic to · · · → X −n → · · · → X −1 → X 0 → 0,

The Heart of a t-Structure Induced by a n-Tilting Module

17

with X i ∈ T , for all i. Thus, if we let U = {X ∈ D(R) | HomD(R) (T [i], X ) = 0, for all i < 0} and

U ⊥ [1] = {Y ∈ D(R) | HomD(R) (T [i], Y ) = 0, for all i > 0},

the pair (U, U ⊥ [1]) in D(R) is a t-structure whose heart is H = {Z ∈ D(R) | HomD(R) (T [i], Y ) = 0, for all i = 0}. Remark 4 If T is a 1-tilting module, the above t-structure coincides with the tstructure induced by the torsion pair associated to T as in the definition given by Happel et al. in [2]. In this case, the heart of the t-structure can be easily described: it consists of the complexes Z such that Hi (Z ) = 0 for i = 0, −1, H0 (Z ) is a torsion object, and H−1 (Z ) is torsion-free. Note that T is a projective generator of the abelian category H. Also, if T is a classical n-tilting module, then the t-structure is compactly generated (T is a compact object in D(R)) and the heart is a Grothendieck category, even a module category equivalent to Mod-EndH (T ). Our main concern is to answer the following: Question 5 When is the heart of the t-structure induced by an n-tilting module a Grothendieck category? For the 1-tilting case we have a complete answer.

4 The 1-Tilting Case Theorem 6 (Parra–Saorín, [5]) The heart H of the t-structure induced by a 1-tilting module T is a Grothendieck category if and only if the torsion-free class F in the torsion pair corresponding to T is closed under direct limits. Question 7 (Saorín) If T is a 1-tilting module such that the heart of the t-structure induced by T is Grothendieck, is then T equivalent to a classical tilting module? (Two tilting modules T, T  are equivalent if they define the same tilting class.) A first key result to help answering the above question is given by: Proposition 8 Let T be a 1-tilting module and let F be the associated torsion-free class. Then F is closed under direct limits if and only if T is a pure projective module, i.e., T is a direct summand of a direct sum of finitely presented modules.

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S. Bazzoni

Proposition 9 Let T be a pure projective 1-tilting module. Then, up to equivalence,  An with An finitely presented modules of p.d ≤ 1 and such that T ⊥ = {An }⊥ . T ≤ ⊕ n∈N

An important result about direct sum decompositions which is very useful in our context is the following famous result. Theorem 10 (Azumaya–Crawley-Jónsson–Warfield) If M = ⊕i∈I Mi , Mi indecomposable countably generated with local endomorphism ring, then any other decomposition of M refines to a decomposition isomorphic to this. In particular, any direct summand of M is a direct sum of modules, each isomorphic to one of the summands Mi . Applying the above theorem we can solve our problem in the case of a commutative ring R. Proposition 11 If R is a commutative ring and T is a pure projective 1-tilting module, then T is projective, thus equivalent to R. In general, the answer to Saorin’s question is negative. In fact, Proposition 12 There exist pure projective 1-tilting modules not equivalent to classical tilting modules. Sketch of the Proof Let R be a nearly simple uniserial domain, that is, a uniserial domain with only one non-trivial two-sided ideal, which is J (R). The ring for which there is a pure projective tilting module not equivalent to a classical one will be S = End(R/a R), for 0 = a ∈ J (R). If R is a nearly simple uniserial domain then, for 0 = a, b ∈ J (R), R/a R ∼ = R/b R, and, for every X = R/a R and every finitely generated A < X , A ∼ = =X∼ X/A. Now fix X , an exact sequence g

f

0 → X → X → X → 0, and let S = End R (X ). Then S has only three non-trivial two-sided ideals which are all idempotent, namely I (the ideal of non-monomorphisms), K (the ideal of nonepimorphisms), and J (the Jacobson radical, which coincides with I K ). Observe that (i) I S = gS is uniserial ( S K = S f is uniserial); (ii) S I is not finitely generated (K S is not finitely generated); (iii) S/I (S/K ) is a division ring; (iv) the simple left S-module S/I is flat, but not injective ( S S/K is injective, but not flat); and (v) the right multiplication by f and g induce an exact sequence ·

−g

·

−f

0 → S −→ I −→ J → 0.

(1)

The Heart of a t-Structure Induced by a n-Tilting Module

19

Thus, S I is pure in S and a Mittag-Leffler module (J = I K = I f ); the finitely presented S-modules are (finite) direct sums of S S, S K = S f ∼ = S/Sg, and S/K ; and p.d.K = 1, Sg ∼ = S. Theorem 13 The class T I = { S M | I M = M} = { S M | g M = M} = (S/Sg)⊥ is a tilting torsion class whose corresponding torsion-free class is F I = { S Y | I Y = 0} = (S/I )-Mod, which is also a torsion class. Furthermore, (i) F I is closed under direct limits (even epimorphic images); (ii) TI is not the tilting class of a finitely generated tilting module since the finitely presented left S-modules S, K are not in T I and p.d.S/K = 2; (iii) I is idempotent, hence (T I , F I , Z I ) is a TTF, that is a torsion torsion-free triple. If J (S) is countably generated (for instance, if R is countable) then I ⊕ J = T is a tilting module such that T ⊥ = T I . Moreover, I is countably generated and projective, J is pure projective, I has no finitely generated summands, and the sequence (1) is a special T I -pre-envelope of S. Remark 14 Recently, there have been important developments on the subject, since the existence of non-classical pure projective 1-tilting modules over two-sided Noetherian rings has been proved.

References 1. A.A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers. Astérisque 100 (1982) 2. D. Happel, I. Reiten, S.O. Smalø, Tilting in Abelian Categories and Quasitilted Algebras, Memoirs American Mathematical Society (1996) 3. M. Hovey, Model Categories, Mathematical Surveys and Monographs, vol. 63. American Mathematical Society (1999) 4. M. Hovey, Cotorsion pairs, model category structures, and representation theory. Math. Z. 241(3), 553–592 (2002) 5. C.E. Parra, M. Saorín, Direct limits in the heart of a t-structure: the case of a forsion pair. arXiv:1311.6166

On Some Local Cohomology Spectral Sequences Alberto F. Boix

Abstract The purpose of this report is to introduce a formalism to produce two collections of spectral sequences. On one hand, a collection is made up by spectral sequences which involve in their second page the left derived functors of the colimit on a certain finite poset. On the other hand, the other is made up by spectral sequences which involve in their second page the right derived functors of the limit on a certain finite poset. In both cases, we provide sufficient conditions to ensure their degeneration at the second page. Finally, we see how to use our second collection of spectral sequences to produce a decomposition of local cohomology modules which can be regarded as a generalization of the classical Hochster formula for the local cohomology of a Stanley–Reisner ring.

1 General Conventions and Notations The content of this report is based in a work in progress with Josep Àlvarez Montaner and Santiago Zarzuela. In what follows, A will denote a commutative Noetherian ring, A is the category of A-modules, and I ⊆ A is an ideal with primary decomposition given by I =  the poset P ∪ {0 P, 1 P} I1 ∩ · · · ∩ In . Moreover, given a poset P, we denote by P obtained by adding to P a minimal (resp., maximal) element 0 P (resp., 1 P). On the other hand, N will denote a finitely generated A-module with finite projective dimension. Finally, given a prime ideal p of A, E(A/p) will denote the injective hull of the residue field of A/p.

To Joaquina Cebrian Melguizo (1924–2015), my grandmother. A.F. Boix (B) Department of Economics and Business, Universitat Pompeu Fabra, Jaume I Building, Ramon Trias Fargas 25-27, 08005 Barcelona, Spain e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_4

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A.F. Boix

2 Homological Spectral Sequences The purpose of this part is to produce a formalism to construct spectral sequences involving in their second page the left derived functors of the colimit (for this reason, we refer to them as homological); moreover, we also provide sufficient conditions to guarantee their degeneration at this second page, mostly in the spirit of [1]. Throughout this section, P is the poset given by all the possible sums of the ideals Ii ’s ordered by reverse inclusion. × A Construction 1 Let P following requirements:

T[∗]

A be an additive bi-functor which verifies the

 T p is a covariant, left exact functor commuting with arbitrary (i) for any p ∈ P, direct sums; (ii) if p ≤ q then there exists a natural transformation of derived functors Ri Tq . Ri T p In addition, we also need to suppose that T[∗] verifies one (and only one) of the following two assumptions: (i) for any p ∈ Spec(A) and for any maximal ideal m of A, there exists an A-module  X such that, for any p ∈ P,  T p (E(A/p))m =

X, if p ∈ W(I p , J ) and p ⊆ m, 0, otherwise.

It must be mentioned that X may depend on p and m, but not on p. Moreover,   W(I p , J ) := q ∈ Spec(A) | I pn ⊆ q + J for some integer n ≥ 1 , and J is an ideal of A which does not depend on any of the previous choices; (ii) for any p ∈ Spec(A) and for any maximal ideal m of A, there exists an A-module  Y such that, for any p ∈ P,  T p (E(A/p))m =

Y, if p ∈ / W(I p , J ) and p ⊆ m, 0, otherwise.

It must be mentioned that Y may depend on p and m, but not on p. Once again, J is an ideal of A which does not depend on any of the previous choices. Next, we give examples of functors verifying the assumptions from Construction 1: (i) the generalized torsion functor    I p (N , −) := colimk∈N Hom A N /I pk N , − ;

On Some Local Cohomology Spectral Sequences

23

(ii) the generalized ideal transform   D I p (N , −) := colimk∈N Hom A I pk N , − ; (iii) the torsion functor with respect to pairs of ideals  I p ,J (M) := {x ∈ M | I p x ⊆ J x for some  ∈ N}. Next statement is the main result of this section. Theorem 2 Let M be any A-module. Then, there is an spectral sequence −i, j

E2

= Li colim p∈P R j T p (M)

i

R j−i T (M)

in the category of A-modules, where T := T1 P . Remark 3 When T p =  I p (N , −), the spectral sequence obtained in Theorem 2 recovers and extends the Mayer–Vietoris spectral sequence provided by Lyubeznik in [3, Theorem 2.1]. On the other hand, when T p =  I p ,J and I = I1 ∩ I2 , our spectral sequence boils down to the Mayer–Vietoris long exact sequence of local cohomology modules with respect to pairs of ideals which, at the best of our knowledge, had not been still established. A natural issue is whether the spectral sequence produced in Theorem 2 degenerates at its second page; in the next result, we provide sufficient conditions to ensure this fact. Theorem 4 Assume that A contains a field, and let M be an A-module such that, for any p ∈ P, R j T p (M) = 0 up to a unique value of j (namely, h p ), and that, for any p, q ∈ P with p = q, Hom A (Rh p T p (M), Rh q Tq (M)) = 0. Then, the spectral sequence obtained in Theorem 2 degenerates at the second page.

3 Cohomological Spectral Sequences Our goal now is to build spectral sequences which involve in their second page the right derived functors of the limit (for this reason, we refer to them as cohomological); as in the previous section, we also provide sufficient conditions to guarantee their degeneration at this second page. In this section, P will denote the poset given by all the possible quotients M/aM, where a runs over all the possible different sums of the ideals I1 , . . . , In , ordered by reverse inclusion, and Inv(P, A) is the category of contra-variant functors with source P and target A. Finally, J , K will denote ideals of A.

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A.F. Boix T

Construction 5 Let A A be a covariant, left exact functor commuting with direct sums. Building from T , we produce the following endo-functor on Inv(P, A): T

Inv(P, A) −→ Inv(P, A) M = (M p ) p∈P −→ T (M) := (T (M p )) p∈P . Moreover, suppose that T verifies one (and only one) of the following two assumptions: (i) for any p ∈ Spec(A) and for any maximal ideal m of A,  T (E (A/p))m =

X, if p ∈ W(J, K ) and p ⊆ m, 0, otherwise,

where X is an A-module which only depends on p and m; (ii) for any p ∈ Spec(A) and for any maximal ideal m of A,  T (E (A/p))m =

Y, if p ∈ / W(J, K ) and p ⊆ m, 0, otherwise,

where Y is an A-module which only depends on p and m. The assumptions established in Construction 5 are satisfied, for instance, in the following examples: (i) (ii) (iii) (iv)

the covariant hom functor Hom A (N , −); the generalized torsion functor  J (N , −); the generalized ideal transform functor D J (N , −); the torsion functor with respect to pairs of ideals  J,K .

Now, we are ready to establish the main result of this section. Theorem 6 If the inverse system A/[∗] := (A/I p ) p∈P is flasque and we have a natural equivalence of functors lim p∈P ◦T ∼ = T ◦ lim p∈P , then there exists a spectral sequence i, j Ri+ j T (A/I ) . E 2 = Ri lim R j T (A/[∗]) p∈P

i

Remark 7 It is worth noting that A/[∗] = (A/I p ) p∈P is flasque if, for instance, the ideal I defines an arrangement of linear varieties; another example is when I is a squarefree monomial ideal inside an affine, normal semigroup ring. Now, we are in position to state the main result about the degeneration of the spectral sequences of this section.

On Some Local Cohomology Spectral Sequences

25

Theorem 8 Preserving the assumptions of Theorem 6, suppose, in addition, that A ∼ contains a field K , and that there is a natural isomorphism A/I  = lim p∈P A/I p . Moreover, we also have to assume that, for any p ∈ P, R j T A/I p = 0 up to a dp dq single value of j (namely, d p ) and that, for any p = q, Hom A (R T A/I p , R T  A/Iq ) = 0. Then, there is a natural isomorphism Ri lim (R j T (A/[∗])) ∼ = p∈P



Rdq T (A/Iq )⊕m i ,

j=dq

i−1 ((q, 1 P); K ) and (q, 1 P) := { p ∈ P | q < p}. Furtherwhere m i := dim K H more, the spectral sequence obtained in Theorem 6 degenerates at the second page.

4 A Hochster Type Decomposition Now, suppose that A contains a field K , and that I1 + · · · + In is contained in a maxj imal ideal m of A; moreover, assume that, for any p ∈ P, Hm (A/I p ) = 0 for all j = dim(A/I p ) dim(A/Iq ) dim(A/I p ), that for any p = q ∈ P, Hom A (Hm (A/I p ), Hm (A/Iq )) = 0, and that A/[∗] is flasque. Under these assumptions, we obtain: Theorem 9 There is a K -vector space isomorphism Hmj (A/I ) ∼ =



dim(A/Iq )

Hm

(A/Iq )⊕m j .

q∈P

Sketch of the Proof Under our assumptions, the spectral sequence produced in Theorem 6 (in this case, T = m ) degenerates at its second page; therefore, this degeneration gives rise to a filtration whose successive quotients are given by the second page. In this way, the formula follows from the fact that all the possible short exact sequences given by this filtration split as K -vector spaces.  The reader will easily note that, when A is a polynomial ring, and I is a squarefree monomial ideal, the formula obtained in Theorem 9 is just the classical Hochster’s decomposition of the local cohomology of a Stanley–Reisner ring (see, among other places, [4, Theorem 13.13]); moreover, this formula partially recovers more general decompositions provided in Brun–Bruns–Römer [2, Theorems 1.1 and 1.3]. Acknowledgments The author is partially supported by MTM2013-40775-P.

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References 1. J. Àlvarez Montaner, R. García López, S. Zarzuela Armengou, Local cohomology, arrangements of subspaces and monomial ideals. Adv. Math. 174(1), 35–56 (2003) 2. M. Brun, W. Bruns, T. Römer, Cohomology of partially ordered sets and local cohomology of section rings. Adv. Math. 208(1), 210–235 (2007) 3. G. Lyubeznik, On some local cohomology modules. Adv. Math. 213(2), 621–643 (2007) 4. E. Miller, B. Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227 (Springer, New York, 2005)

Homotopy Representations of Classifying Spaces Natàlia Castellana

Abstract In the theory of compact Lie groups, the existence of faithful unitary representations for every compact Lie group is a consequence of the Peter–Weyl theorem. The existence of such representations imposes finiteness properties at the level of the cohomology of the classifying spaces. The proof involves analytic techniques which are not available for classifying spaces of homotopy theoretical structures such as pcompact groups and p-local finite groups. Understanding maps between classifying spaces is part of the program for developing an homotopy representation theory. In this paper I will describe progress made in this direction (joint work with L. Morales and J. Cantarero).

5.1 Interactions Between Algebraic Topology, Representation Theory and Commutative Algebra Let G be a finite group. The classifying space BG of G is a topological space which classifies principal G-bundles. This construction was introduced by Milnor in the 1950s. It is a connected topological space (in fact a C W -complex) with a contractible universal cover and π1 (BG) = G. The classifying space construction builds a bridge between algebraic topology, commutative algebra and representation theory as we will try to explain now. On the one hand, the cohomology of a finite group with coefficients in a commutative ring R is algebraically defined as H ∗ (G; R) := Ext∗RG (R, R), but on the other hand, there is a natural isomorphism H ∗ (G; R) ∼ = H ∗ (BG; R) with the singular or cellular cohomology of the classifying space. A relevant property from the point of view of commutative algebra is the fact that the cohomology of a finite group with coefficients in a Noetherian ring is a Noetherian graded commutative ring. Another important result, due to Cartan–Eilenberg [5] is the fact that the mod p cohomology is computed by stable elements. That is, H ∗ (G; F p ) ∼ = lim P≤G H ∗ (P; F p ) where N. Castellana (B) Universitat Autònoma de Barcelona, Barcelona, Spain e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_5

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the inverse limit is taken with respect to restrictions and conjugations by elements of G. The fact that H ∗ (G; F p ) is Noetherian was proven by Evens [10] algebraically, but it was also proved using classifying spaces and fibrations by Venkov [15]. The idea is to use the existence of a faithful complex representation G → SU (n) to produce a fibration BG → B SU (n) whose homotopy fiber SU (n)/G has the homotopy type of a finite C W -complex. The Serre spectral sequence shows then that H ∗ (BG; F p ) is a finitely generated module over H ∗ (B SU (n); F p ) which is a finitely generated polynomial algebra. From the point of view of commutative algebra, the program of understanding properties of H ∗ (G; R) from the group structure of G was initiated by Quillen. He showed in [14] that the Krull dimension of H ∗ (G; F p ) is the rank of the maximal elementary abelian p-subgroups of G (in fact, up to nilpotent elements H ∗ (G; F p ) is detected by all H ∗ (V ; F p ), where V ≤ G is an elementary abelian p-subgroup). In recent work of Dwyer–Greenlees–Iyengar [8], the authors develop commutative algebra in the context of homotopy theory and they unify different types of duality, including Poincaré duality and Benson–Carlson duality. They introduce the Gorenstein condition at the level of cochains (considered as DGA or S-algebras). In this context, given a space X , we say that R := C ∗ (X ; F p ) → F p is Gorenstein of shift a is H om R (F p , R) is equivalent to  a F p as a left F p -module plus some technical conditions. Theorem 1 (Dwyer–Greenlees–Iyengar [8]) Let G be a finite group. Then, the Salgebra C ∗ (BG; F p ) → F p is Gorenstein. The idea of the proof of Theorem 1 also uses the fibration introduced in the proof of the fact that the cohomology of G is Noetherian by Venkov, by introducing a new element: SU (n)/G is a finite Poincaré duality C W -complex. One immediate corollary of the previous theorem is the Benson–Carlson duality for the cohomology of finite groups. Naively, the fact that C ∗ (X ; F p ) → F p is Gorenstein implies the existence of a local cohomology spectral sequence. When H ∗ (BG; F p ) is Cohen– Macaulay, the spectral sequence degenerates at the E 2 term and the consequence is that the cohomology is also Gorenstein. Theorem 2 (Benson–Carlson duality [2]) Let G be a finite group and p a prime number. If H ∗ (G; F p ) is Cohen–Macaulay then it is also Gorenstein. Question 3 If H ∗ (G; F p ) is computed by stable elements using p-subgroups and conjugacy relations, can the Gorenstein property be also detected by using local information on p-subgroups and conjugacy relations? Representation theory of finite groups gets into the picture by looking at another cohomology theory, topological K -theory. This generalized cohomology theory is represented by the infinite loop space BU , the classifying space of the infinite unitary group. There is a natural homomorphism from the complex representation ring R(G) to K -theory, R(G) → K (BG).

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Theorem 4 (Atiyah [1]) Let G be a finite group, and let I be the augmentation ideal of R(G). There is an isomorphism R(G)∧I ∼ = K (BG). If X is a finite C W -complex then K (X ) is the Grothendieck group on complex vector bundles on X , but BG is not a finite C W -complex. In that case, the Grothedieck group of complex vector bundles, denoted by K(BG), was studied by Jackowski– Oliver showing that it has a description similar to the stable elements formula for mod p cohomology. Theorem 5 (Jackowski–Oilver [11]) There is a natural isomorphism ∼ =

K(BG) → lim R(P), P≤G

where the inverse limit is taken over all p-subgroups of G under restrictions and conjugations by elements of G. Since the right hand side of the isomorphism in Theorem 5 only depends on local information of the group, one could ask whether this is true for the proof, since the original proof makes a strong use of the theory of G-spaces.

5.2 Local Models for Classifying Spaces of Finite Groups Let p be a prime number. Let G be a finite group and S a fixed Sylow p-subgroup. The p-local structure of G formally consists of S together with the conjugacy relations between its subgroups induced by G. The notions of a fusion system and p-local structures in finite groups have been of interest since last century, having a starting point in the work of Burnside. Puig was the first, in the early 1970s, to consider the category F S (G) to encode the p-local structure of a finite group G. The category F S (G) is the category with objects the set of all subgroups of S, and the morphisms between two subgroups are given by conjugacy by an element of G. This notion was generalized to the concept of saturated fusion system by Puig [13] in the 1990s. Definition 6 Let S be a finite p-group. A saturated fusion system over S is a subcategory F of the category of groups with objects the set of all subgroups of S, and whose morphisms satisfy the following properties. For all P, Q ≤ S, (i) Hom S (P, Q) ⊂ HomF (P, Q) ⊂ Inj(P, Q); and (ii) each ϕ ∈ HomF (P, Q) is the composite of an isomorphism in F followed by an inclusion. For all P ≤ S and all P ≤ S which is F-conjugate to P (P and P are isomorphic as objects of F) we have the following:

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(i) for all P ≤ S which is fully normalized in F (|N S (P)| ≥ |C S (P )|), P is fully centralized in F (|C S (P)| ≥ |C S (P )|), and Out S (P) ∈ Syl p (Out F (P)); (ii) if P ≤ S and ϕ ∈ HomF (P, S) are such that ϕ(P) is fully centralized, setting Nϕ = {g ∈ N S (P) | ϕcg ϕ −1 ∈ Aut S (ϕ(P))}, ¯ P = ϕ. there is ϕ¯ ∈ HomF (Nϕ , S) such that ϕ| The standard example of a saturated fusion system is F S (G), where G is a finite group and S a fixed Sylow p-subgroup. Exotic examples of saturated fusion systems (that is, not of the form F S (G) for any finite group G) are described by several authors; see [7, 12]. If one wants to recover the classifying space at a prime p, one needs to generalize the association between F S (G) and BG ∧p , the Bousfield–Kan pcompletion of the classifying space. With this purpose, Broto–Levi–Oliver [3] define the notion of a centric linking associated to a saturated fusion system. Definition 7 Let F be a saturated fusion system over a finite p-group S. A centric linking system associated to F is a category L whose objects are the F-centric subgroups of S (i.e., the subgroups P ≤ S such that C S (P) = Z (P)), together with a functor π : L → F c , and “distinguished” monomorphisms δ P : P → Aut L (P) for each F-centric subgroup P ≤ S, satisfying the following conditions: (i) π is the identity on objects and surjective on morphisms. For each pair of objects P, Q ≤ L, Z (P) acts freely on Mor L (P, Q) by composition (upon identifying Z (P) with δ P (Z (P)) ≤ Aut L (P)), and π induces a bijection ∼ =

Mor L (P, Q)/Z (P) → HomF (P, Q). (ii) For each F-centric subgroup P ≤ S and each g ∈ P, π sends (δ P (g)) ∈ AutL (P) to cg ∈ Aut F (P). (iii) For each f ∈ Mor L (P, Q) and each g ∈ P, f ◦ δ P (g) = δ Q (π f (g)) ◦ f . Recently, in 2013, A. Chermak proved the existence and uniqueness of centric linking systems associated to saturated fusion systems. Theorem 8 (Chermak [6]) Let F be a saturated fusion system on a finite p-group S. Then there exists a centric linking system L associated to F. Furthermore, L is uniquely determined by F. Definition 9 The classifying space BF of a saturated fusion system (S, F) is the Bousfield–Kan p-completion of the nerve of the associated linking system |L|∧p . With this new setting we can now formulate in a more precise way the questions we addressed in the previous section.

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Question 10 We formulate the following questions: (i) Given a saturated fusion system (S, F), does Benson–Carlson duality hold for H ∗ (BF; F p )? that is, is C ∗ (BF; F p ) → F p Goresntein in the sense of Dwyer– Greenlees–Iyengar? (ii) Can we describe the Grothendieck group of vector bundles K(BF) using the representation rings of p-subgroups of S? The strategy to answer these questions is to construct maps BF → BU (n)∧p whose homotopy fiber has the desired properties: its mod p cohomology is a finite Poincaré duality algebra.

5.3 Homotopy Representations and Duality for p-local Finite Groups We start our program by introducing the notion of homotopy representation for a saturated fusion system. Definition 11 Let (S, F) be a saturated fusion system. A complex homotopy representation of F is a map f : BF → BU (n)∧p . We say f is faithful if the homotopy fiber of f has finite mod p cohomology. Restriction to S and to its subgroups using distinguished monomorphisms in the structure of the linking system, gives rise to a map [BF, BU (n)∧p ] −→ lim[B P, BU (n)∧p ] ∼ = lim Rep(P, U (n)) ⊂ Rep(S, U (n)). F

F

Due to Dywer–Zabrodsky [9], the classifying map functor for representations into unitary groups induces an isomorphism Rep(P, U (n)) ∼ = [B P, BU (n)∧p ] when P is a p-group. First, we want to describe the inverse limit of representations of subgroups, and then study the injectivity and exhaustivity of the resulting map. Definition 12 Let (S, F) be a fusion system. An n-dimensional complex representation ρ of S is fusion-preserving if for any P ≤ S and f ∈ HomF (P, S), ρ| P ∼ = ρ| f (P) ◦ f ∈ Rep(P, U (n)). Theorem 13 (Cantarero–Castellana–Morales [4]) Let (S, F) be a saturated fusion system. Let μr eg be the regular representation of S. Then, (i) the inverse limit limF Rep(P, U (n)) is the set of fusion preserving n-dimensional representations of S; (ii) if ρ : S → U (n) is a fusion preserving representation, there exists N ≥ 0 and f ∈ [BF, BU (M)∧p ], such that f restricted to B S is B(ρ ⊕ N μr eg ), where M = n + N |S|;

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(iii) given two maps f, g : |L|∧p → BU (n)∧p such that its restrictions to B S are homotopy equivalent, there exists N ≥ 0 such that f ⊕ N μr eg g ⊕ N μr eg as maps |L|∧p → BU (n + N |S|)∧p . Two important corollaries are derived from Theorem 13. Corollary 14 Let (S, F) be a saturated fusion system. Then, there is an isomorphism K(BF) ∼ = lim R(P). F

Since the regular representation is a faithful complex fusion preserving representation, we obtain the existence of faithful complex homotopy representations applying Theorem 13. Corollary 15 There exists a complex homotopy representation f : BF → BU (n)∧p whose restriction to S is a direct sum of copies of the regular representation, and such that the mod p cohomology of the fiber is a finite Poincaré duality algebra. In view of the results obtained in the previous section, if the homotopy fiber of f is a Poincaré duality algebra, then classifying spaces of saturated fusion systems satisfy Benson–Carlson duality.

References 1. M.F. Atiyah, Characters and cohomology of finite groups. Inst. Hautes Études Sci. Publ. Math. 9, 23–64 (1961) 2. D.J. Benson, J.F. Carlson, Projective resolutions and Poincaré duality complexes. Trans. Am. Math. Soc. 342(2), 447–488 (1994) 3. C. Broto, R. Levi, B. Oliver, The homotopy theory of fusion systems. J. Am. Math. Soc. 16(4), 779–856 (2003) 4. J. Cantarero, N. Castellana, L. Morales, Complex homotopy representations for fusion systems (in preparation) 5. H. Cartan, S. Eilenberg, Homological Algebra, Princeton Landmarks in Mathematics (Princeton University Press, Princeton, NJ, 1999) 6. A. Chermak, Fusion systems and localities. Acta Math. 211(1), 47–139 (2013) 7. A. Díaz, A. Ruiz, A. Viruel, All p-local finite groups of rank two for odd prim p. Trans. Am. Math. 359(4), 1725–1764 (2007) 8. W.G. Dwyer, J.P.C. Greenlees, S. Iyengar, Duality in algebra and topology. Adv. Math. 200, 357–402 (2006) 9. W. Dwyer, A. Zabrodsky, Maps between classifying spaces, in Algebraic Topology, Barcelona, 1986, Lecture Notes in Mathematics, vol. 1298, pp. 106–119 (Springer, 1987) 10. L. Evens, The Cohomology of Groups, Oxford Mathematical Monographs (The Clarendon Press, Oxford University Press, New York, 1991) 11. S. Jackowski, B. Oliver, Vector bundles over classifying spaces of compact Lie groups. Acta Mathematica 176, 109–143 (1996) 12. R. Levi, B. Oliver, Construction of 2-local finite groups of a type studied by Solomon and Benson. Geom. Topol. 6, 917–990 (2002) 13. L. Puig, Frobenius categories. J. Algebra 303, 309–357 (2006)

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14. D. Quillen, The spectrum of an equivariant cohomology ring, I, II. Ann. Math. 94 (2), 549–572 (1971); Ibid. 94, 573–602 (1971) 15. B. Venkov, Cohomology algebras for some classifying spaces. Dokl. Akad. Nauk. SSR 127, 943–944 (1959)

Decomposing Gorenstein Rings as Connected Sums Hariharan Ananthnarayan, Ela Celikbas and Zheng Yang

Abstract In 2012 Ananthnarayan, Avramov and Moore gave a new construction of Gorenstein rings. They defined a connected sum of two Gorenstein local rings as an appropriate quotient of their fibre product. Given a Gorenstein ring, one would like to know whether it can be decomposed as a connected sum and if so, what are its components. We answer these questions in the case of a Gorenstein Artin local algebra over a field.

In topology, amalgamating two manifolds near a chosen point on each creates another manifold, called a “connected sum”. This concept plays a significant role in the classification of closed surfaces. Connected sums in algebra are related to connected sums in topology through an expression of the cohomology algebras. The work of Sah in the ’70s is one of the earliest discussions of connected sums in commutative ring theory. Gorenstein rings, due to their various kinds of symmetries and duality properties, form an important and ubiquitous class of rings. In 2012, Ananthnarayan, Avramov and Moore introduce a new construction of Gorenstein rings, called a connected sum. For Cohen–Macaulay local rings R, S and k of the same dimension, and ring πS πR homomorphisms R −→ k ←− S, they first consider the fibre product (or pullback) R ×k S = {(r, s) ∈ R × S : π(r ) = π(s)} and then define a connected sum of R and S over k as an appropriate quotient of R ×k S; see [2]. In [3], we study connected sums over a field in the Artinian case: Definition 1 Let (R, m R , k) and (S, m S , k) be Gorenstein Artin local rings different from k. Let soc(R) = δ R , soc(S) = δ S , and identify δ R with (δ R , 0) and δ S with H. Ananthnarayan (B) Department of Mathematics, I.I.T. Bombay, Powai 400076, Mumbai, India e-mail: [email protected] E. Celikbas Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA e-mail: [email protected] Z. Yang Department of Mathematics, University of Nebraska, Lincoln, NE 68588, USA e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_6

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(0, δ S ). Let π R and π S be the natural projections from R and S respectively onto k. A connected sum of R and S over k, denoted R#k S, is the ring R#k S = (R ×k S)/δ R − δ S  where R ×k S = {(r, s) ∈ R × S : π R (r ) = π S (s)} is the fibre product of R and S over k. If neither of R and S in the definition above is a field, then their fibre product R ×k S is not Gorenstein. However, a connected sum R#k S of R and S is a Gorenstein ring. For more details, see [1, Chapter 4] and [2, Section 2]. Remark 2 Connected sums of R and S over k depend on the generators of the socle δ R and δ S chosen. For example, the connected sums Q 1 = (R ×k S)/y 2 − z 2  and Q 2 = (R ×k S)/y 2 − 5z 2  of R = Q[Y ]/Y 3  and S = Q[Z ]/Z 3  are not isomorphic as rings, as shown in [2, Ex. 3.1]. In [3], we give the following characterization of connected sums of k-algebras: Theorem 3 (Connected sums in equicharacteristic) Let Q be a Gorenstein Artin local k-algebra with (Q) ≥ 1, where (Q) := max{n : mn = 0} denotes the Loewy length of Q. For positive integers m and n, Y and Z denote the sets of indeterminates {Y1 , . . . , Ym } and {Z 1 , . . . , Z n } respectively, and Y · Z denotes {Yi Z j : 1 ≤ i ≤ m, 1 ≤ j ≤ n}. Then the following are equivalent: (a) Q can be decomposed nontrivially as a connected sum over k; (b) Q k[Y1 , . . . , Ym , Z 1 . . . , Z n ]/I Q for m, n ≥ 1 with Y · Z ⊂ I Q ⊂ Y, Z2 . If the above conditions are satisfied, then we can write Q R#k S, where R = k[Y]/I R , S = k[Z]/I S with I R = I Q ∩ k[Y], and I S = I Q ∩ k[Z]. Furthermore, the following holds: (i) λ(Q) = λ(R) + λ(S) − 2 and edim(Q) = edim(R) + edim(S), where λ(−) and μ(−) respectively denote the length and the minimal number of generators of the respective modules, and edim(−) denotes the embedding  dimension;  + 1, where (ii) for 0 < i < min{(S), (R)}, HQ (i) = H R (i) + HS (i) ≤ m+n 2 H(−) denotes the Hilbert function of the respective rings; (iii) I Q = I Re + I Se + Y · Z +  R −  S , where  R ∈ k[Y] and  S ∈ k[Z] are such that their respective images δ R ∈ R and δ S ∈ S generate the respective socles, and I Re and I Se denote the extension of I R and I S to k[Y, Z] via the natural inclusions; 1 and = P R1(t) + P S1(t) − (iv) μ(I Q ) = μ(I R ) + μ(I S ) + mn + φ(m, n), P Q (t) 1 − φ(m, n)t 2 , where P(−) (t) denotes the Poincar´e series of the respective rings, and φ(m, n) is 1 when m, n ≥ 2, φ(1, 1) = −1, and φ(m, n) = 0 otherwise; (v) if (R) > (S) ≥ 2, gr(Q) gr(R) ×k gr (S/ soc(S)) where gr(−) denotes the associated graded ring of the corresponding rings. From now on, assume notations as in Theorem 3. We say that a Gorenstein Artin ring (Q, m, k) decomposes as a connected sum over k if Q R#k S for some Gorenstein Artin local rings R and S such that R  Q  S.

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In this case, we call R and S the components in a connected sum decomposition of Q, and Q R#k S a non-trivial decomposition. Every Gorenstein Artin local ring, which is not a field, is trivially a connected sum over its residue field. That is, if (R, m R , k) is a Gorenstein Artin local ring with (R) ≥ 1 and S is a k-algebra of length two (this forces S to be Gorenstein), then one can check that R#k S R. If Q cannot be decomposed as a connected sum over k, we say that Q is indecomposable as a connected sum over k. If R and S are Gorenstein Artin local rings with (R), (S) ≥ 2, then Q = R#k S is a non-trivial decomposition of Q as a connected sum over k. One would like to know whether a Gorenstein Artin ring Q can be decomposed as a connected sum over its residue field. Smith–Stong in [7, Section 4] studied this question from a geometric point of view for projective bundle ideals. Buczy´nska– Buczy´nski–Kleppe–Teitler [4] also studied the same problem in terms of inverse systems using polynomials that are direct sums and corresponding apolar Gorenstein algebras. In [3], we give the following results that provide some conditions under which Q is indecomposable as a connected sum: Theorem 4 A Gorenstein Artin k-algebra (Q, m Q , k) is indecomposable as a connected sum over k when one of the following holds: (i) edim(Q) = 3 and μ(I Q ) = 5; (ii) edim(Q) = 4 and μ(I Q ) is an even number; (iii) edim(Q) ≥ 3 and Q is a complete intersection ring. Theorem 5 Let Q be a Gorenstein Artin local k-algebra with edim(Q) = d. If, for  some i ≥ 2, HQ (i) ≥ d−2+i + 2 then Q is indecomposable as a connected sum i over k. Example 6 The following examples (i) and (ii) show that Theorem 4 does not give necessary conditions for indecomposibility. (i) Let Q Q[Y1 , Y2 , Y3 ]/I Q , where I Q = Y14 , Y24 , Y34 , Y12 Y22 − Y12 Y32 , Y12 Y22 − Y22 Y32. Then Q is Gorenstein with d = edim(Q) = 3. Furthermore, HQ (2) = 6 ≥ d2 + 2 hence, by Theorem 5, Q is indecomposable as connected sum over Q. However, μ(I Q ) = 5. (ii) Let Q = Q[Y1 , Y2 , Y3 , Y4 ]/I Q , where I Q = Y14 , Y24 , Y34 , Y44 , Y12 Y22 − Yi2 Y j2 : 1 ≤ i < j ≤ 4. Then Q is Gorenstein, d = edim(Q) = 4, μ(I Q ) = 9 is odd, and Q is indecomposable as a connected sum since HQ (2) = 10 ≥ d2 + 2. 2 3 2 (iii) Let Q = Q[X 1 , X 2 ]/X d 1 X 2 , X 1 − X 2 . Then Q is Gorenstein, d = edim(Q) = 2 and HQ (2) = 2 < 2 + 2. However, Q is indecomposable as a connected sum over Q, which can be seen as follows: Suppose Q R#Q S for some Gorenstein Artin Q-algebras R and S. Then, by Theorem 3, one can find elements Y and Z in X 1 , X 2  \ X 1 , X 2 2 such that Y · Z ∈ I Q and Y, Z  = X 1 , X 2 . Write Y = a1 X 1 + a2 X 2 + F and Z = b1 X 1 +

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b2 X 2 + G, where a1 , a2 , b1 , b2 ∈ Q, and F, G ∈ X 1 , X 2 2 . Then Y · Z ∈ I Q implies that Y Z + cX 22 ∈ X 1 , X 2 3 for some c ∈ Q. If c = 0, then Y Z ∈ X 1 , X 2 3 gives a contradiction, and c = 0 forces a1 = 0 = b1 , contradicting Y, Z  = X 1 , X 2 .   Remark 7 Example 6(iii) shows that the condition HQ (2) ≥ d2 + 2 is not necessary for indecomposibility. A Gorenstein Artin local ring (Q, m Q , k) is almost stretched if μ(m2Q ) ≤ 2. In a private conversation, Paolo Mantero asked if almost stretched rings are decomposable as connected sums. Example 6(iii) also shows that Q is almost stretched, but is indecomposable as a connected sum over k. A Gorenstein Artin k-algebra Q is said to be compressed if it has a maximum possible Hilbert function given the embedding dimension d and Loewy length s, i.e.,  d+s−i−1 , s−i }. if the Hilbert function of Q is HQ (i) = min{ d+i−1 i Corollary 8 If Q is a compressed Gorenstein k-algebra with (Q) ≥ 4, then Q is indecomposable as a connected sum over k. We also investigate the connections between associated graded rings and connected sums. Theorem 10, contained in [3], is our main result relating associated graded rings with connected sums. It shows conditions on the associated graded ring of a Gorenstein Artin k-algebra which force it to be decomposable as a connected sum. Definition 9 We say that a graded Artinian k-algebra G is Gorenstein up to linear socle if the socle of G in degree two and higher is a one-dimensional vector space over k, i.e., dimk (soc(G) ∩ (G + )2 ) = 1. Theorem 10 Let (Q, m Q , k) be a Gorenstein Artin k-algebra and set s = (Q). If G = gr(Q) is Gorenstein up to linear socle and s ≥ 3, then there exist Gorenstein Artin local k-algebras R and S such that Q R#k S, and where (i) edim(S) = type(G) − 1 and (S) ≤ 2; moreover, if G is not Gorenstein, then (S) = 2; (ii) gr(R) G/soc(G) ∩ G 1 ; in particular, (R) = s and H R (i) = HQ (i) for 2 ≤ i ≤ s; (iii) if m = edim(R) and n = edim(S), then [P R (t)]−1 = [P Q (t)]−1 − [P S (t)]−1 + 1 + φ(m, n)t 2 , where φ(m, n) is given as in Theorem 3(iv) and, P S (t) is as in [3, Remark 1.6.c]. Thus, P Q (t) is rational in t if and only if P R (t) is so. In particular, if G is not Gorenstein then Q is decomposable as a connected sum. Sally [6] proves a structure theorem for stretched Gorenstein rings and Elias– Rossi [5] give a similar structure theorem for short Gorenstein rings under some assumptions on the residue field. When such a short or stretched Gorenstein ring Q is an algebra over a field, these structure theorems show that Q can be written as a

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connected sum of a graded Gorenstein Artin ring R with the same Loewy length as Q, and a Gorenstein Artin ring S with Loewy length less than three. In either case, using a construction of Iarrobino, we see that the associated graded ring of Q has the property that, in degrees two or higher, its socle is one-dimensional, that is, gr(Q) is Gorenstein up to linear socle. Thus we use Theorem 10 to give applications to short and stretched Gorenstein k-algebras. In particular, we show that these rings, when they are not graded, are non-trivial connected sums, and derive some consequences without any restrictions on the residue field. Theorem 11 Let (Q, m Q , k) be a Gorenstein Artin k-algebra with (Q) ≥ 3. Then Q is stretched if and only if G = gr(Q) is Gorenstein up to linear socle and type(G) = edim(Q). In particular, Q is a connected sum with [P Q (t)]−1 = 1 − edim(Q)t + t 2 when edim(Q) ≥ 2. Theorem 12 Let (Q, m Q , k) be a short Gorenstein Artin k-algebra with Hilbert function HQ = (1, h, n, 1). Then Q is a connected sum. Furthermore, if n ≤ 4, then P Q (t) is rational.

References 1. H. Ananthnarayan, Approximating Artinian rings by Gorenstein rings and three-standardness of the maximal ideal. Ph.D. Thesis, University of Kansas (2009) 2. H. Ananthnarayan, L.L. Avramov, W.F. Moore, Connected sums of Gorenstein local rings. J. Reine Angew. Math. 667, 149–176 (2012) 3. H. Ananthnarayan, E. Celikbas, Z. Yang, Decomposing Gorenstein rings as connected sums (Preprint) 4. W. Buczy´nska, J. Buczy´nski, J. Kleppe, Z. Teitler, Apolarity and direct sum decomposability of polynomials. Preprint available at arXiv:1307.3314v2 5. J. Elias, M.E. Rossi, Isomorphism classes of short Gorenstein local rings via Macaulay’s inverse system. Trans. Amer. Math. Soc. 364, 4589–4604 (2012) 6. J. Sally, Stretched Gorenstein rings. J. London Math. Soc. 20(2), 19–26 (1979) 7. L. Smith, R.E. Stong, Projective bundle ideals and Poincaré duality algebras. J. Pure Appl. Algebra 215, 609–627 (2011)

Rigid and Test Modules Olgur Celikbas

Abstract We discuss a class of modules, called test modules, that are, in principle, those detecting finite homological dimensions. The main purpose of this mote is to report the following result: if a commutative Noetherian local ring (R, m, k) admits a test module (e.g., the residue field k) of finite Gorenstein dimension, then R is Gorenstein. This extended abstract is based on joint works with Celikbas et al. (Kyoto J Math 54:295–310, 2014) [9], Celikbas et al. (Homological dimensions of rigid modules) [10], and Celikbas and Wagstaff (Testing for the Gorenstein property) [11].

1 Definition and Motivation Throughout, R is a commutative Noetherian local ring with unique maximal ideal m and residue field k. With modR we denote the category of all finitely generated R-modules. For an ideal I of R, we set I to be the integral closure of I . Theorem 1 (Auslander–Buchsbaum–Serre; see [7]) If pd(k) < ∞, then R is regular. Theorem 2 (Burch [8]) Let I be an ideal of R such that I = I and depth(R/I ) = 0. R If M ∈ mod(R) and Tor nR (M, R/I ) = Tor n+1 (M, R/I ) = 0 for some n ≥ 0, then pd(M) ≤ n. In particular, if pd(R/I ) < ∞, then R is regular. Theorem 3 (Corso–Huneke–Katz–Vasconcelos [14]) Let I be an m-primary ideal of R such that I = I . If M ∈ mod(R) and Tor nR (M, R/I ) = 0 for some n ≥ 1, then pd(M) ≤ n − 1. Motivated by Theorems 3 and 11, we give the following definition: Definition 4 (Celikbas–Dao–Takahashi [9]) Let M ∈ mod(R). Then M is called a pd-test module (test module as defined in [9]) if, whenever N ∈ mod(R) and O. Celikbas (B) Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, USA e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_7

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pd(N ) = ∞, we have that ToriR (M, N ) = 0 for infinitely many i. In other words, if N ∈ modR and ToriR (M, N ) = 0 for all i  0, then pd(N ) < ∞. Remark 5 Similar definitions of test modules were given and studied by Jotthilingham [21], Majadas [22] and Ramras [23], but those are different from Definition 4. Notation 6 T(R) = {M ∈ (R) : M is pd − test over R}.

2 Some Examples of Test Modules Here, we record some examples of test modules; see [10, Appendix A] for further ones. Example 7 If I is an m-primary ideal of R such that I = I (e.g., I = m), then R/I ∈ T(R); see Theorem 3. The following example may be deduced from [19, 3.5]: Example 8 Let R = k[[x, y, z]]/(x 2 − y 2 , x 2 − z 2 , x y, x z, yz). Then M ∈ T(R) if and only if pd(M) = ∞, i.e., M is not free. The complexity of M, is defined as cx(M) = inf{r ∈ N ∪ {0} | ∃A ∈ R such that βnR (M) ≤ A · n r −1 for all n  0}. Here, βnR (M) denotes the n-th Betti number of M. Note that pd(M) < ∞ if and only if cx(M) = 0. In general, complexity may not be finite, but it cannot exceed the codimension of R in case R is a complete intersection ring; see [4]. Theorem 9 (i) (Huneke–Wiegand [20]) Let R be a hypersurface, e.g., let R = k[[x1 , . . . , xn ]]/( f ) where 0 = f ∈ (x1 , . . . , xn )2 . Then, M ∈ T(R) if and only if pd(M) = ∞, i.e., cx(M) = 1. (ii) (Avramov–Buchweitz [5]) Let R be a complete intersection of codimension c ≥ 1. Then, M ∈ T(R) if and only if cx(M) = c. Example 10 If R = k[[x, y, z, u]]/(x y, zu) and M = R/(y, u), then cx(M) = 2 so M ∈ T(R).

3 Characterizations of Gorenstein Rings in Terms of Test Modules The Gorenstein dimension G-dim(M) of M cannot exceed the projective dimension of M, i.e., G-dim(M) ≤ pd(M), with equality if pd(M) < ∞; see [2] for details. Similar to Theorem 1, there is a characterization of Gorenstein rings in terms of Gorenstein dimension of the residue field:

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Theorem 11 (Auslander–Bridger [3]) If G-dim(k) < ∞, then R is Gorenstein. Theorem 11 has been improved under some mild conditions; recall that the maximal ideal m of R is an example of an integrally closed m-primary ideal. Theorem 12 (Goto–Hayasaka [16]) Let I be an m-primary ideal of R such that I = I . Assume R satisfies Serre’s condition (S1 ), or I contains a NZD of R (e.g., R is a domain). If G-dim(R/I ) < ∞, then R is Gorenstein. If I is an m-primary ideal such that I = I and G-dim(R/I ) < ∞ then, under some mild conditions, R is Gorenstein; see Example 7 and Theorem 12. This raises the following question: Question 13 (Celikbas–Dao–Takahashi [9]) If 0 = M ∈ T(R) and G-dim(M) < ∞, then must R be Gorenstein? Theorem 14 (Celikbas–Dao–Takahashi [9]) Let R be a commutative Noetherian ring (not necessarily local) with a dualizing complex, and let 0 = M, N ∈ modR. Assume M ∈ T(R) and that ExtiR (M, N ) = 0 for all i  0. Then, (i) id(N ) < ∞; (ii) if Supp(N ) = Spec(R), then R is Cohen-Macaulay; (iii) if Supp(N ) = Spec(R) and pd(N ) < ∞, then R is Gorenstein. Corollary 15 Assume R has a dualizing complex, e.g., R is complete (with respect to the m-adic topology). Assume 0 = M ∈ T(R) and G-dim(M) < ∞. Then R is Gorenstein. Proof It follows that ExtiR (M, R) = 0 for all i  0 and hence, id(R) < ∞ by Theorem 14.  Theorem 16 (Celikbas–Wagstaff [11]) Let (R, m) → (S, n) be a local flat ring homomorphism, i.e., S is a commutative Noetherian local ring that is flat as an Rmodule via the map f , and the closed fiber S/mS is nontrivial. Assume the induced map R/m → S/mS is a finite field extension, i.e., mS = n and the induced map R/m → S/mS is finite. If M ∈ T(R), then S ⊗ R M ∈ T(S). Here is an example indicating that the hypothesis mS = n is essential in Theorem 16. Example 17 Let R = k[[x]] → S = k[[x, y]]/(y 2 ). This is a finite free map since S is free over R with R-basis {1, y}. Let M = R/x R. Then M ∈ T(R) but N = / T(S) since pd S (N ) = 1 and S is not regular. M ⊗ R S = S/x S ∈ The next corollary gives an affirmative answer to Question 13. Corollary 18 (Celikbas–Wagstaff [11]) Let 0 = M ∈ T(R), e.g., M is a syzygy module of R/I for an m-primary ideal I with I = I . If G-dim(M) < ∞, then R is Gorenstein.

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 ⊗ R M ∈ T( R),  where R  is the m-adic comProof We have, by Theorem 16, that R  ⊗ R M) = G-dim(M) < ∞, it follows from Corollary pletion of R. Since G-dim R( R  is Gorenstein. Therefore, R is Gorenstein. 15 that R  Remark 19 Definition 4 was extended to homologically bounded complexes of Rmodules. Moreover, Corollary 18 was improved significantly by obtaining the same conclusion for G-dim test complexes; see [11] for details.

4 Rigid-Test Modules Definition 20 (Auslander [1]) A module M ∈ modR is called rigid if, whenever n is a nonnegative integer, N ∈ modR and Tor nR (M, N ) = 0, we have that R (M, N ) = 0. Tor n+1 Example 21 Let M ∈ modR. If pd(M) = 1, or pd(M) = 2 and M is torsion, then M is rigid; see, for example, [12]. Definition 22 (Celikbas–Gheibi–Zargar–Sadeghi [10]) A module M ∈ T(R) is called rigid-test provided that M is rigid. The next result shows that rigid-test modules can be used, instead of the residue field k, to detect the projective dimension: Theorem 23 (Celikbas–Gheibi–Zargar–Sadeghi [10]) Let 0 = M, N ∈ modR. Assume N is a rigid-test module (e.g., N = k). Then, (i) if Ext nR (M, N ) = 0 for some n ≥ depth(N ), then pd(M) ≤ n − 1; (ii) pd(M) = sup{i ∈ Z : ExtiR (M, N ) = 0}; (iii) if pd(N ) < ∞ or id(N ) < ∞, then R is regular. Rigidity assumption is essential for Theorem 23: Example 24 (Celikbas–Gheibi–Sadeghi–Zargar [10]) Let R = k[[x, y, z]]/(y 2 − x z, x 2 y − z 2 , x 3 − yz), M = m and N be the canonical module of R. Then N ∈ T(R) but N is not rigid-test. Moreover, ExtiR (M, N ) = 0 for all i ≥ 1 and pd(M) = ∞. The Gorenstein injective dimension Gid (see [15]) is a refinement of the usual injective dimension: if M ∈ modR, then Gid(M) ≤ id(M), with equality provided that id(M) < ∞. It is known that, if either Gid(R) or Gid(k) is finite, then R is Gorenstein; see [13, 18]. As an application of Theorem 23, we show that the Gorensteinness of R is also determined by the finiteness of the Gorenstein injective dimension Gid(m) of m. Corollary 25 (Celikbas–Gheibi–Sadeghi–Zargar [10]) Let I be an m-primary ideal of R such that I = I (e.g., I = m). Assume Gid(I ) < ∞. Then R is Gorenstein.

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Proof It follows from [24, 1.3] that R is Cohen–Macaulay. Pick 0 = X ∈ modR such that id(X ) < ∞. Then, by [17, 2.22], we have ExtiR (X, I ) = 0 for all i  0. Now it follows from Theorems 3 and 23 that pd(X ) < ∞. Hence, R is Gorenstein by a result of Foxby; see [7].  Question 26 Let 0 = M be a rigid-test module. Assume Gid(M) < ∞. Then must R be Gorenstein? The complete intersection dimension CI-dim (see [6]) is a refinement of the usual projective dimension: if M ∈ modR, then CI-dim(M) ≤ pd(M), with equality provided that pd(M) < ∞. Moreover, if CI-dim(k) < ∞, then R is a complete intersection ring; see [6]. Hence we ask: Question 27 Let 0 = M ∈ T(R). Assume CI-dim(M) < ∞. Then must R be a complete intersection ring? What if M is a rigid-test module? Acknowledgements We would like to thank Ryo Takahashi for his valuable comments and for explaining us Example 17.

References 1. M. Auslander, Modules over unramified regular local rings. Illinois J. Math. 5, 631–647 (1961) 2. M. Auslander, Anneaux de Gorenstein, et torsion en algèbre commutative. Séminaire d’Algèbre Commutative dirigé par Pierre Samuel, vol. 1966/67 (Secrétariat mathématique, Paris, 1967) 3. M. Auslander, M. Bridger, Stable module theory. Memoirs Am. Math. Soc. 94, (1969) (American Mathematical Society, Providence, RI) 4. L.L. Avramov, Modules of finite virtual projective dimension. Invent. Math. 96(1), 71–101 (1989) 5. L.L. Avramov, R.O. Buchweitz, Support varieties and cohomology over complete intersections. Invent. Math. 142(2), 285–318 (2000) 6. L.L. Avramov, V. Gasharov, I. Peeva, Complete intersection dimension. Inst. Hautes Études Sci. Publ. Math. 86, 67–114 (1998) 7. W. Bruns, J. Herzog, Cohen–Macaulay rings. Cambridge Studies in Advanced Mathematics, vol. 39 (Cambridge University Press, Cambridge, 1993) 8. L. Burch, On ideals of finite homological dimension in local rings. Proc. Cambridge Philos. Soc. 64, 941–948 (1968) 9. O. Celikbas, H. Dao, R. Takahashi, Modules that detect finite homological dimensions. Kyoto J. Math. 54(2), 295–310 (2014) 10. O. Celikbas, M. Gheibi, A. Sadeghi, M.R. Zargar, Homological dimensions of rigid modules. Preprint available at arXiv:1405.5188 11. O. Celikbas, S.S. Wagstaff, Testing for the Gorenstein property. Preprint available at arXiv:1504.08014 12. O. Celikbas, R. Wiegand, Vanishing of Tor, and why we care about it. J. Pure Appl. Algebra 219(3), 429–448 (2015) 13. L.W. Christensen, Gorenstein dimensions, vol. 1747. Lecture Notes in Mathematics (Springer, Berlin, 2000) 14. A. Corso, C. Huneke, D. Katz, W.V. Vasconcelos, Integral closure of ideals and annihilators of homology. Commutative algebra, Lecture Notes Pure Application Mathematics, vol. 244 (Chapman & Hall/CRC, Boca Raton, FL, 2006), pp. 33–48

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15. E.E. Enochs, O.M.G. Jenda, Gorenstein injective and projective modules. Math. Z. 220(4), 611–633 (1995) 16. S. Goto, F. Hayasaka, Finite homological dimension and primes associated to integrally closed ideals. Proc. Am. Math. Soc. 130(11), 3159–3164 (2002) (electronic) 17. H. Holm, Gorenstein homological dimensions. J. Pure Appl. Algebra 189(1–3), 167–193 (2004) 18. H. Holm, Rings with finite Gorenstein injective dimension. Proc. Am. Math. Soc. 132(5), 1279–1283 (2004) (electronic) 19. C. Huneke, D.A. Jorgensen, Symmetry in the vanishing of Ext over Gorenstein rings. Math. Scand. 93(2), 161–184 (2003) 20. C. Huneke, R. Wiegand, Tensor products of modules and the rigidity of Tor. Math. Ann. 299(3), 449–476 (1994) 21. P. Jothilingam, Test modules for projectivity. Proc. Am. Math. Soc. 94(4), 593–596 (1985) 22. J. Majadas, On test modules for flat dimension. J. Algebra Appl. 13(3), 1350107, 6 pp (2014) 23. M. Ramras, On the vanishing of Ext. Proc. Am. Math. Soc. 27, 457–462 (1971) 24. S. Yassemi, A generalization of a theorem of Bass. Comm. Algebra 35(1), 249–251 (2007)

Tate Homology Beyond Gorenstein Rings Lars Winther Christensen

Abstract Tate homology, originally defined for modules over group algebras, has a straightforward generalization to Iwanaga–Gorenstein rings, and a far-reaching generalization to associative rings. We report on progress in understanding the latter.

1 Introduction The theories of Tate homology and Tate cohomology go back to the early 1950s, and they were originally introduced as (co)homology theories for (modules over) group algebras. The underlying construction has since evolved through a series of generalizations to yield a theory for Iwanaga–Gorenstein rings; that is, Noetherian rings with finite self-injective dimension on either side. This process was started by Tadasi Nakayama [13] already in 1957, and the most recent developments, due to Avramov–Martsinkovsky [3], Veliche [14], and Iacob [12], date from the 2000s. Here is the central piece of technology. Definition 1 A complex T of projective right R-modules is called totally acyclic if one has H(T ) = 0 = H(Hom R (T, P)) for every projective right R-module P. A  π → M, where T is complete resolution of a right R-module M is a diagram T −→ P − a totally acyclic complex of projective right R-modules, π is a projective resolution, and i is an isomorphism for i  0. It is not evident, but every module over an Iwanaga–Gorenstein ring has a complete resolution [9, 14] (and that characterizes these rings). The next definition thus defines Tate (co)homology for every pair of modules over an Iwanaga–Gorenstein ring. Definition 2 Let M be a right R-module with a complete resolution T → P → M. For a left R-module N , Tate homology  Tor ∗R (M, N ) is the homology of the complex T ⊗ R N , and for a right R-module N , Tate cohomology Ext ∗R (M, N ) is the cohomology of the complex Hom R (T, N ). L.W. Christensen (B) Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_8

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It is not only possible to take Tate homology beyond group algebras, it is also useful. Here is an example due to Christensen–Jorgensen [7]. (A commutative local ring is Iwanaga–Gorenstein if and only if it is Gorenstein in the commutative algebra sense.) Theorem 3 Let R be a commutative Gorenstein local ring. For R-modules M and N with Tor ∗R (M, N ) = 0 one has depth R (M ⊗LR N ) = depth R M + depth R N − depth R. The symbol M ⊗LR N denotes the derived tensor product of M and N . It can be computed as P ⊗ R N , where P → M is a projective resolution. The homology of M ⊗LR N is Tor ∗R (M, N ), and the theorem thus generalizes Auslander’s [1] depth formula depth R (M ⊗ R N ) = depth R M + depth R N − depth R for modules R (M, N ) = 0. with Tor >0

2 Stabilization of (Co)Homology A view of Tate (co)homology as a stabilization of ordinary (co)homology emerged in the 1980 s and early 1990 s in a work of P. Vogel (published by Goichot [10]) and in the work Benson–Carlson [5]. The stable cohomology Ext ∗R (M, N ) of a pair of right R-modules is computed as follows: let P → M and Q → N be projective resolutions. The total Hom-complex Hom R (P, Q) is the product totalization of the double complex (Hom R (Pi , Q j ))i, j≥0 . The direct sum totalization of the same double complex yields a subcomplex of Hom R (P, Q); the quotient complex is denoted  H om R (P, Q), and its cohomology is stable cohomology Ext ∗R (M, N ). It was proved by Cornick–Kropholler [8] that there is an isomorphism  Ext ∗R (M, −) ∼ Ext ∗R (M, −) = whenever Tate cohomology  Ext ∗R (M, −) is defined, i.e., M has a complete resolution. This establishes stable cohomology as a wide-ranging generalization of Tate cohomology, available over every associative ring. Moreover, a detailed study by Avramov–Veliche [4] of stable cohomology over commutative local rings has shown that the theory carries useful information beyond the setting of Gorenstein rings. Now, what about the homological side? A theory of stable homology, also due to P. Vogel, is included in [10]. Here is the definition. Definition 4 Let M be a right R-module and N be a left R-module. Let P → M be a projective resolution and let N → I be an injective resolution. The tensor product P ⊗ R I is the direct sum totalization of the double complex (Pi ⊗ R I j )i, j≥0 . It is a  R I of the same double complex, and the subcomplex of the product totalization P ⊗ homology of the quotient complex is stable homology Tor ∗R (M, N ).

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While it is proved in [10] that stable homology, indeed, coincides with Tate homology over group algebras, little has been known about the general stable homology theory. The purpose of this note is to report on progress in this direction that has been achieved in the recent work Celikbas–Christensen–Liang–Piepmeyer [6].

3 Stable Homology To simplify the statements, R is now assumed to be Noetherian (on either side) and all modules are tacitly assumed to be finitely generated. First of all, stable homology agrees with Tate homology whenever the latter is defined. Theorem 5 Let M be a right R-module with a complete resolution. For every i ∈ Z the stable homology  ToriR (M, −) is naturally isomorphic to Tate homology  ToriR (M, −). One expects a homology theory to detect finiteness of homological dimensions. The next two results reflect the asymmetry in the definition of stable homology. Proposition 6 Let R be an Artin algebra or commutative and local. For a right R-module M, the following conditions are equivalent: (a) (b) (c)

M has finite projective dimension;  ToriR (M, −) = 0 for all i ∈ Z; there is an i ≥ 0 with ToriR (M, −) = 0.

Proposition 7 Let R be an Artin algebra or commutative and local. For a left Rmodule N , the following conditions are equivalent: (a) (b) (c)

N has finite injective dimension;  ToriR (−, N ) = 0 for all i ∈ Z; there is an i ≤ 0 with ToriR (−, N ) = 0.

It is evident from these two vanishing results that stable homology cannot be balanced in the way Tor is balanced. Theorem 8 Let R be an Artin algebra or commutative. The following conditions on R are equivalent: (a) R is Iwanaga–Gorenstein; (b) for all right R-modules M, all left R-modules N , and all i ∈ Z there are isomor◦ phisms ToriR (M, N ) ∼ ToriR (N , M) (here, R ◦ denotes the opposite ring of R). = Stable homology also detects commutative Gorenstein rings in a different, and perhaps surprising, way.

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Theorem 9 Let R be commutative. The following conditions are equivalent: (a) the local ring Rp is Gorenstein for every prime ideal p in R; (b) for every R-module M one has Tor ∗R (M, R) = 0. The proof showcases one way in which to extract information from stable homology, or rather, vanishing of stable homology. Proof By a theorem of Goto [11], the ring R is Gorenstein at every prime ideal if and only if every R-module M has a complete resolution T → P → M. A result of Foxby, published in [2], implies that M has a complete resolution if (and only if) there is an isomorphism M ∼ = RHom R (RHom R (M, R), R) in the derived category over R. (a) =⇒ (b): Let M be an R-module, by Goto’s theorem it has a complete resolution T → P → M, so by Theorem 5 one has  Tor ∗R (M, R) = H(T ⊗ R R) ∼ Tor ∗R (M, R) ∼ = = H(T ) = 0 . (b) =⇒ (a): Let M be an R-module; by assumption one has Tor ∗R (M, R) = 0. Let P → M be a degree-wise finitely generated projective resolution and let R → I be an injective resolution. In the derived category there are isomorphisms M∼ = P ⊗R I ∼ = P ⊗ R Hom R (R, I ) . = P ⊗R R ∼ =P∼ There is a natural morphism of complexes P ⊗ R Hom R (R, I ) −→ Hom R (Hom R (P, R), I ) , and the right-hand complex is isomorphic to RHom R (RHom R (M, R), R) in the derived category. However, the natural morphism is not invertible; that comes down to the left-hand complex being a direct sum totalization as opposed to the right-hand complex which is a product totalization. Now, the assumption   R I in the derived category, and one has Tor ∗R (M, R) = 0 yields P ⊗ R I ∼ =P⊗ ∼ ∼   where the last P ⊗ R I = P ⊗ R Hom R (R, I ) = Hom R (Hom R (P, R), I ), isomorphism holds as both complexes are now product totalizations. Therefore, for every R-module M we deduce that, in the derived category, M∼ = RHom R (RHom R (M, R), R) and it follows from the works of Foxby and Goto that R is Gorenstein at every prime. 

References 1. M. Auslander, Modules over unramified regular local rings. Illinois J. Math. 5, 631–647 (1961) 2. L.L. Avramov, S.B. Iyengar, J. Lipman, Reflexivity and rigidity for complexes. I. Commutative rings. Algebra Number Theory 4(1), 47–86 (2010)

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3. L.L. Avramov, A. Martsinkovsky, Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension. Proc. London Math. Soc. (3) 85(2), 393–440 (2002) 4. L.L. Avramov, O. Veliche, Stable cohomology over local rings. Adv. Math. 213(1), 93–139 (2007) 5. D.J. Benson, J.F. Carlson, Products in negative cohomology. J. Pure Appl. Algebra 82(2), 107–129 (1992) 6. O. Celikbas, L.W. Christensen, L. Liang, G. Piepmeyer, Stable homology over associative rings. Trans. Am. Math. Soc. (to appear) (available at arXiv:1409.3605 [math.RA]) 7. L.W. Christensen, D.A. Jorgensen, Vanishing of Tate homology and depth formulas over local rings. J. Pure Appl. Algebra 219(3), 464–481 (2015) 8. J. Cornick, P.H. Kropholler, On complete resolutions. Topology Appl. 78(3), 235–250 (1997) 9. E.E. Enochs, O.M.G. Jenda, Relative homological algebra. de Gruyter Expositions in Mathematics 30, (2000) (Walter de Gruyter & Co., Berlin) 10. F. Goichot, Homologie de Tate-Vogel équivariante. J. Pure Appl. Algebra 82(1), 39–64 (1992) 11. S. Goto, Vanishing of ExtiA (M, A). J. Math. Kyoto Univ. 22(3), 481–484 (1982/83) 12. A. Iacob, Absolute, Gorenstein, and Tate torsion modules. Comm. Algebra 35(5), 1589–1606 (2007) 13. T. Nakayama, On the complete cohomology theory of Frobenius algebras. Osaka Math. J. 9, 165–187 (1957) 14. O. Veliche, Gorenstein projective dimension for complexes. Trans. Am. Math. Soc. 358(3), 1257–1283 (2006)

On the Classification of Artin Algebras and the Inverse System of Macaulay Joan Elias

Abstract The aim of this note is to present some recent results on the classification of local Artinian algebras by using the inverse system of Macaulay.

1 Introduction Let (A, m) be an Artin local ring with maximal ideal m and residue field k = A/m. The Hilbert function of A is  n  m HF A (n) = length A mn+1 for n ≥ 0, the multiplicity of A is e(A) = length A (A), and the embedding dimension of A is b = HF A (1). Some basic results on the classification of Artin rings are the following. There are a finite number of isomorphism classes for e(A) ≤ 6. This result was proved by J. Briançon if b = 2 and k = C, extended by G. Mazzola to k = k¯ and char (k) = 2, 3, and finally B. Poonen extended the result to any algebraically closed residue field k; see [8]. On the other hand D.A. Suprunenko proved that if k infinite then there are infinite many isomorphism classes for e(A) ≥ 7; see [9]. Infinite moduli families appear, for instance, in the embedding dimension two case; see [5]. Proposition 1 Let A be a Gorenstein local algebra A = k[[x, y]]/I with Hilbert function HF A = {1, 2, 2, 2, 1, 1, 1}. Then A is isomorphic to one and only one of the following Artin rings k[[x, y]]/J , where J is: I0 = (y 2 − x y − x 4 , x 3 y), I1 = (y 2 − x 3 y − x 4 , x 3 y), or K c = {(y 2 − x 2 y − cx 4 , x 3 y), c ∈ k ∗ . An open problem, that motivated the introduction of the Macaulay’s inverse system device in the classification of Artin algebras, was the classification of Artin algebras with Hilbert function {1, n, n, 1}, even if A is Gorenstein. J. Elias (B) University of Barcelona, Barcelona, Spain e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_9

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2 The Inverse System of Macaulay We assume that the ground field k is of characteristic zero. We write R = k[[x1 , . . . xn ]], m R = (x1 , · · · , xn )R, S = k[x1 , . . . , xn ] and m S = (x1 , · · · , xn )S. If α = (α1 , . . . , αn ) ∈ Nn then we write x α = x1α1 · · · xnαn . Let A = R/I be an Artin ring with maximal ideal m = m R /I . The socle of A is the ideal soc(A) = (0 : A m), the Cohen–Macaulay type of A is τ (A) = dimk (soc(A)) and the socle-degree of A is the integer s(A) = Max{ j | m j = 0}. Definition 2 A is Gorenstein if and only if dimk (soc(A)) = 1. And A is level if and only if soc(A) = ms , s = s(A). Let R–mod.N oeth (resp., R–mod.Ar tin) be the category of Noetherian Rmodules (resp., Artinian R-modules). If we denote by E R (k) the injective hull of k as R-module then Matlis proved the following duality; see [1]. Proposition 3 Let A = R/I be an Artin ring. The contravariant functor ·∨ = hom R (·, E R (k)) : R–mod.N oeth −→ R–mod.Ar tin is additive, exact and it is an anti-equivalence with (·∨ )2 = I d in both directions. Gabriel proved that the injective hull of the residue field k as R-module is isomorphic to the polynomial ring S; see [6]. The R-module structure on S is defined by the derivation R × S −→ S α

β

 α

β

(x , x ) → x ◦ x =

β! x β−α (β−α)!

0,

β≥α other wise,

n αi !. where α, β ∈ Nn , and α! = i=1 As a corollary of Matlis duality we get Macaulay’s correspondence; see [7]. Proposition 4 There exists a one-to-one set correspondence: {A = R/I Artin} ←→ {f.g. sub–R–modules of S} A = R/I → A∨ = hom R ( RI , S) L = hom R (L , S) ← L . ∨

Furthermore, A∨ = hom R ( RI , S) = {h ∈ S | I ◦ h = 0} and L ∨ = hom R (L , S) = R/Ann(L), where Ann(L) = { f ∈ R | f ◦ L = 0}. Definition 5 The Macaulay’s inverse system of I is I ⊥ = {h ∈ S | I ◦ h = 0}. By using Macaulay’s correspondence, we can deduce the following result.

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Corollary 6 Let A = R/I be an Artin algebra. Then, (i) e(A) = dimk (I ⊥);  ⊥ (ii) HF A (n) = dimk I ∩SS≤n W ; (2) K (πn W, n) > W ; K (π1 W, 1)  W ; (3) Pn W  W and, in particular,  (4) C(W ) = C(ZW ) = C( k≥0 K (πk W, k)). Before stating our main cellular inequality, we describe how polyGEMs are preserved by general cellularizations. The proof of this result is a consequence of the previous proposition. Theorem 9 If X is a polyGEM, then so is cell A X for any space A. This result can be particularized for nilpotent Postnikov stages: Corollary 10 If X is a nilpotent n-Postnikov stage, then so is cell A X for any A. For more general spaces we have obtained the following cellular inequality, probably the main result of our work: Theorem 11 If X is a nilpotent n-Postnikov stage, then so is cell A X for any A.

4 Applications In this last section we deal with various consequences of our previous results. The first one is a characterization of the cellular approximation of K (G, 1) for a nilpotent group G. This problem was the first motivation for our study, and the statement makes a big difference with the general non-nilpotent case: Proposition 12 Assume that G is a nilpotent group and A a connected space. Then cell A K (G, 1)  K (cellπ1 A G, 1), where cellπ1 A G is the group theoremretical π1 Acellularization of G.

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For finite p-groups, we obtain a sharper description: Corollary 13 Let P be a finite p-group. Then cell A K (P, 1) is a K (P  , 1) for some subgroup P  of P. It is interesting to remark here that not every subgroup of a finite p-group G gives rise to a cellullarization of a finite p-group G, not even up to isomorphism. For example, for any n ≥ 2 the are only two possible cellularizations of K (D2n , 1), a contractible space or the whole K (D2n , 1). The following is a direct consequence of Theorem 9 above: Corollary 14 For any space X and any n ≥ 1, K (π1 X/ n π1 X, 1)  X . In particular, if π1 (X ) is nilpotent, then K (π1 X, 1)  X . The nature of the next application of our cellular inequalities is different from the previous one, and solves a different problem. It is a consequence of Bousfield Key Lemma [1] that if X is simply connected, then the map πn : map∗ (X, X ) → Hom(πn X, πn X ) is a weak equivalence if and only if X is weakly equivalent to K (πn X, n). Hence, it is a natural conjecture to ask if this is true for non-simply connected spaces. It is proved in [6] that if X is the classifying space of a finite group, there are plenty of counterexamples to the conjecture, X = B3 being the simplest one. However, all these counterexamples were constructed for non-nilpotent spaces. Our next result states in particular that in this context, nilpotent spaces behave like simply connected spaces. Theorem 15 Let X be a connected space whose fundamental group π1 X is nilpotent. Assume that the map π1 : map∗ (X, X ) → Hom(π1 X, π1 X ) is a weak equivalence. Then X is weakly equivalent to K (π1 X, 1). Our last result is a Serre type statement, that was completely unexpected to us, and describes a global relation between the integral homology groups and the homotopy groups of a nilpotent space. It is remarkable that no spectral sequence is needed in our proof.  Proposition 16 If X is nilpotent, then C (Z(X )) = C( k≥0 K (πk X, k)).

References 1. A.K. Bousfield, Homotopical localizations of spaces. Am. J. Math. 119, 1321–1354 (1997) 2. A.K. Bousfield, D.M. Kan, Homotopy Limits, Completions and Localizations. Lecture Notes in Mathematics, vol. 304 (Springer-Verlag, Berlin, 1972) 3. W. Chachólski, E.D. Farjoun, R. Flores, J. Scherer, Cellular properties of nilpotent spaces. To appear in Geom. Topol 4. E.D. Farjoun, Cellular Spaces, Null Spaces and Homotopy Localization (Springer-Verlag, Berlin, 1996)

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5. E.D. Farjoun, Two completion towers for generalized homology, from Une dégustation topologique [Topological morsels]: homotopy theoremry in the Swiss Alps (Arolla, 1999). Contemp. Math. 265, Am. Math. Soc. 27–39 (Providence, RI, 2000) 6. R. Flores, J. Scherer, Cellularization of classifying spaces and fusion properties of finite groups. J. Lond. Math. Soc. 76(2), 41–56 (2007)

Decomposition Spaces and Incidence (Co)Algebras Imma Gálvez-Carrillo

Abstract Decomposition spaces are simplicial ∞-groupoids with an exactness property giving coherent associativity of its objective incidence (co)algebra. Our theory encompasses the Connes–Kreimer algebra, (derived) Hall algebras and Möbius inversion. This note describes joint work with J. Kock (Universitat Autònoma de Barcelona) and A. Tonks (Leicester).

1 Simplicial ∞-Groupoids and Objective (Co)Algebra Let  be the simplicial category, with objects the finite non-empty standard ordinals [n] = {0, 1, . . . , n} and morphisms the order-preserving maps between them. These are generated by ∂ i : [n − 1] → [n] and σ i : [n + 1] → [n], n ≥ 0, i ∈ [n], that is, the injection skipping the value i, and the surjection repeating the value i. A simplicial groupoid is a functor X : op → Groupoids. One writes X n = X ([n]). The arrows ∂ i , σ i in  induce face maps and degeneracy maps di := X (∂ i ) : X n → X n−1 , and si := X (σ i ) : X n → X n+1 . A simplicial map X → Y is a natural transformation: a family of maps (X n → Yn )n≥0 commuting with face and degeneracy maps. We often use simplicial ∞-groupoids instead of simplicial groupoids. One model for the notion of ∞-groupoid is that of Kan complex. The corresponding model for the notion of ∞-category is that of quasi-category; see [4, 5]. Let LIN be the monoidal ∞-category with objects all slices S/I of the ∞-category S of ∞-groupoids, morphisms the linear functors S/I  S/J , and S/I ⊗ S/J := S/I ×J . A slice S/B is a generalised ‘vector space with specified basis’ π0 B: any X → B is a homotopy linear combination 

b

X b · (1 −−→ B).

I. Gálvez-Carrillo (B) Departament de Matemàtiques, Universitat Politècnica de Catalunya, ESEIAAT, 08222 Terrassa, Spain e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_12

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Postcomposition and pullback along f : B  → B define adjoint functors between p q slice categories f ! : S/B   S/B : f ∗ . A span I ←− M −→ J defines a linear funcp∗ q! tor S/I −→ S/M −→ S/J . Composition is ‘matrix multiplication’: by the Beck– Chevalley condition, the composite of linear functors defined by I ← M → J and J ← N → K is defined by the pullback span I ← M × J N → K . A coalgebra in LIN is a slice S/I , equipped with linear functors δ0 : S/I  S (counit) and δ2 : S/I  S/I ⊗ S/I (comultiplication), defined by spans I ← M → 1 and I ← N → I × I . These must be coherently counital and coassociative. The idea is expressed by the following diagrams commuting up to specified (coherent) cells:

(id ⊗ δ0 )δ2 = id = (δ0 ⊗ id)δ2 _id⊗δ _ 0_/ S/I S⊗2 /I O }> O  }  δ2 δ ⊗id id  0  } } S/I _ _ _/ S⊗2

/I ? O   id _  o N I  / M×I   _  id       / I 2, N I o

/I

δ2

(id ⊗ δ2 )δ2 = (δ2 ⊗ id)δ2 _id⊗δ _ 2_/ S⊗3 S⊗2 /I /I O O  δ ⊗id  δ2  2  S/I _ _ _/ S⊗2 /I δ2

IO2 o

I ×O M

IO2 o

I ×O N

/ I3 ? O   _  / N ×I N o P  _       / I 2. N I o

The cardinality of a suitably finite ∞-groupoid X is |X | =



∞ 

|πi (X, x0 )|(−1) ∈ Q, i

x0 ∈0 X i=1

the cardinality of a suitably finite span S ← M → T is a matrix |M| : Q π0 S → Q π0 T , and the cardinality of a suitably finite coalgebra in LIN is a (classical) Q-coalgebra.

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2 Incidence Coalgebras in LIN and Decomposition Spaces For any simplicial ∞-groupoid X , there are comultiplication maps, i.e., linear functors δn : S/ X 1  S/ X 1 ⊗n , n ≥ 0, defined by the canonical spans X 1 ← X n → X 1 n . Then, to prove coassociativity of δ2 , for example, we need the following pullback diagrams: XO 1 o

d1

d1

_

d1

X2 o

d2

_



(d2 ,d0 )

 X1 × X1 o

(d2 ,d0 )

XO 2 X3

id ×d1



d1 ×id

/ X2 × X1

(d3 ,d0 d0 )

(d2 ,d0 )×id

(d22 ,d0 )

 X1 × X2

/ X1 × X1 O

id ×(d2 ,d0 )

 / X1 × X1 × X1

Theorem 1 (The incidence coalgebra C(X ) of a Segal space X ) If X is a Segal space, (e.g., the nerve of a poset, monoid or category) then the linear functors δ0 , δ2 above are counital and coassociative, that is, C(X ) := S/ X 1 is a coalgebra in LIN. We can prove the counital and coassociative properties under weaker hypotheses. First, some terminology: the generic (or end-point preserving) maps g : [n] → [q] in  are generated by inner coface and codegeneracy maps, and free (or distancepreserving) maps f : [n]  [m] in  are generated by outer coface maps ∂ ⊥ = ∂ 0 , ∂ = ∂n : ∂ 2 / ∂ o [2] [3] [2] o [1] O _ O 1 ⊥ ⊥ ∂ ∂1 ∂ _ O∂ O _ _ / [3] o  o [2] [1] 1 [2] ∂

∂

Pushouts of generic and free maps exist in , and the results are generic and free. The two examples we have given are essential to understanding coassociativity: their images under a functor X : op → S are pullbacks precisely when the squares () above are. Definition 2 A decomposition space X is a simplicial ∞-groupoid X : op → S that sends generic/free pushouts in  to pullbacks in S, ⎛

/ ⎜ [n] ⎜ ⎜g ⎜ _ ⎝ [q] /

f

/ [m]

⎟ ⎟ ⎟ g _ _ ⎟ / [ p] ⎠ 

f

⎛

⎞ X

−→

o ⎜ XO n ⎜ ∗ ⎜g ⎜ ⎝ Xq o

f∗

f ∗

⎞ XO m ⎟ ⎟ g ∗ ⎟ . _ ⎟ ⎠ Xp

This notion agrees with that of unital 2-Segal space formulated independently in [2].

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Theorem 3 (Coherent coassociativity) If X is a decomposition space then any linear functor S/ X 1  S/ X 1 ⊗n given by composites of tensors of comultiplication maps is again a comultiplication map. In particular, (id ⊗δ0 )δ2 = id = (δ0 ⊗ id)δ2 , (id ⊗δ2 )δ2 = δ3 = (δ2 ⊗ id)δ2 and so, C(X ) := S/ X 1 is a (counital, coassociative) coalgebra in LIN. For the proof, we extend  to a larger category with ‘external’ direct sums, so that images under X of sums of spans induce multilinear functors, e.g., X 12 ← X 2 × X 3 → X 15

δ2 ⊗ δ3 : S/ X 1 ⊗2  S/ X 1 ⊗5 .



A map of simplicial ∞-groupoids is cULF if it is both conservative and ULF (Unique Lifting of Factorisations), that is, it is a cartesian natural transformation on the degeneracy and the generic face maps, respectively. Lemma 4 (Functoriality for cULF maps of decomposition spaces) If F : X → X  is a cULF map between decomposition spaces then F! : S/ X 1 → S/ X 1 commutes with the comultiplication maps, and hence defines a map of coalgebras F! : C(X ) → C(X  ). X1 o

Fn

F1

 X 1 o

δn S/ X 1 _ _ _/ S⊗n / X1

/ X 1n

Xn

_



 X n

F1n

F1!



/ X n 1



δn

F1! ⊗n

S/ X 1 _ _ _/ S⊗n / X

1

Recall that the Decalage functors Dec⊥ and Dec on simplicial objects forget, respectively, the bottom and top face and degeneracy maps.

XO d⊥

o

XO 0 o

d1 s0 d0

Dec X

/ X1 o O o

d2 d1

s1 s0

X1 o

o

d2 s1 d1

o o

/ X2 o

d3

o

d2 d1

s2 s1

d2

/ XO 2 o o

d0

o

d3

/

d0

d0

d0 ⊥

o

d1 d0

/

s2

/ X3 / O

s1 s0

···

d0

o

/

o

d4

/ X3 o

d3

o

d2 d1

/

s3 s2 s1

/

/ X4

···

Lemma 5 A simplicial ∞-groupoid X : op → S is a decomposition space if and only if both Dec (X ) and Dec⊥ (X ) are Segal spaces and the corresponding natural transformations d and d⊥ are cULF.

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3 Example: The Connes–Kreimer Coalgebra Consider the nerve F of the category of forests and the root-preserving inclusions between them. The Connes–Kreimer coalgebra [2] was introduced previously in [1] as a quotient of the incidence coalgebra of the Segal space F by identifying two inclusions if their complements are isomorphic. Alternatively, consider the groupoid H2 of forests with an admissible cut, the groupoid H1 of forests, and the span H1 ← H2 → H1 × H1 where the map H1 ← H2 forgets the cut and H2 → H1 × H1 performs it; see also [3].

The span induces a comultiplication whose cardinality is the Connes–Kreimer comultiplication. This structure is part of a decomposition space H, where Hi is the groupoid of forests with i − 1 admissible cuts. It is not a Segal space, but its decalage is the Segal space F of forests and root-preserving inclusions. The induced comparison map C(F) → C(H) is just Dür’s identification of inclusions with isomorphic complements. This is one of a long list of examples, which also includes Schmitt restriction species, [6].

4 Example: Hall Coalgebras Consider the nerve W of the category vect of finite-dimensional Fq -vector spaces and linear injections. Dür obtained the q-binomial coalgebra from C(W) by identifying two injections if their cokernels are isomorphic. Let V0 = 1, V1 be the maximal subgroupoid of vect, and V2 be the groupoid of short exact sequences. The spans V1 o Eo 

V2  [E  → E → E  ]

/ V1 × V1 , / (E  , E  )

V1 o [0] o

V0 ∗

/1

/∗

define a coalgebra in LIN, with cardinality the q-binomial coalgebra. In fact, the groupoids Vi are part of a decomposition space V defined by Waldhausen’s S• (vect). It is not a Segal space, but Dec⊥ (V) = W. The canonical map C(W) → C(V) gives Dür’s identification. Theorem 6 Waldhausen’s S• construction of an abelian category A is a decomposition space X , whose decalage is the Segal space Y of monomorphisms in A. The Ringel–Hall coalgebra H (A) is the incidence coalgebra of X , and may be obtained from that of Y by the reduction C(Y )  C(X ) = H (A) induced by the cULF map d⊥ : Y = Dec⊥ (X ) −→ X .

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Hall coalgebras were an important motivation for the work of Dyckerhoff and Kapranov. The above theorem was first proved there, independently, for exact ∞categories.

5 Incidence Algebras and Möbius Inversion Without Additive Inverses If S/I is a coalgebra in LIN, then the linear functors S/I  S form a dual algebra with convolution product F ∗ G = (F ⊗ G) δ2 and unit δ0 : S/I  S. The zeta = functor ζ : S/I  S is defined by the span I ←− I → 1. Let X be a decomposition space in which s0 : X 0 → X 1 is mono, and let Xr be the subgroupoid of X r of nondegenerate simplices. The incidence algebra of X is the convolution algebra dual to S/ X 1 , and we aim to finding an objective version of classical Möbius inversion, for the zeta functor. Consider the linearfunctors μr : S/ X 1   S defined by the spans X 1 ←− Xr −→ 1, and let μeven = r even μr and μodd = r odd μr . Theorem 7 We have ζ ∗ μeven = δ0 + ζ ∗ μodd and μeven ∗ ζ = δ0 + μodd ∗ ζ. After taking cardinalities, we can write μ = μeven − μodd and obtain ζ ∗ μ = δ0 = μ ∗ ζ.

References 1. A. Dür, Möbius functions, incidence algebras and power series representations. Lecture Notes in Mathematics, vol. 1202 (Springer-Verlag, Berlin, 1986) 2. T. Dyckerhoff, M. Kapranov, Higher Segal spaces I. Preprint. arXiv:1212.3563 3. I. Gálvez-Carrillo, J. Kock, A. Tonks, Groupoids and Faàdi Bruno formulae for Green functions in bialgebras of trees. Adv. Math. 254, 79–117 (2014) 4. A. Joyal, Quasi-categories and Kan complexes. J. Pure Appl. Algebra 175, 207–222 (2002) 5. J. Lurie, Higher topos theory. Ann. Math. Stud. 170 (2009) 6. W.R. Schmitt, Hopf algebras of combinatorial structures. Can. J. Math. 45, 412–428 (1993)

Cellular Approximations for Fusion Systems Alberto Gavira-Romero

Abstract In this joint work with N. Castellana, we compute the cellularization of classifying spaces of saturated fusion systems with respect to classifying spaces of finite p-groups.

1 Cellular Approximations Let A be a pointed space. In [7], E. Dror-Farjoun formalized and introduced the concept of A-homotopy. Let C(A) denote the smallest collection of pointed spaces that contains A and is closed under weak equivalences and pointed homotopy colimits. A pointed space X ∈ C(A) is called an A-cellular space. Theorem 1 (Dror-Farjoun [7]) Let A and X be pointed spaces. Then there exists a pointed map c : C W A X → X such that C W A X is an A-cellular space and c is an A-equivalence, i.e., it induces a weak equivalence in pointed mapping spaces, c∗ : map∗ (A, C W A X )

/

map∗ (A, X ).

The space C W A X is called the A-cellularization or A-cellular approximation of X . Example 2 If A = S n , then C W S n X = X n − 1 → X is the (n − 1)-connected cover of X .

A. Gavira-Romero (B) Dep. de Computación, Álxebra, Universidade da Coruña, Campus Elviña, s/n, 15071 A Coruña, Spain e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_13

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BZ/ p-Cellularization of BG

Let G be a finite group and p a prime number. In [8–10], a thorough study of the space C W BZ/ p (BG) is carried out. In particular, the authors establish the relationship between the p-fusion data of G and the homotopy type of the BZ/ p-cellularization of BG. Let (·)∧p denote the Bousfield–Kan p-completion, i.e., an endofunctor (·)∧p : Spaces∗ → Spaces∗ that isolates the p-primary information in spaces, in the sense  that a pointed map f : X → Y induces a homotopy equivalence f p∧ : X ∧p / Y p∧ if and only if it induces an isomorphism in homology H∗ ( f ; F p ) : H∗ (X ; F p ) see [2] for details.

∼ =

/ H∗ (Y ; F p ) ;

Definition 3 Let G be a finite group and S a Sylow p-subgroup of G. A p-subgroup K ≤ S is strongly closed in G if, whenever k ∈ K and g ∈ G are such that gkg −1 ∈ S, then gkg −1 ∈ K . Since the intersection of strongly closed subgroups is also strongly closed, we may let Cl S (Z/ p) ≤ S denote the smallest strongly closed subgroup in G that contains all elements of order p in S. Theorem 4 (Flores and Foote [9]) Let G be a finite group generated by its elements of order p, and let S be a Sylow p-subgroup of G.  (i) If Cl S (Z/ p) = S, then C W BZ/ p (BG) → BG → q= p BG q∧ is a homotopy fibration.  (ii) If Cl S (Z/ p) = S, then C W BZ/ p (BG) → BG → B ∧p × q= p BG q∧ is a homotopy fibration, where  is a finite group (which depends on G and Cl S (Z/ p)). Remark 5 If G is a finite p-group generated by elements of order p (that is, G = S), then Cl S (Z/ p) = S and BG q∧  ∗ for all q = p. Hence, BG ∈ C(BZ/ p). Remark 6 By construction, H˜ ∗ (C W BZ/ p (BG); R) ∼ = 0 if R = Q or Fq for q = p where, in this case, H∗ (C W BZ/ p (BG); F p ) ∼ = H∗ (C W BZ/ p (BG)∧p ; F p ). This situation suggests a natural question: how is C W BZ/ p (BG)∧p related to BG ∧p ? Proposition 7 (Castellana and Flores [4]) Let G be a finite group generated by its elements of order p. Then, C W BZ/ p (BG)∧p  C W BZ/ p (BG ∧p )∧p . Remark 8 The space BG ∧p is nilpotent, then C W BZ/ p (BG ∧p ) is also nilpotent. Hence, using Sullivan Arithmetic Square, we get C W BZ/ p (BG ∧p )∧p  C W BZ/ p (BG ∧p ). The homotopy type of BG ∧p has been extensively studied through homology decompositions (see [11, 12]). The important role of conjugacy relations among subgroups indicated that an axiomatic description from the work of Puig [14] could be useful in this context.

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Definition 9 Let S be a finite p-group. A fusion system over S is a subcategory F of the category of groups such that Ob(F) = {P ≤ S} and, for all P, Q ≤ S, Hom S (P, Q) ⊂ HomF (P, Q) ⊂ Inj(P, Q), and any morphism ϕ ∈ Hom S (P, Q) is the composite of an isomorphism in F followed by an inclusion. A fusion system F is saturated if it verifies the saturation axioms; see [3, Definition 1.2]. In [3], the authors introduced the notion of centric linking system L over a saturated fusion system F, as a category whose objects are F-centric subgroups of S with a functor L → F c satisfying certain compatibility relations with morphisms in F; see [3, Definition 1.7] for more details. Definition 10 Let (S, F) be a saturated fusion system and L be a centric linking system over F. The classifying space of F is the p-completion of the nerve BF := |L|∧p . Remark 11 The classifying space BF is p-complete and  BZ/ p-null (i.e., (BF)∧p  BF and map∗ ( BZ/ p, BF)  ∗). Furthermore, it is nilpotent. Recently, A. Chermak proved the existence and uniqueness of centric linking systems over saturated fusion systems. Theorem 12 (Chermak [6]) Given a saturated fusion system (S, F), there is a unique centric linking system L over F, up to isomorphism. Example 13 Let G be a finite group and let S be a Sylow p-subgroup of G. Let F S (G) be the category whose objects are Ob(F S (G)) = {P ≤ S} and morphisms are HomFS (G) (P, Q) = {ϕ ∈ Hom(P, Q) | ϕ = cg for some g ∈ G}. Then F S (G) is a saturated fusion system over S and there exists a centric linking system L S (G) such that BF S (G) = L S (G)∧p  BG ∧p .

3

B P-Cellularization of BF

Similarly to the case of finite groups, a subgroup closed by the fusion will determine the cellular structure of the classifying space. Definition 14 Let (S, F) be a saturated fusion system. A subgroup K ≤ S is strongly F-closed if for all P ≤ K and ϕ ∈ HomF (P, S), then ϕ(P) ≤ K . The intersection of strongly F-closed subgroups is again a strongly F-closed subgroup. Then, given a finite p-group P, ClF (P) denotes the smallest strongly F-closed subgroup of S that contains the normal subgroup generated by {Im(α) | α ∈ Hom(P, S)}.

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Theorem 15 Let F be a saturated fusion system over a finite p-group S. Then, BF is B P-cellular if and only if ClF (P) = S. Corollary 16 Let F be a saturated fusion system over a finite p-group S. Then, (i) the classifying space BF ∈ C(B S); (ii) if S is generated by its elements of order pi with i ≤ r , then BF ∈ C(BZ/ p m ) for all m ≥ r . Next, we describe which is the strategy to prove Theorem 15. Consider the Chachólski fibration to compute C W B P (BF) given in [5], C W B P (BF)

/ BF

r

/ P B P C,

where P B P denotes the nullification functor with respect to  B P (see [1]) and C  is the homotopy cofibre of the evaluation map ev : [B P,BF ]∗ B P → BF. Lemma 17 In this case, C W B P (BF)  BF if and only if r ∧p  ∗. So, to prove that BF is cellular is equivalent to prove that the map r ∧p is nullhomotopic. For that, we use the notion of kernels of maps from a classifying space developed by D. Notbohm in the context of compact Lie groups. Definition 18 (Notbohm [13]) Let (S, F) be a saturated fusion system. Let f : BF → Z be a pointed map, where Z is a p-complete and  BZ/ p-null space. Then, ker( f ) := {x ∈ S | f | Bx  ∗}. Proposition 19 Let (S, F) be a saturated fusion system. Let f : BF → Z be a pointed map, where Z is a p-complete and  BZ/ p-null space. Then, (i) f  ∗ if and only if ker( f ) = S; (ii) ker( f )  S is strongly F-closed; (iii) if K ≤ S is strongly F-closed, then there are an integer N ≥ 0 and a map f : BF → (B N )∧p such that ker( f ) = K . Finally, we compute the kernel of r ∧p . Proposition 20 Let F be a saturated fusion system over a finite p-group S. Then, ClF (P) = ker(r ∧p ).

4 Examples We describe some examples to show situations where the cellularization can be explicitly described.

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Example 21 Consider G = 3 . Notice that 3 is generated by elements of order 2 / C(Z/3). (in fact, by transpositions), but not by elements of order 3. We have 3 ∈ The classifying space B3 is not BZ/3-cellular, actually C W BZ/3 (B3 ) is homotopy equivalent to BZ/3. Also, since Z/3 ∈ Syl3 (3 ), (B3 )∧3 ∈ C(BZ/3r ) for all r ≥ 1 by Corollary 16. In some cases we can compute the cellularization of the classifying space of a saturated fusion system (S, F) when ClF (P) = S; for instance, when ClF (P) is normal in F. In this case, there exist a saturated fusion system (S/ClF (P), F/ClF (P)) and a “quotient” map BF → B(F/ClF (P)). Proposition 22 Let (S, F) be a saturated fusion system. If ClF (P)  F, then C W B P (BF) → BF → B(F/ClF (P)) is a homotopy fibration. Example 23 Let G = Z/ p n  Z/q for p = q and n ≥ 1. The Sylow p-subgroup of G is S = (Z/ p n )q  G. Then, ClFS (G) (Z/ pr ) = (Z/ pr )q  F S (G) for all r ≥ 1. Finally, Theorem 15 and Proposition 22 give us the following equivalence C W BZ/ pr (BG ∧p )

 

if r ≥ n, BG ∧p hofib(BG ∧p → B(G/Z/ pr )∧p ) if r < n.

References 1. A.K. Bousfield, Localization and periodicity in unstable homotopy theory. J. Am. Math. Soc. 7(4), 831–873 (1994) 2. A.K. Bousfield, D.M. Kan, Homotopy Limits, Completions and Localizations. Lecture Notes in Mathematics, vol. 304 (Springer-Verlag, Berlin, 1994) 3. C. Broto, R. Levi, B. Oliver, The homotopy theory of fusion systems. Trans. Am. Math. Soc. 16(4), 779–856 (2003) 4. N. Castellana, R.J. Flores, Homotopy idempotent functors on classifying spaces. Trans. Am. Math. Soc. 367(2), 1217–1245 (2015) 5. W. Chachólski, On the functors C W A and PA . Duke Math. J. 84(3), 599–631 (1996) 6. A. Chermak, Fusion systems and localities. Acta Math. 211(1), 47–139 (2013) 7. E. Dror-Farjoun, Cellular Spaces, Null Spaces and Homotopy Localization. Lecture Notes in Mathematics, vol. 1622 (Springer-Verlag, Berlin, 1996) 8. R.J. Flores, Nullification and cellularization of classifying spaces of finite groups. Trans. Am. Math. Soc. 359(4), 1791–1816 (2007) 9. R.J. Flores, R.M. Foote, The cellular structure of the classifying spaces of finite groups. Isr. J. Math. 184, 129–156 (2011) 10. R.J. Flores, J. Scherer, Cellularization of classifying spaces and fusion properties of finite groups. J. Lond. Math. Soc. (2) 76(1), 41–56 (2007) 11. S. Jackowski, J. McClure, Homotopy descomposition of classifying spaces via elementary abelian subgroups. Topology 31, 113–132 (1992)

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12. S. Jackowski, J. McClure, B. Oliver, Homotopy classification of self-maps of BG via G-actions. Ann. Math. 135, 184–270 (1992) 13. D. Notbohm, Kernels of maps between classifying spaces. Isr. J. Math. 87(1–3), 243–256 (1994) 14. L. Puig, Frobenius categories. Trans. Am. Math. Soc. 303(1), 309–357 (2006)

Homological Epimorphisms and the Lie Bracket in Hochschild Cohomology Reiner Hermann

Abstract In 2009, Koenig–Nagase established a long exact sequence relating the Hochschild cohomology of an algebra with the Hochschild cohomology of the quotient of the algebra by a stratifying ideal. It is well-known that the morphisms in this long exact sequence are multiplicative. In this exposition, we will argue that those morphisms preserve the Lie bracket (and the squaring map) as well. It will turn out that this really just has to do with the fact, that the canonical map from the algebra to its quotient is a (surjective) homological epimorphism in the sense of Geigle– Lenzing. Our considerations substantially rely on a generalisation of Schwede’s homotopy theoretical interpretation of the Lie bracket in Hochschild cohomology. A brief reminder thereof will be given, too.

1 The Loop Bracket Throughout, we let K be a commutative ring and ⊗ = ⊗ K . The following overview is based on preprint [7]. Let A be a K -algebra, and Aev = A ⊗ Aop . In [12], Stefan Schwede described the Lie bracket in Hochschild cohomology (see [5]) in terms of bimodule extensions. More precisely, Schwede took advantage of the fact that the monoidal category (Mod(Aev ), ⊗ A , A) is, in general, highly asymmetric in order to produce, for given m- and n-self extensions S and T of A with Yoneda composite S ◦ T , a loop

(S, T )

≡

S A ww w w {ww S ◦ TcG GG GG G

TG GG GG # (−1); mn T ◦ S w ww ww (−1)mn T  A S

R. Hermann (B) Institutt for matematiske fag, NTNU, 7491 Trondheim, Norway e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_14

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ev in the category E xt Am+n ev (A, A) of (m + n)-self extensions of A over A , that is, an m+n element in the fundamental group π1 (E xt Aev (A, A), S ◦ T ). This loop identifies (A, A) thanks to Vladimir Retakh (see [10, 11]) who with an element in Ext m+n−1 Aev proved the existence of an isomorphism ∼

n  Ext n−1 R (U, V ) −→ π1 (E xt R (U, V ), S )

(for a ring R and U, V ∈ Mod(R)) which is, in an appropriate sense, independent of the taken base point S  . Schwede’s main theorem in this context is now the following. Theorem 1 (see [12, Theorem 3.1]) Let A be a K -projective K -algebra and m, n  1 be integers. Then, for all elements α ∈ HHm (A) and β ∈ HHn (A), represented by extensions S = S(α) and T = T (α) respectively, the Lie bracket (−1)n {α, β} A of α (A, A). and β identifies with the image of the loop (S, T ) in Ext m+n−1 Aev Note that Schwede’s construction, and hence also his main result, covers the degrees m, n  1; see [6] for the missing cases m  0, n = 0.

2 Homological Epimorphisms and the Main Theorem 2.1 A Compatibility Result In [8], we generalised Schwede’s construction to “suitable” exact monoidal categories. One class of such categories are exact monoidal K -categories (C, ⊗, 1) which are closed under kernels of epimorphisms (that is, the class of admissible epimorphisms coincides with the class of epimorphisms in C) such that − ⊗ X : C → C is an exact functor for all X ∈ Ob C. In this setting, we can m+n−1 n (1, 1) which speprovide a map [−, −]C : Ext m C (1, 1) × Ext C (1, 1) → Ext C cialises to Schwede’s map, and hence the Lie bracket, when C is taken to be the full subcategory of Mod(Aev ) whose objects are Aev -modules which are projective on either side. For exact and colax monoidal functors between exact monoidal categories of the above form, we proved the result below (see [1] to recall the definition of a colax monoidal functor). Theorem 2 Let F : (C, ⊗C , 1C ) → (D, ⊗D , 1D ) be an exact and colax monoidal ∼ functor whose unit morphism F1C − → 1D is an isomorphism. Then the induced graded K -algebra homomorphism F

∼

→ Ext∗D (1D , 1D ) Ext ∗C (1C , 1C ) −→ Ext ∗D (F1C , F1C ) − takes [−, −]C to [−, −]D .

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2.2 Homological Epimorphisms Recall that a ring homomorphism f : R → S is an epimorphism if and only if the restriction functor f  : Mod(S) → Mod(R) is full and faithful. Due to a classical result of Silver, this is the same as saying that the multiplication map S ⊗ R S → S is an isomorphism of S-bimodules. The case where the derived restriction functor D( f  ) defines a full and faithful functor D(Mod(S)) → D(Mod(R)) has been studied by Geigle–Lenzing; see [4]. It is clear that f will have to be an epimorphism in that case. By adding the condition ToriR (S, S) = 0 for all i > 0, one obtains a precise characterisation of this situation. Epimorphisms satisfying the latter Tor-vanishing condition are called homological epimorphisms.

2.3 The Main Theorem Let us fix two K -algebras A and B which are projective when considered as K modules. Let further q : B → A be a K -linear homological epimorphism. The induced K -algebra homomorphism q ev = q ⊗ q op : B ev → Aev remains a homological epimorphism, and the left adjoint to the restriction functor D(qev ) has remarkable properties, by Theorem 2: Theorem 3 The graded K -algebra homomorphism A ⊗LB (−) ⊗LB A : HH∗ (B) = HomD(B ev ) (B, B[∗]) −→ HomD(Aev ) (A, A[∗]) = HH∗ (A)

also preserves the Lie bracket. Indeed, the crucial observation for the proof of the above statement is that q : B → A gives rise to a colax monoidal functor A = A ⊗ B (−) ⊗ B A : (Mod(B ev ), ⊗ B , B) −→ (Mod(Aev ), ⊗ A , A). To produce the desired exact and monoidal subcategories between which A defines an exact functor, one starts with the category of bimodules being projective on either side, and successively removes those modules that do not contribute to the exactness of A, in such a way that the resulting category remains exact and closed under ⊗ A .

3 An Application to a Koenig–Nagase Long Exact Sequence 3.1 Stratifying Idempotents Let R be a ring. The notion of a stratifying idempotent in R goes back to Cline– Parshall–Scott; see [3]. For such an idempotent e ∈ R it is, by definition, required

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that the multiplication map Re ⊗e Re e R − → Re R is an isomorphism, and Torie Re (Re, e R) = 0 for i > 0. If I = Re R, S = R/I and q : R  S, one has Tor 1R (S, S) = I /I 2 = 0 and R (I, S) ToriR (S, S) ∼ = Tori−1 R ∼ (Re ⊗e Re e R, S) = Tori−1 e Re ∼ (Re, e R ⊗ R S) = Tori−1 ∼ = 0,

(by a long exact sequence for − ⊗ R S) (as Re ⊗e Re e R ∼ = Re R) (by [2, Chap. 9, Theorem 2.8]) (as e annihilates S)

for i > 1, whence q : R → S = R/Re R is a (surjective) homological epimorphism. Given K -algebras A and B, and an (A ⊗ B op )-bimodule M, the idempotents 

10 e= 00



  00 and f = 01

are stratifying in the algebra T of upper triangular matrices 

 AM T = . 0 B Therefore, T  T /T eT = B and T  T /T f T = A are homological epimorphisms.

3.2 A Cohomological Long Exact Sequence Let B be a K -projective K -algebra and e ∈ B a stratifying idempotent such that A = B/BeB is K -projective. From the adjunction isomorphism Hom Aev (A ⊗ B BB ⊗ B A, A) ∼ = Hom B ev (BB, A), where BB is the bar resolution of B, one obtains ∼ → Ext ∗B ev (B, A). Therefore, applying Hom B ev (B, −) to the canonical short HH∗ (A) − exact sequence q → 0, 0− → BeB − → B −→ B/BeB − induces a cohomological long exact sequence γn

· · · −→ Ext nB ev (B, BeB) −→ HHn (B) −−→ HHn (A) −→ · · ·

(1)

as observed by Koenig–Nagase in [9]. The following Lemma implies, when combined with Theorem 3, that the map γ∗ : HH∗ (B) → HH∗ (A) in the long exact sequence (1) preserves the cup product (see also [9]) and the Lie bracket. Lemma 4 The map γ∗ agrees with A ⊗LB (−) ⊗LB A : HH∗ (B) → HH∗ (A). In this context, we like to raise the following question.

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Question  5 Let K be anarbitrary field of characteristic 2.If Bdenotes the algebra K (Z2 × Z2 ) K Z2 00 B= , with stratifying idempotent e = , what is the precise 0 K 01 ∗ Lie algebra structure of HH (B)? Can the above sequence (1) be used to determine it ? (the Lie structure of HH∗ (B/BeB) is well understood). Note that the above algebra B is Xu’s initial counterexample to the finite generation conjecture of Snashall–Solberg; see [13, 14]. Hence, we further ask: Question 6 Keep the assumptions of Question 5, and let N ⊆ HH∗ (B) denote the set of all nilpotent homogeneous elements. Let further G(N ) be the weak Gerstenhaber ideal generated by N (that is, G(N ) is the smallest graded ideal containing N and being closed under {−, −}). Is the graded quotient algebra HH∗ (B)/G(N ) finitely generated over K ? If so, is this in fact a general phenomenon, that is, do we have finite generation for all finite dimensional algebras B?

References 1. M. Aguiar, S. Mahajan, Monoidal Functors, Species and Hopf Algebras. CRM Monograph Series, vol. 29 (American Mathematical Society, Providence, RI, 2010) 2. H. Cartan, S. Eilenberg, Homological Algebra (Princeton University Press, Princeton, NJ, 1956) 3. E. Cline, B. Parshall, L. Scott, Stratifying Endomorphism Algebras, vol. 124 (American Mathematical Society, 1996), viii+119 pp 4. W. Geigle, H. Lenzing, Perpendicular categories with applications to representations and sheaves. J. Algebra 144(2), 273–343 (1991) 5. M. Gerstenhaber, The cohomology structure of an associative ring. Ann. Math. 78(2), 267–288 (1963) 6. R. Hermann, Exact sequences, Hochschild cohomology and the Lie module structure over the M-relative center. Preprint (2014). http://arxiv.org/abs/1407.2497 7. R. Hermann, Homological epimorphisms, recollements and Hochschild cohomology—with a conjecture by Snashall–Solberg in view. Preprint (2014). http://arxiv.org/abs/1411.0836 8. R. Hermann, Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology. Mem. Am. Math. Soc. (2016). http://arxiv.org/abs/1403.3597 9. S. Koenig, H. Nagase, Hochschild cohomology and stratifying ideals. J. Pure Appl. Algebra 213(5), 886–891 (2009) 10. A. Neeman, V. Retakh, Extension categories and their homotopy. Compositio Math. 102(2), 203–242 (1996) 11. V.S. Retakh, Homotopy properties of categories of extensions. Uspekhi Mat. Nauk 41(6), 179–180 (1986) 12. S. Schwede, An exact sequence interpretation of the Lie bracket in Hochschild cohomology. J. Reine Angew. Math. 498, 153–172 (1998) 13. N. Snashall, O. Solberg, Support varieties and Hochschild cohomology rings. Proc. Lond. Math. Soc. 88(3), 705–732 (2004) 14. F. Xu, Hochschild and ordinary cohomology rings of small categories. Adv. Math. 219(6), 1872–1893 (2008)

Gorenstein Projective Precovers Alina Iacob

Abstract The existence of Gorenstein projective precovers is one of the main open problems in Gorenstein homological algebra. We prove that the class of Gorenstein projective modules is a special precovering over any right coherent and left n-perfect ring. This is joint work with S. Estrada and S. Odaba¸si, and more details can be found in Estrada et al. (Gorenstein flat and projective (pre)covers. Preprint [2]).

The class of Gorenstein projective modules is one of the key elements in Gorenstein homological algebra. So it is natural to consider the question of existence of the Gorenstein projective precovers. As useful as the Gorenstein homological methods have proved to be, they can only be used when the appropriate resolutions exist. The existence of Gorenstein projective precovers over Gorenstein rings is known; see [1]. Then, Jørgensen proved their existence over commutative Noetherian rings with dualizing complexes (2007). More recently (2011), Murfet–Salarian [3] proved the existence of Gorenstein projective precovers over commutative Noetherian rings of finite Krull dimension. We extend their result by: 1. Relaxing the conditions on the ring: we work with a right coherent ring such that every flat left R-module has finite projective dimension. In this case there exists an integer n ≥ 0 such that the projective dimension of any flat left R-module is less than or equal to n. Such a ring R is called a left n-perfect ring. 2. Proving the existence of the special Gorenstein projective precovers in the category of complexes of left R-modules, Ch(R), over a right coherent and left n-perfect ring R; as a corollary, we prove that over such a ring, the class of Gorenstein projective modules is special precovering in R–Mod. We recall that a module M is Gorenstein projective if there exists an exact and H om(−, Pr oj) exact complex of projective modules P = · · · → P1 → P0 → P−1 → · · · such that M = K er (P0 → P−1 ). Such a complex P is called a totally acyclic complex of projective modules. We will use the notation GP for the class of Gorenstein projective modules. A. Iacob (B) Department of Mathematical Sciences, Georgia Southern University, Statesboro, USA e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_15

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As usual, we will denote by GP ⊥ , the right orthogonal class of the Gorenstein projective modules, i.e., GP ⊥ = {L | E xt 1 (G  , L) = 0, for all Gorenstein projective modules G  }. We also recall the definitions for Gorenstein projective precovers, covers, and special precovers. Definition 1 A homomorphism φ : G → M is a Gorenstein projective precover of M if G is Gorenstein projective and if, for any Gorenstein projective module G  and any φ  : G  → M, there exists u ∈ H om(G  , G) such that φ  = φu. A Gorenstein projective precover φ is said to be a cover if any v ∈ End R (G) such that φ = φv is an automorphism of G. A Gorenstein projective precover φ : G → M is said to be special if K er (φ) is in the right orthogonal class of that of Gorenstein projective modules, GP ⊥ . The importance of Gorenstein projective (pre)covers comes from the fact that, when the class GP is precovering, every R-module M has a Gorenstein projective resolution, i.e., a H om(GP, −) exact complex · · · → G 1 → G 0 → M → 0 with G 0 → M and G i → K er (G i−1 → G i−2 ) Gorenstein projective precovers. Such a complex is unique up to homotopy, so it can be used to compute right derived functors of H om. The Gorenstein projective complexes are defined in a similar manner as the Gorenstein projective modules (simply replace projective module in the definition with projective complex). The Gorenstein projective precovers and covers for complexes are defined similarly as for modules. There is another useful characterization of the Gorenstein projective complexes: Proposition 2 (Yang–Liu–Liang [4]) A complex G is Gorenstein projective if and only if each G n is a Gorenstein projective module. In the proof of our main result we also use Gorenstein flat modules and complexes. They are defined in terms of the tensor product: Definition 3 A module G is Gorenstein flat if there exists an exact complex of flat modules F = · · · → F1 → F0 → F−1 → · · · such that I ⊗ F is still exact for any injective right R-module I, and such that G = K er (F0 → F−1 ). Such a complex F is called F-totally acyclic. We will use GF to denote the class of Gorenstein flat modules. The Gorenstein flat precovers, covers and resolutions are defined in a similar manner as the Gorenstein projective ones (simply replace GP with GF in the definition). The definition of a Gorenstein flat complex follows the one of Gorenstein flat modules, by replacing flat module with flat complex and injective module with injective complex. In the case when R is a coherent ring, there is another useful characterization of the Gorenstein flat complexes: Proposition 4 (Yang–Liu–Liang [4]) Let R be a coherent ring. A complex H is Gorenstein flat if and only if each Hn is a Gorenstein flat module.

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Our first result shows the existence of Gorenstein flat covers in the category of complexes of R-modules, Ch(R), over any right coherent ring R: Proposition 5 Let R be a right coherent ring. Then every complex of left R-modules has a Gorenstein flat cover. Then we prove that, when R is a right coherent and left n-perfect ring, any Ftotally acyclic complex of projective modules is a totally acyclic complex. We use this result to prove the following: Proposition 6 Let R be a right coherent and left n-perfect ring. Then, every Gorenstein flat R-module M has Gorenstein projective dimension less than or equal to n. We use Proposition 6 to prove: Proposition 7 Over a right coherent and left n-perfect ring, every Gorenstein flat complex G has a special Gorenstein projective precover. Our main result is the following: Theorem 8 If R is a right coherent and left n-perfect ring, then the class of Gorenstein projective complexes, GP(C), is a special precovering in Ch(R). The proof of Theorem 8 uses the fact that, by Proposition 5, any complex X φ

→ X . Since the complex G is Gorenstein flat, by has a Gorenstein flat cover, G − ψ

Proposition 7, G has a special Gorenstein projective precover, P − → G. Then we φψ

prove that P −→ X is a special Gorenstein projective precover of X . From Theorem 8 we immediately obtain the following corollaries: Corollary 9 If R is right coherent and left n-perfect, then the class of Gorenstein projective modules, GP, is a special precovering in R-Mod. Corollary 10 If R is a right coherent and left n-perfect ring then (GP, GP ⊥ ) is a complete hereditary cotorsion pair. Examples of right coherent and left n-perfect rings include, but are not limited to Gorenstein rings and commutative Noetherian rings of finite Krull dimension. Jørgensen proved in 2005 that, given any right Noetherian ring with a dualizing complex, there exists an integer n ≥ 0 such that the projective dimension of flat left R-modules is bounded by n. So, any such ring R is both right coherent and left n-perfect for some n, and therefore the class of Gorenstein projective complexes is a special precovering in Ch(R) over such a ring.

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References 1. E.E. Enochs, O.M.G. Jenda, Relative Homological Algebra (Walter de Gruyter, 2000) 2. S. Estrada, A. Lacob, S. Odabasi, Gorenstein flat and projective (pre)covers. Preprint 3. D. Murfet, S. Salarian, Totally acyclic complexes over Noetherian schemes. Adv. Math. 226(2), 1096–1133 (2011) 4. G. Yang, Z. Liu, L. Liang, On Gorenstein flat preenvelopes of complexes. Rend. Sem. Mat. Univ. Padova 129, 171–187 (2013)

Hochschild Homology on Schemes and Fundamental Class Ana Jeremías López

Abstract One of the main features of Grothendieck Duality is the interplay between concrete and abstract aspects of the cohomology theory for quasi-coherent sheaves in Algebraic Geometry. The fundamental class of a scheme-map is the link between these two aspects leading to explicit versions in terms of differentials and residues of local and global duality and the relations between them. In this context, the role of Hochschild homology and cohomology is clear from Lipman (Contemporary Math 61, 1987) [5]. In algebraic geometry, Hochschild homology has been mostly studied in the special case of smooth varieties over a field of characteristic 0. Its importance relies in the connection with Hodge and De Rham cohomology theories through the Hochschild–Konstant–Rosenberg isomorphism. We will describe the first steps of a program for exploring Hochschild homology and cohomology on a general Noetherian base scheme. The original motivation is to define a suitable context for studying the properties of the fundamental class in Grothendieck Duality. The results are joint work with Alonso and Lipman; see Alonso et al. (Asian J Math 15:451–498, 2011) [2], Alonso et al. (Adv Math 257:365–461) [1].

1 Bivariant Hochschild Theory By Poincaré duality, the homology and cohomology of a nonsingular space convey the same information. In the case of singular spaces, MacPherson observed that cohomology and homology play different roles, both important: homology supports characteristic classes, and cohomology is the ring of operations of homology. The bivariant theories constitute a general framework that makes sense of these observations; they were defined and developed in Fulton–MacPherson [4].

A.J. López (B) Departamento de Álxebra, Facultade de Matemáticas, 15782 Santiago de Compostela, Spain e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_16

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1.1 Data for a Bivariant Hochschild Theory Our base scheme will be a Noetherian scheme S. The underlying category Se is the category of separated, flat and essentially finite type schemes over S. The proper maps of Se constitute the class of confined maps. The class of independent squares is formed by those oriented fiber squares in Se X f

Y

g

X f

d g

Y

such that the bottom is an essentially étale morphism. To every scheme X in Se we can associate its derived category Dqc (X ) of complexes of sheaves with quasi-coherent homology. The category Dqc (X ) is monoidal closed. Given a map f in Se , the usual adjunction L f ∗  R f ∗ factors through Dqc . The assignments R()∗ and L()∗ are pseudo-functors. Grothendieck duality provides the fifth operation: the twisted inverse image. It is a D+ qc -valued contravariant pseudofunctor ()!+ on Se obtained by patching two pseudo-functors: the inverse image for essentially étale morphisms and the right adjoint of the direct image for proper maps. The fact that ()!+ makes sense as a pseudo-functor and its basic properties is a non-trivial theory developed by Grothendieck, Hartshorne, Deligne and Verdier, and put up to date and clarified by Lipman, Neeman, Nayak and Conrad, among others. Recently, Neeman [7] has proposed a general base change theorem allowing to define the functor ( )! on Dqc , and proved its functorial properties.

1.2 Definition of a Bivariant Hochschild Theory To define the bivariant Hochschild theory we consider a derived avatar of the Hochschild complex. For each S-scheme x : X → S, let δx : X → X × S X be the diagonal embedding. The complex H X := Lδx∗ Rδx∗ O X is called the pre-Hochschild complex of X . If x is flat, the homologies of H X are the sheafied Hochschild homologies. In the flat case we call H X the Hochschild complex of X . Let H := ⊕i≥0 H i (S, O S ) and let Gr-H be the category of symmetric graded H -modules. The bivariant Hochschild theory HH : Arr(Se ) −→ Gr-H, is defined for each Se -morphism f : X → Y by HHi ( f ) := HomD(X ) (H X , f ! HY [i]), i ∈ Z.

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Any Se -morphism f : X → Y has associated maps f  : HY → f ∗ H X and its étale morphism then f  is an isomoradjoint f  : f ∗ HY → H X . If f is an essentially  phism. These maps, jointly with the counit f : R f ∗ f ! OY → OY attached to a proper finite flat dimension Se -morphism f , are the ingredients to define the operations for the theory HH: given two composable maps f : X → Y and h : Y → Z and elements α ∈ HHi ( f ) and β ∈ HH j (h) their product is an element α · β ∈ HHi+ j (h f ); if f is proper, the push forward homomorphism f  : HHi (h f ) −→ HHi (h) assigns to each α ∈ HHi (h f ), an element f  α ∈ HHi (h); and each independent square d (as in page xxx) has associated the pull back homomorphism g  : HH( f ) → HH( f  ). Theorem 1 The triple B = (Se , Gr-H, HH) is a bivariant theory in the sense of Fulton–MacPherson.

1.3 Bivariant Hochschild Cohomology and Homology Let X ∈ Se . The bivariant Hochschild cohomology of X is defined by id

HHi (X ) := HHi (X − → X ) = ExtiX (H X , H X ) (i ∈ Z). HH∗ is a ring-valued contravariant functor for essentially étale morphisms. The bivariant Hochschild homology of X is defined by x

! → S) = Ext −i HHi (X ) := HH−i (X − X (H X , x O S ) (i ∈ Z);

HH∗ is a graded module over the graded ring HH∗ . It is a covariant functor for proper maps.

1.4 Relation to Classical Hochschild Homology and Cohomology Let x : X → S be a smooth S-scheme of relative dimension n. In [3], C˘aald˘araru defines the Hoschschild homology of x : X → S by HHiCal (X ) := HomD(X ×X ) (δx! O X , δx∗ O X [−i]), i ∈ Z. Unravelling C˘ald˘araru’s definition δx! O X = δx∗ ω−1 X [−n]. Then, having in mind that in the smooth case ω X [n] = x ! O S , and using the properties of Grothendieck operations, we conclude that our bivariant Hochschild homology agrees with the one ∼ ald˘araru’s Hochschild cohomoldefined by C˘ald˘araru: HHCal ∗ (X ) = HH∗ (X ). But C˘ ogy (whose definition goes back to Kontsevich) is a retract of the bivariant Hochschild cohomology.

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2 Definition and Meaning of the Fundamental Class Let f : X → Y be a flat map in Se . We define a natural map c f : δx ∗ δx ∗ f ∗ −→ f ! δ y ∗ δ y∗ combining canonical maps constructed using the duality trace, pseudo functorialities and base change. The natural transformation c f attaches to OY an element of the bivariant group HH( f ), the fundamental class of f c f := c f (OY ) : H X −→ f ! HY ∈ HH0 ( f ). Consider now the absolute case. For X ∈ Se , we get the fundamental class of X , cx : H X −→ x ! O S , as the composition: via

 δx

Bch

ps∗

∼

∼

H X −−−→ δx∗ p2! x ∗ O S −−→ δx∗ p1∗ x ! O S −→ x ! O S . To grasp the significance of this map, let us assume that x : X → S is equidimensional of relative dimension n, and take its −nth homology: cx−n : H −n (H X ) = HHn (X ) −→ ω X |S := H −n (x ! O S ). We may compose with the canonical map nX |S → HHn (X ) to get a derived-category map C X |S : nX |S [n] → x ! O S . This map is the inverse of Verdier’s isomorphism (see [6]), when X is essentially smooth over S, and it is fundamental to understand abstract Grothendieck duality. The bivariant Hochschild theory B = (Se , GrR-Mod, HH) provides us with the right framework to settle the key properties of the fundamental class: • Transitivity: If f : X → Y and g : Y → Z are flat maps in Se , then cg f = c f · cg . • Base change: Let d be an independent square in Se , as in p. xxx. If f is flat, then g  (c f ) = c f  . As a consequence of transitivity, the fundamental class is an orientation for the class of flat maps in the bivariant Hochschild theory.

2.1 The Dual of the Hochschild Bivariant Theory The natural transformation fundamental class gives an oriented bivariant theory B for flat maps. Interchanging the roles of the fundamental class and the Hochschild localization morphism, we construct its dual bivariant theory B. Both B and B associate to X ∈ Se the same cohomology groups, but the homology groups with respect to B are the classical Hochschild homology groups.

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By composing the fundamental class cx : H X → x ! O S with the canonical map H X ⊗ H X → H X , we obtain a pairing px : H X ⊗ H X −→ x ! O S . If x : X → S is essentially smooth, the duality map dx : H X −→ R Hom X (H X , x ! O S ) is an isomorphism, compatible with étale localization. As a consequence, the bivariant groups associated by B and B to a flat map of essentially smooth S-schemes are isomorphic. Acknowledgments This work has been partially supported by Spain’s MINECO and E.U.’s FEDER research projects MTM2011-26088, MTM2014-59456 and Xunta de Galicia’s GRC2013045 2000.

References 1. L. Alonso Tarrío, A. Jeremías López, J. Lipman, Grothendieck duality and Hochschild homology, II: the fundamental class of a flat scheme-map. Adv. Math. 257, 365–461 2. L. Alonso, A. Tarrío, Jeremías López, J. Lipman, Bivariance, Grothendieck duality and Hochschild homology, I: construction of a bivariant theory. Asian J. Math. 15, 451–498 (2011) 3. A. C˘ald˘araru, S. Willerton, The Mukai pairing, I: a categorical approach. New York J. Math. 16, 61–98 (2010) 4. W. Fulton, R. MacPherson, Categorical framework for the study of singular spaces. Mem. Am. Math. Soc. 31(243) (1981) 5. J. Lipman, Residues and traces of differential forms via Hochschild homology. Contemp. Math. 61 (1987) (American Mathematical Society, Providence, RI) 6. J. Lipman, A. Neeman, On the fundamental class of an essentially smooth scheme-map. arXiv:1501.00954v1 7. A. Neeman, An improvement on the base-change theorem and the functor f ! . arXiv:1406.7599v1

A Remark on Leclerc’s Frobenius Categories Martin Kalck

Abstract Leclerc recently studied certain Frobenius categories in connection with cluster algebra structures on coordinate rings of intersections of opposite Schubert cells. We show that these categories admit a description as Gorenstein projective modules over an Iwanaga–Gorenstein ring of virtual dimension at most two. This is based on a Morita type result for Frobenius categories.

1 Motivation Let G be a complex simple Lie group of type Q = A, D, or E (e.g., G = SLn+1 (C) for Q = An ) with Borel subgroup B ⊂ G (e.g., B = {upper triangular matrices}) and Weyl group W (e.g., W ∼ = Sn+1 given by permutation matrices). For a Weyl group element w ∈ W there are associated subvarieties Cw (Schubert cell) and C w (opposite Schubert cell) in the flag variety G/B. On the other hand, there is a torsion pair (Cw , C w ) in the category of finite dimensional modules over the preprojective algebra  := (Q), and the categories Cw , C w are Frobenius and have projective generators (in fact, the latter statements may be deduced from Proposition 5). These Frobenius categories were used by Geiß–Leclerc–Schröer [3] to categorify cluster algebra structures on coordinate rings of the corresponding (opposite) Schubert cells. Let v ∈ W . The intersections Cv,w := C v ∩ Cw are known as open Richardson varieties and have been studied by Kazhdan–Lusztig in connection with KLpolynomials. Generalizing the aforementioned work [3], Leclerc [5] categorifies a cluster subalgebra of the coordinate rings of Cv,w using the intersection Cv,w of a torsion free part C v with a torsion part Cw of two torsion pairs mentioned above. Under some finiteness assumptions, he obtains a cluster algebra structure on the whole coordinate ring and he conjectures that this holds in general. The subcategories Cv,w ⊆ mod  inherit an exact structure which is again Frobenius. Our aim is to explain this in a more abstract setting and give equivalent descripM. Kalck (B) University of Edinburgh, Scotland, UK e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_17

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tions of Cv,w . This is summarized in the following Proposition which is a special case of Proposition 5. Proposition 1 Let Cv,w := Cw ∩ C v ⊆ mod . Then, (i) Cv,w is a Frobenius category with proj Cv,w = add f v tw () = add tw f v () =: add Pv,w , where tu (−) denotes the torsion radical and f u (−) := (−)/tu (−) for a torsion pair (Cu , C u );   HomCv,w (Pv,w ,−) (ii) Cv,w −−−−−−−−−→ GP v,w is an exact equivalence, where v,w is the Iwanaga–Gorenstein ring v,w := EndCv,w (Pv,w ), of virtual dimension at most two; (iii) in particular, Cv,w is equivalent to the subcategory of second syzygies of finite dimensional v,w -modules; (iv) the functors f v and tw induce ring homomorphisms w := EndCw (tw ()) → v,w and v := EndC v ( f v ()) → v,w , which are surjective if Cv ⊆ Cw ; in turn, this condition is equivalent to w = v  v with l(w) = l(v  ) + l(v), called condition (P) in Leclerc [5, 5.1]; (v) ([1, 5.16]) if condition (P) holds, then v,w is Morita equivalent to v ; therefore, v,w has the same virtual dimension as v which is at most 1 (see [2]). Remark 2 Let w := /Iw be the algebra considered in [2]. Then, there are algebra −1 op isomorphisms w ∼ = w0 w ∼ = w−1 , where w0 denotes the longest Weyl group element.

2 A Morita Type Result for Frobenius Categories Definition 3 A two-sided Noetherian ring R is called Iwanaga-Gorenstein, when both right and left injective dimensions are finite, inj.dim R R < ∞ and inj.dim R R < ∞. It is well-known that this implies inj.dim R R = d = inj.dim R R . We call d =: vir.dim R the virtual dimension of R. In this case the category of Gorenstein-projective R-modules GP(R) := {M ∈ mod R | ExtiR (M, R) = 0 for all i > 0} is a Frobenius category with subcategory of projective-injective objects proj R. Equivalently, GP(R) is the subcategory of d-th syzygies of finitely generated Rmodules GP(R) ∼ = d (mod R) := {d (M) | M ∈ mod R}. If R is a local commutative Noetherian ring, Gorenstein projective R-modules are precisely maximal Cohen–Macaulay R-modules and inj.dim R R = kr.dim R. Our aim is to characterize the categories of Gorenstein projective modules GP(R) over Iwanaga–Gorenstein rings R among all Frobenius categories.

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For an additive category B, we denote by mod B the category of finitely presented contravariant additive functors B → Ab. We first list properties of the categories E := GP(R) for R Iwanaga–Gorenstein: (i) proj E = add P (= proj R) for some P ∈ E and EndE (P) (∼ = End R (R)) is two-sided Noetherian; (ii) E is idempotent complete (since E ⊆ mod R is closed under direct summands); (iii) E is Frobenius (use exact duality Hom R (−, R) : GP(R) → GP(R op )); (iv) E has weak kernels and cokernels (use Auslander–Buchweitz approximation); (v) gl.dim mod E, gl.dim mod E op ≤ n (= max{2, inj.dim R}). The following result may be interpreted as an analogue of Morita theory for Frobenius categories. The implication (b) ⇒ (a) is well-known. The converse is the special case proj E = add P, M = E from [4, 2.8], which is due to Iyama and inspired by a stable version of Dong Yang and the author [4, 2.15]. Proposition 4 Let E be an exact category and let P ∈ E. The following are equivalent: (a) E and P satisfy the conditions (i)-(v) above; (b) R = EndE (P) is Iwanaga–Gorenstein, HomE (P, −) : E → GP(R) is an exact equivalence, and vir.dim R ≤ gl.dim mod E.

3 From Pairs of Torsion Pairs to Frobenius Categories Let (T , F) be a torsion pair in an abelian category A. In particular, there is a short exact sequence 0 → t (X ) → X → f (X ) → 0 for all X in A. This gives rise to functors t : A → T and f : A → F, which are right (resp. left) adjoint to the canonical inclusions. Proposition 5 Let A be an abelian category with torsion pairs (T1 , F1 ) and (T2 , F2 ) and set C12 := T1 ∩ F2 . Then the following statements hold: (i) C12 is extension closed and idempotent complete, since T1 and F2 are. In particular, C12 inherits a natural exact structure from A. (ii) C12 has kernels and cokernels. In other words, C12 is a preabelian category. In particular, the categories of finitely presented additive functors mod C12 and op mod C12 are abelian and have global dimension at most 2. For example, the f

→ Y is a composition of the canonical inclusions t1 (ker f ) → ker f → X − kernel of f (here, ker f denotes the kernel of f in A). (iii) If T1 has enough projectives and F2 has enough injectives, then C12 has enough injectives (= t1 (inj F2 )) and projectives (= f 2 (proj T1 )). (iv) If additionally Ext1C12 (X, Y ) = 0 ⇔ Ext1C12 (Y, X ) = 0, then C12 is Frobenius. For example, this is satisfied if A or Db (A) are 2-Calabi–Yau. This in turn is  where Q is a quiver without loops and known to hold for A = fdmod((Q)),

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 is the m-adic completion of its preprojective algebra, where m denotes (Q) the ideal generated by all arrows. (v) Assume, additionally, that proj T1 = add P and injF2 = add I , then proj C12 = add f 2 (P) = add t1 (I ). If 12 := EndC12 ( f 2 (P)) is two-sided Noetherian,   HomC12 ( f 2 (P),−) then there is an exact equivalence C12 −−−−−−−−−→ GP 12 , and 12 is Iwanaga–Gorenstein of virtual dimension at most 2. (vi) In the situation of (v) the functors f 2 and t1 induce ring homomorphisms ϕ2 : EndT1 (P) → 12 and τ1 : EndF2 (I ) → 12 with kernels given by the ideals of morphisms factoring over t2 (P) and f 1 (I ), respectively. The ring homomorphisms are surjective if T2 ⊆ T1 . In Example 7, ϕ2 is injective but not surjective. Remark 6 This is an analogue of Buan–Iyama–Reiten–Scott [2] dual description of Geiß–Leclerc–Schröer’s categories Cw , as categories of submodules of projective modules over the algebra w ; see also [3, Theorem 2.8]. Since w is Iwanaga– Gorenstein of virtual dimension 1, Gorenstein projective modules are first syzygies, which in turn are just submodules of projective modules. See also [4, Section 6] for a further discussion.

4 Examples, Remarks and Questions Example 7 We consider the situation of [5, 3.16], i.e., Q is of type A3 , w = s1 s3 s2 s1 s3 and v = s2 . Then, ϕ2 : w := EndCw (tw ()) → v,w is injective and its cokernel in the category of vector spaces is isomorphic to C. Moreover, v,w is the Auslander algebra of the preprojective algebra of type A2 and therefore is of global (and virtual) dimension 2. Remark 8 (Duality) Let Q be a Dynkin quiver and let D := Homk (−, k) be the standard duality. It is well-known that there is an algebra isomorphism ψ :  ∼ = op , D

ψ∗

→ mod op − → mod . Using the notawhich gives rise to a duality : mod  − tion in Leclerc [5, § 3.2], one can check that (Pv,w ) ∼ = Pw0−1 w,w0 v holds, where w0 denotes the longest Weyl group element. In particular, induces an algebra isoop op op morphism v,w ∼ = w−1 w,w v . Thus, v ∼ = v,w0 ∼ = id,w0 v ∼ = w0 v for the algebras 0 0 appearing in Proposition 1(iv). Open Problem 9 Find a ‘combinatorial description’ of v,w , e.g., as a quiver with relations. Remark 10 The number of isoclasses of indecomposable projective v,w -modules seems to be unknown in general. It is not always bounded above by |Q 0 |; see Example 7. Question 11 (Leclerc) How does the virtual dimension of v,w depend on Q, v, w and how is this number related to the geometry of the open Richardson variety Cv,w ?

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By Remark 2 and [2], vir.dim v , w ≤ 1. They are zero if and only if C v (resp., Cw ) are exact abelian subcategories of mod , which are then equivalent to mod Pi/e (e ∈  idempotent). Thus, if vir.dim v , w = 0, then vir.dim v,w = 0 (since Cv,w is abelian). If one of v and w has virtual dimension zero, then Cv,w is the torsion (or torsion-free) part of a torsion pair in mod /e. By Mizuno [6] and Buan–Iyama–  Reiten–Scott [2], v,w ∼ = (/e)w and is therefore of virtual dimension = (/e)v or ∼ at most 1. Also, gl.dim v = n ≤ 1 (or gl.dim w = m ≤ 1) implies gl.dim v,w ≤ min{n, m}. If both v , w have infinite global dimension and virtual dimension 1, then virtual dimensions 0, 1, 2 occur for v,w . Remark 12 (Commutativity) It follows from Mizuno [6], that all torsion pairs in mod  are of the form (Cw , C w ) for some Weyl group element w. In particular, there are only finitely many torsion pairs, which is very surprising given the size of mod . The explicit description of the associated functors tw and f w (see, e.g., Leclerc [5, § 3.2]) shows that f v tw (M) ∼ = tw f v (M) for Weyl group elements v, w ∈ W and M ∈ mod . This seems very unusual for a pair of torsion pairs in general abelian categories and fails already for mod U2 (k), where U2 (k) denotes the ring of 2 × 2 upper triangular matrices. Acknowledgements This material grew out of a discussion with Bernard Leclerc and Henning Krause after a talk of Leclerc on the results of [5] in spring 2014. I am grateful to Bernard Leclerc for his inspiring work, his interest and discussions. Moreover, I would like to thank Henning Krause for insisting that these considerations might be of interest and Osamu Iyama for very inspiring discussions. In particular, I learned parts of Proposition 5 from him. I am very grateful to Michael Wemyss for lots of stimulating questions on this topic. I also had fruitful discussions with Sergio Estrada, Mikhail Gorsky, Frederik Marks, Yuya Mizuno and Milen Yakimov. Thanks to the organizers of this conference for the opportunity to present this work and to the participants for their interest and questions. I am grateful to EPSRC for financial support (EP/L017962/1).

References 1. P. Baumann, J. Kamnitzer, P. Tingley, Affine Mirkovi´c-Vilonen polytopes. Publ. Math. Inst. Hautes Étud. Sci. 120, 113–205 (2014) 2. A. Bakke Buan, O. Iyama, I. Reiten, J. Scott, Cluster structures for 2-Calabi–Yau categories and unipotent groups. Compos. Math. 145(4), 1035–1079 (2009) 3. C. Geiß, B. Leclerc, J. Schröer, Kac-Moody groups and cluster algebras. Adv. Math. 228, 329– 433 (2011) 4. M. Kalck, O. Iyama, M. Wemyss, D. Yang, Frobenius categories, Gorenstein algebras and rational surface singularities. Compos. Math. 151(3), 502–534 (2015) 5. B. Leclerc, Cluster structures on strata of flag varieties. Adv. Math. arXiv:1402.4435 6. Y. Mizuno, Classifying τ -tilting modules over preprojective algebras of Dynkin type. Math Z. 277(3–4), 665–690 (2014)

Atom-Molecule Correspondence in Grothendieck Categories Ryo Kanda

Abstract For a one-sided Noetherian ring, Gabriel constructed two maps between the isomorphism classes of indecomposable injective modules and the two-sided prime ideals. In this note, we provide a categorical reformulation of Gabriel’s maps and investigate further properties of them.

1 Introduction For a one-sided Noetherian ring, Gabriel [1] described the relationship between indecomposable injective modules and two-sided prime ideals as follows. Theorem 1 (Gabriel [1]) Let Λ be a right Noetherian ring. Then, we have two maps ϕ

{indecomposable injective right Λ -modules}/∼ =  {two-sided prime ideals of Λ} ψ

characterized by the following properties: (i) for each indecomposable injective right Λ-module I , the only associated (twosided) prime of I is ϕ(I ); (ii) for each two-sided prime ideal P of Λ, the injective envelope E(Λ/P) of the right Λ-module Λ/P is the direct sum of a finite number of copies of the indecomposable injective Λ-module ψ(P). Moreover, the composite ϕψ is the identity map. In this note, we generalize Theorem 1 to a certain class of Grothendieck categories as maps between the atom spectrum and the molecule spectrum of a Grothendieck category. Moreover, by using naturally defined partial orders on these spectra, we establish a bijection between the minimal elements of the atom spectrum and those of the molecule spectrum. R. Kanda (B) Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya-shi, Aichi-ken 464-8602, Japan e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_18

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2 Atom Spectrum Throughout this note, let A be a Grothendieck category. Definition 2 A nonzero object H in A is called monoform if, for each nonzero sub-object L of H , no nonzero sub-object of H is isomorphic to a sub-object of H/L. For monoform objects H1 and H2 in A, we say that H1 is atom-equivalent to H2 if there exists a nonzero sub-object of H1 which is isomorphic to a sub-object of H2 . The atom equivalence is, in fact, an equivalence relation among the monoform objects in A. Proposition 3 (Storrer [4, p. 626]) Let R be a commutative ring. Then a nonzero object H in Mod R is monoform if and only if there exist p ∈ Spec R and a monomorphism H → k(p) in Mod R, where k(p) = Rp /pRp . Definition 4 The atom spectrum ASpec A of A is the quotient set of the set of monoform objects in A by the atom equivalence. Each element of ASpec A is called an atom in A. For each monoform object H in A, the equivalence class of H is denoted by H . The notion of atoms was originally introduced by Storrer [4] for module categories and generalized to arbitrary abelian categories by Kanda [2]. The next result shows that the atom spectrum of a Grothendieck category is a generalization of the prime spectrum of a commutative ring. Proposition 5 (Storrer [4, p. 631]) Let R be a commutative ring. Then the map Spec R → ASpec(Mod R) given by p → R/p is bijective. Definition 6 Let M be an object in A. Define the subset AAss M of ASpec A by AAss M = {α ∈ ASpec A | α = H for some monoform sub-object H of M}. We call each element of AAss M an associated atom of M. Similarly, define the subset ASupp M of ASpec A by ASupp M = {α ∈ ASpec A | α = H for some monoform sub-quotient H of M}. We call it the atom support of M. Proposition 7 Let R be a commutative ring, and let M be an R-module. The bijection Spec R → ASpec(Mod R) in Proposition 5 induces bijections Ass M → AAss M and Supp M → ASupp M. Definition 8 For α, β ∈ ASpec A, we write α ≤ β if β ∈ ASupp H holds for each monoform object H in A with H = α.

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Proposition 9 Let R be a commutative ring. The bijection Spec R → ASpec (Mod R) in Proposition 5 is an isomorphism between the partially ordered sets (Spec R, ⊂) and (ASpec(Mod R), ≤). Let I and J be indecomposable injective objects in A. We write I ≤ J if ⊥ I ⊃ ⊥ J , where ⊥ I := { M ∈ A | HomA (M, I ) = 0 }. In general, this is a partial preorder among the isomorphism classes of indecomposable injective objects in A. Theorem 10 Let A be a locally Noetherian Grothendieck category. Then we have a poset isomorphism ∼ (ASpec A, ≤) − → ({indecomposable injective objects in A}/∼ =, ≤) given by H → E(H ). In particular, the partial preorder among the isomorphism classes of indecomposable injective objects in A is a partial order. Note that for a right Noetherian ring Λ, the category Mod Λ of right Λ-modules is a locally Noetherian Grothendieck category, and hence Theorem 10 can be applied to Mod Λ.

3 Molecule Spectrum In this section, we introduce a new spectrum of a Grothendieck category, which we call the molecule spectrum. Definition 11 A full subcategory C of A is called closed if C is closed under subobjects, quotient objects, arbitrary direct sums, and arbitrary direct products. For closed subcategories C and D of A, denote by C ∗ D the full subcategory of A consisting of all objects M in A for which there exists an exact sequence 0→L→M →N →0 with L ∈ C and N ∈ D. The following well-known result shows that closed subcategories of a Grothendieck category should be considered analogously to the two-sided ideals of a ring. Proposition 12 Let Λ be a ring. (i) We have a poset isomorphism ∼ ({two-sided ideals of Λ}, ⊂) − → ({closed subcategories of Mod Λ}, ⊃) given by I → Mod(Λ/I ), where Mod(Λ/I ) is identified with the full subcategory {M ∈ Mod Λ | M I = 0} of Mod Λ.

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(ii) Let I and J be two-sided ideals of Λ. Then we have Mod

Λ Λ Λ = Mod ∗ Mod IJ J I

as a full subcategory of Mod Λ, i.e., the isomorphism in (i) induces an isomorphism of monoids, ∼ ({two-sided ideals of Λ}, ·) − → ({closed subcategories of Mod Λ}, ∗). (iii) Let M be a right Λ-module. Then the two-sided ideal AnnΛ (M) corresponds, by (i), to the smallest closed subcategory M closed of A containing M. Definition 13 A nonzero closed subcategory P of A is called prime if for each closed subcategories C and D satisfying P ⊂ C ∗ D, we have P ⊂ C or P ⊂ D. Proposition 14 Let Λ be a ring. Then the isomorphism in Proposition 12(i) induces a poset isomorphism ∼ ({two-sided prime ideals of Λ}, ⊂) − → ({prime closed subcategories of Mod Λ}, ⊃).

Definition 15 A nonzero object H in A is called prime if, for each nonzero subobject L of H , it holds that L closed = H closed . For prime objects H1 and H2 in A, we say that H1 is molecule-equivalent to H2 if H1 closed = H2 closed . Definition 16 The molecule spectrum MSpec A of A is the quotient set of the set of prime objects in A by the molecule equivalence. Each element of MSpec A is called a molecule in A. For each prime object H in A, the equivalence class of H is . denoted by H Proposition 17 Let A be a Grothendieck category with a Noetherian generator. Then, we have a bijection ∼ MSpec A − → {prime closed subcategories of A}  → H closed . For each ρ = H  ∈ MSpec A, the prime closed subcategory given by H H closed corresponding to ρ is denoted by ρ closed . Definition 18 Let A be a Grothendieck category with a Noetherian generator. For ρ, σ ∈ MSpec A, we write ρ ≤ σ if ρ closed ⊃ σ closed holds. The partial order on MSpec A can be also defined for Mod Λ, where Λ is an arbitrary ring, and we can show that there exists a poset isomorphism between the set of two-sided prime ideals of Λ and MSpec(Mod Λ).

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4 Atom-Molecule Correspondence Let A be a Grothendieck category having a Noetherian generator and satisfying the Ab4* condition, that is, direct product preserves exactness. For a right Noetherian ring Λ, the category Mod Λ satisfies this assumption. Denote by AMin A (resp., MMin A) the set of minimal elements of ASpec A (resp., MSpec A). As shown in [3, Proposition 4.7], the set AMin A is not empty unless the Grothendieck category is zero. The following theorem is our main result. Theorem 19 (i) The poset homomorphism ϕ : ASpec A → MSpec A given by , where H is taken to be a prime monoform object in A, is surjective. H → H ∼ (ii) The map ϕ induces a bijection AMin A − → MMin A. (iii) There exists an injective poset homomorphism ψ : MSpec A → ASpec A satisfying the following properties: (1) the composite ϕψ is the identity map on MSpec A; (2) for each ρ, σ ∈ MSpec A, we have ρ ≤ σ if and only if ψ(ρ) ≤ ψ(σ ); (3) for each α ∈ ASpec A and ρ ∈ MSpec A, we have ψ(ρ) ≤ α if and only if ρ ≤ ϕ(α). Acknowledgements The author is a Research Fellow of Japan Society for the Promotion of Science. This work is supported by Grant-in-Aid for JSPS Fellows 25·249.

References 1. P. Gabriel, Des catégories abéliennes. Bull. Soc. Math. France 90, 323–448 (1962) 2. R. Kanda, Classifying Serre subcategories via atom spectrum. Adv. Math. 231(3–4), 1572–1588 (2012) 3. R. Kanda, Specialization orders on atom spectra of Grothendieck categories. J. Pure Appl. Algebra 219(11), 4907–4952 (2015) 4. H.H. Storrer, On Goldman’s primary decomposition. Lectures on rings and modules, Lecture Notes in Mathematics, vol 246 (Springer, Berlin-Heidelberg-New York, 1972), pp. 617–661

Proalgebraic Crossed Modules of Quasirational Presentations Andrey Mikhovich

Abstract For every quasirational (pro- p) relation module R, we construct the so Q p = lim R/[R, R called p-adic rationalization, which is the pro-fd module R ⊗ ← − ∧ ∼  Mn ] ⊗ Q p , and prove the isomorphism R ⊗Q p = Rw (Q p ), where Rw∧ (Q p ) stands for the rational points of the abelianization of the continuous p-adic Malcev completion of R. We show how Rw∧ embeds into a sequence which arises from a certain prounipotent crossed module. The latter can be seen as concrete examples of proalgebraic homotopy types. We provide the Identity Theorem for pro- p-groups, answering a question of Serre.

1 Schematization and Proalgebraic Crossed Modules What is homotopy theory? After Quillen, we may regard this as a formal setting for model categories and their equivalences. The Quillen equivalence between homotopy categories of compactly generated Hausdorff spaces and simplicial sets is the brightest example (hence, the importance of simplicial sets). However, Grothendieck’s meditative dream was that combinatorial homotopy theory lies not far from geometry in its schematic reincarnation (just as homotopical invariants of smooth manifolds). Recently, this idea was realized in B. Toën’s theory of schematic homotopy types. The constructive form gives for a pair (X, k) (where X is a connected simplicial set and k is a field) some schematic homotopy type, which is a simplicial proal∧ ∧ , where G is a Kan loop-group functor, and (G X )alg is the gebraic group (G X )alg proalgebraic completion of a free simplicial group G X . Lets look at old problems of two-dimensional combinatorial homotopy theory using schematic glasses. First, we apply Kan loop-group functor to connected two-dimensional simplicial sets and obtain a free simplicial group degenerated in dimensions greater than one. Kan results A. Mikhovich (B) Moscow State University, Moscow, Russia e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_19

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give a C W -basis for this free simplicial group, so we get some free simplicial group also degenerated in dimensions greater than one, o

//

/ F(X ∪ Y )

s0

/ F(X ) ,

d0 d1

/

(1)

where d0 , d1 , s0 , on x ∈ X , y ∈ Y , r y ∈ F(X ) are defined by d1 (x) = x, d1 (y) = r y , d0 (x) = x, d0 (y) = 1, s0 (x) = x. The corresponding 2-reduced simplicial group o F(X ∪ Y )

s0

/ F(X )

d0

/

d1

is the standard object for study in combinatorial group theory. We use the proalgebraic completion to jump into reduced schematic homotopy types

F(X ∪

o

s0 ∧ / F(X )alg .

d0

∧ Y )alg

/

d1

∧ (k) here are too big for practical purposes, and we But the groups of k-points Falg need to find a reasonable approximation, which is sufficiently rich for 2-dimensional combinatorial homotopy. Old constructions due to Quillen (see Quillen’s formula in [20, Cor. 21]) and Magnus (see below) contain the helpful hint.

Remark 1 (Magnus embedding) Several important homotopical invariants in the theory of groups and geometric topology are defined using the embedding of the group ring of a free group F into the algebra of formal power series on noncommutative indeterminates: μ : ZF(x1 , . . . , xn ) → Z tx1 , . . . , txn

x i  → 1 + txi , where μ(xi−1 ) = 1 − txi + tx2i − · · · . Indeed, integral coefficients do not play a principal role, and we use the embedding F(x1 , . . . , xn ) → k tx1 , . . . , txn with coefficients in any field k. Now Pontriagin–VanKampen duality gives a possibility to look at defining relations as linear functions on a certain commutative Hopf algebra; see [20, 3]. This all means, by practical reasons, that we must restrict ourselves to the prounipotent completions and to corresponding prounipotent homotopy types. In the case of 2-homotopy we will work with 2-reduced simplicial prounipotent groups,

F(X ∪ Y )∧u

o

s0

/ F(X )∧u .

d0 d1

/

(2)

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Let Ru = d1 (ker (d0 )) be the corresponding Zarissky normal closure of defining relations and G u := Fu (X )/Ru . Unless otherwise stated, we will consider only finite presentations (|X | < ∞, |Y | < ∞), because we still have no sufficient understanding of what F(X )∧u is in the infinite case. The choice of prounipotent affine group schemes is not occasional, there is an entirely schematic explanation. In fact, (G/H )(k) ∼ = G(k)/H (k) (char (k) = 0) and so, group-theoretic settings are compatible with schematic ones. The simplicial identity d0 s0 = id Fu (X ) (saving notions) implies that (as in the discrete case) we have the formula Fu (X ∪ Y ) ∼ = K er d0  s0 Fu (X ). Now, one can collect necessary homotopical information from d1

the study of K er d0 − → Fu (X ) with the action of Fu (X ) through s0 by conjugation on K er d0 . (K er d0 , Fu (X ), d1 ) is a particular example of a prounipotent precrossed module. The construction of prounipotent (pre)crossed modules in general (and their category as well) is standard, and we refer the reader to [1]. To obtain a bridge between ordinary combinatorial group theory and prounipotent (pre)crossed modules we need the following definition from [7, 16, 17]. The group of Q p -points of any affine group scheme G has the p-adic topology. Indeed, [2] shows that G can be expressed as a filtered inverse limit G = lim G α of ← − linear algebraic groups. Each G α (Q p ) has a canonical p-adic topology induced by the embedding G α → G L n . Define the topology on G(Q p ) by G(Q p ) = lim G α (Q p ). ← − Definition 2 Fix a topological group G (with the pro- p topology in our further considerations). Define the p-adic Malcev completion of G by the following universal diagram, where ρ is a continuous Zarisky-dense homomorphism of G into the Q p points of a prounipotent affine group G ∧w , G ∧w (Q p ) 8 q q q q q qq τ G MM MMM M χ M&  H (Q p ) . ρ

We require that, for every continuous Zarisky-dense homomorphism χ into the Q p points of a prounipotent affine group H , there is a unique homomorphism τ of prounipotent groups, making the diagram commutative. Remark 3 Finite cardinality of Z is a sufficient condition to identify a free prounipotent group Fu (Z ) with F(Z )∧w and F(Z )∧u . However, we use different letters “u” or “w” to emphasize the source (‘w” in the case of pro- p continuous Malcev completions, “u” in the case of prounipotent presentations). If (2) comes from a pro- p presentation (since any homomorphism of a finitely generated pro- p group into the Q p -points of a prounipotent group is continuous, see [7, Lemma A.7]) we also have the isomorphism G u ∼ = G ∧w .

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Definition 4 ([14]) A free prounipotent (pre)crossed G 2 -module on X is a prounipotent crossed module (G 1 , G 2 , d) with a function ν : X → G 1 (Q p ) which has a Zarisky dense group closure ν(X ) ∈ G 1 (Q p ) such that, for every prounipotent (pre)crossed G 2 -module (G 1 , G 2 , d ) and any function ν : X → G 1 (Q p ) having a Zarisky dense group closure ν (X ) ∈ G 1 (Q p ), there is a homomorphism of prounipotent (pre)crossed G 2 -modules ϕ : (G 1 , G 2 , d) → (G 1 , G 2 , d ) such that ν = ϕk ν, where ϕk is the corresponding homomorphism of groups of k-points ϕk : G 1 (Q p ) → G 1 (Q p ). d1

→ Fu (X ) is a free prounipotent precrossed Lemma 5 (Mikhovich [14]) K er d0 − module on Y , where d1 comes from the presentation (2). In [14], we construct the prounipotent crossed module (Cu , Fu (X ), d) of the d

d0 − → Fu (X ), where Pu =< d(b) a = prounipotent presentation (2) as follows: K er Pu −1 bab > is the Zarisky normal closure of Peiffer commutators and d arises from d1 in the presentation (2). The equality Pu (A) = [K er d0 , K er d1 ](A) holds for any Q p -algebra A, which certainly implies:

Lemma 6 (Mikhovich [14]) There is an isomorphism of prounipotent crossed modules ( K erPu d0 , Fu (X ), d) ∼ = ( [K erKd0er,Kd0er d1 ] , Fu (X ), d). Lemma 7 (Mikhovich [14]) Ru acts trivially on C u , where C u is a factor of Cu by the derived subgroup and hence, C u (Q p ) is a O(G u )∗ -module, where O(G u )∗ is the Pontryagin–VanKampen dual of the representing Hopf algebra O(G u ) of G u .

2 Quasirational Presentations For pro- p groups, fix a prime p > 0 throughout the paper; see [19] for details on pro- p groups. For discrete groups, p will vary. Let G be a (pro- p) group which has a (pro- p) presentation of finite type 1 −→ R −→ F −→ G −→ 1.

(3)

Let R = R/[R, R] be the corresponding relation G-module, where [R, R] is a (closed) commutator subgroup (in the pro- p case). Then, denote Mn the corresponding Zassenhaus p-filtration of F, which is defined by the rule Mn = { f ∈ F | f − 1 ∈ np , p = ker (F p F → F p )}; see [9, 7.4] for details. Definition 8 The presentation (3) is quasirational if, for every n > 0 and each prime p > 0, the F/RMn -module R/[R, RMn ] has no p-torsion ( p is fixed for pro- p groups, and runs through all primes p > 0 and corresponding p-Zassenhaus filtrations in the discrete case). The relation modules of such presentations will be called quasirational relation modules.

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Proposition 9 (Mikhovich [13]) Suppose (3) is a presentation of a pro- p group G with a single defining relation. Then (3) is quasirational. Quasirational presentations may be studied by passing to rationalized completions Q p := lim R/[R, RMn ] ⊗ Q p . Since lim is left exact for quasirational (pro- p) R⊗ ← − ← − Q p ) in the spirit of presentations, we have an embedding of abelian groups R → R ⊗ Gaschütz theory; see [4]. We modify the definition of topological module in [5] by a Q p is crucial: stronger topology on M ⊗ A. Then, the following description of R ⊗ Theorem 10 (Infinite Gaschütz Theory [14]) Let (3) be a Q R- (pro- p) presentation, then there is a diagram of topological O(G u )∗ -modules γ

/ R∧ Cu A w AA AA AA τ A 

Ru ,

where Cu , Rw∧ , Ru are the abelianizations of the corresponding prounipotent groups. Furthermore, ∗ |Y |  , where Qp (i) Cu (Q p ) ∼ = O(G u )∗ |Y | ∼ = Q p G u |Y | ∼ = UP(O(G u) G u := EndG u (O(G u )); (ii) the structure of topological O(G u )∗ -module on Rw∧ (Q p ) comes from the isomorQ p . phism Rw∧ (Q p ) ∼ = R⊗ Remark 11 The celebrated Lyndon Identity Theorem states that relation modules of one-relator presentations of discrete groups are induced from cyclic subgroups, i.e., if (3) is one-relator, then R = R/[R, R] ∼ = Z ⊗u ZG, where R = (u m ) F and u is not a proper power. We provide the following analog for pro- p groups with a single defining relation (i.e., for pro- p groups G with dim F p H 2 (G, F p ) = 1); this is a problem from [18, 10.2]. Corollary 12 (Identity Theorem for pro- p groups, [15]) If (3) is a one-relator (not necessary finitely generated) pro-p group, then there is an isomorphism of topological Q p ∼ O(G u )∗ -modules R ⊗ = O(G u )∗ . Proof It is enough to prove our statement for finitely generated one-relator pro- p groups, because of the decomposition of F into the inverse limit of finitely generated free pro- p groups in [3]. There are two cases: (i) The corresponding prounipotent presentation is degenerated, i.e., G u is free prounipotent, then we prove the isomorphism R u (Q p ) ∼ = O(G u )∗ directly by showing that Ru is a free precrossed G u -module; see [15]. (ii) In the case of proper prounipotent presentations [10, 3.10] we use the modern version of Hochschild cohomology of affine groups [8] (a concise introduction may be found in [6]) to generalize the results from [10, 11] for k = k, and to provide the equality dim k H 2 (G u , k) = 1.

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Further generalization of [10, Theorem 3.14] implies that cd(G u ) = 2 and, reproving [11, Prop. 3.13] and using Pontryagin–VanKampen duality, we obtain Ru (Q p ) ∼ =  O(G u )∗ . Finally, Infinite Gaschütz Theory gives the required isomorphism. Many ideas and results discussed here could be developed for a field of positive characteristic. Anyway the task to compare p-adic pro-algebraic and pro- p completions seems very interesting, and several results were obtained in [17]. From this general perspective, quasirationality emphasizes a space where deep interactions between positive and zero characteristics are possible. We just mention the construction of conjurings [15] (Amitsur–Lewitzky elements of noncyclic free pro- p groups, but substantially deforming the p-power structure) elucidating probable absence of asphericity [12] for one-relator pro- p groups in general.

References 1. R. Brown, J. Huebschmann, Identities among relations, low-dimensional topology. London Math. Soc. Lect. Notes Ser. 48, 153–202 (1982) 2. P. Deligne, J.S. Milne, Tannakian categories in Hodge cycles, motives, and Shimura varietes. Lect. Notes Math. 900, 101–228 (1982) 3. D. Gildenhuys, C.K. Lim, Free pro-C groups. Math. Z. 125, 233–254 (1972) 4. K. Gruenberg, Relation modules of finite groups, in Published as the Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 25 (American Mathematical Society, 1976) 5. R. Hain, Algebraic cycles and variations of mixed Hodge structure in Complex geometry and Lie theory. Proc. Symp. Pure Math. 53, 175–221 (1991) 6. R. Hain, Deligne–Beilinson cohomology of affine groups. arXiv:1507.03144 7. R. Hain, M. Matsumoto, Weighted completion of Galois groups and Galois actions on the fundamental group of P1 − {0, 1, ∞}. Compos. Math. 139(2), 119–167 (2003) 8. J.C. Jantzen, Representations of algebraic groups, in AMS Mathematical Survey and Monography, vol. 107, 2nd edn. (2003) 9. H. Koch, Galoissche theorie der p-erweiterungen (Berlin-New York; VEB Deutscher Verlag der Wissenschaften, Berlin, Springer, 1970) 10. A. Lubotzky, A. Magid, Cohomology of unipotent and prounipotent groups. J. Algebra 74, 76–95 (1982) 11. A. Lubotzky, A. Magid, Cohomology, Poincare series, and group algebras of prounipotent groups. Amer. J. Math. 107, 531–553 (1985) 12. O.V. Mel’nikov, Aspherical pro- p-groups. Mat. Sb. 193(11), 71–104 (2002) 13. A. Mikhovich, Quasirational relation modules and p-adic Malcev completions. Topol. Appl. 201, 86–91 (2016) 14. A. Mikhovich, Quasirationality and prounipotent crossed modules. To appear in Math. Notes 15. A. Mikhovich, Identity theorem for pro- p groups 16. J.P. Pridham, Galois actions on homotopy groups. Geom. Topol. 15(1), 501–607 (2011) 17. J.P. Pridham, On the l-adic pro-algebraic and relative pro-l fundamental groups, in Arithmetics of Fundamental Groups. Control in Mathematics and Computer Sciences, vol. 2 (Springer, 2012), pp. 245–279 18. J.P. Serre, Structure de certains pro- p groupes (d’aprs Demushkin). Semin. Bourbaki 252 (1963) 19. J.P. Serre, Galois Cohomology (Springer, Springer Monographs in Mathematics, 1997) 20. A. Vezzani, The pro-unipotent completion (2012), http://users.mat.unimi.it/users/vezzani/ Files/Research/prounipotent.pdf

Some F-Invariants for Quotient Singularities Yusuke Nakajima

Abstract For the case of quotient singularities, we discuss some numerical invariants defined via the Frobenius morphism.

1 Introduction Throughout this note, we suppose that k is an algebraically closed field of prime characteristic p > 0. Let (A, m, k) be a Noetherian local ring with char A = p > 0. Since char A = p > 0, we define the Frobenius morphism F : A → A, (a → a p . For e ∈ N, we can also define the e-times iterated Frobenius morphism F e : A → A, e a → a p . For any A-module N , we denote the module N with its A-module structure pulled back via the e-times iterated Frobenius morphism F e by e N . Namely, e N is just N as an abelian group, and its A-module structure is defined by a · n  F e (a)n = e a p n for all a ∈ A, n ∈ N . We say A is F-finite if 1 A (and hence every e A) is a finitely generated A-module. For example, if A is a complete Noetherian local ring with a perfect residue field k, then A is F-finite. In positive characteristic commutative algebra, we investigate properties of A through the structure of e A (or e N ). However, it is difficult to describe such a structure explicitly. For example, (Q1) What kind of A-module appears in e A (or e N ) as a direct summand? (Q2) Can we understand the asymptotic behavior of e A (or e N )? These kinds of problems are difficult in general. Therefore we will consider these problems for the case where quotient singularities. This abstract is based on the author’s talk given in the worshop “Homological bonds between Commutative Algebra and Representation Theory”.

Y. Nakajima (B) Graduate School of Mathematics, Nagoya University, Chikusaku, Nagoya, Furocho 464-8602, Japan e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_20

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2 The Case of Quotient Singularities Let G be a finite subgroup of GL(d, k) which contains no pseudo-reflections and S  k[[x1 , . . . , xd ]] be a power series ring. We assume that the order of G is coprime to p = char k. We denote the invariant subring of S under the action of G by R  S G . In order to consider the problem (Q1) for an invariant subring R, we use the notion of finite F-representation type. This notion was introduced by Smith–Van den Bergh [9] and the definition is as follows. Definition 1 (Smith–Van den Bergh [9]) Let (A, m, k) be a complete Noetherian local ring with char A = p > 0. We say that A has finite F-representation type (or FFRT for short) by N if there exists a finite set N of isomorphism classes of indecomposable finitely generated A-modules such that for every e ∈ N, the A-module e A is isomorphic to a finite direct sum of elements of N . For example, a power series ring S has FFRT by {S} (cf., Kunz’s theorem) and FFRT is inherited by a direct summand [9]. Thus, an invariant subring R also has FFRT. More explicitly, we have the following proposition. Proposition 2 (Smith–Van den Bergh [9]) Let V0 = k, V1 , . . . , Vn be the complete set of irreducible representations of G and we set Mt  (S ⊗k Vt )G , for t = 0, 1, . . . , n. Then R has finite F-representation type by the finite set {M0 ∼ = R, M1 , . . . , Mn }. ⊕c ⊕c Thus we can write e R as e R ∼ = R ⊕c0,e ⊕ M1 1,e ⊕ · · · ⊕ Mn n,e .

Remark 3 It is known that each Mt is an indecomposable maximal Cohen–Macaulay (= MCM) R-module and Ms  Mt , for s = t, under the assumption that G contains no pseudo-reflections. Moreover, the multiplicities ct,e are determined uniquely in that case (for more details, see [3, Section 2]). From these observations, we could understand (Q1). Thus, we will move to the problem (Q2). Namely, we will consider the asymptotic behavior of each multiplicity ct,e on the order of p ed . Since an invariant subring R has FFRT, it is known that the limit lime→∞ ct,e / p ed exists, for t = 0, 1, . . . , n; see [9, 14]. Especially, for the case where t = 0, this limit is also known as the F-signature of R and is denoted by s(R); c.f. [4]. Thus, we will denote this limit by s(R, Mt ) = lime→∞ ct,e / p ed and call it the generalized F-signature of Mt with respect to R. Before considering the generalized version, we collect some properties of the F-signature. Theorem 4 (cf., [1, 4, 10, 15]) Let A be a d-dimensional reduced F-finite Noetherian local ring with char A = p > 0. Then, (i) (ii) (iii) (iv)

s(A) exists; 0 ≤ s(A) ≤ 1; A is regular if and only if s(A) = 1; A is strongly F-regular if and only if s(A) > 0.

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Especially, we know the explicit value of the F-signature of an invariant subring R. Theorem 5 (Watanabe–Yoshida [11]) Let R be the invariant subring defined in the beginning of this section. Then we have s(R) = 1/|G|. The following is a generalization of Theorem 5 and the answer for the problem (Q2). Theorem 6 (Hashimoto–Nakajima [3]) For t = 0, 1, . . . , n, one has s(R, Mt ) =

dimk Vt rank R Mt = . |G| |G|

Remark 7 A similar result holds for a finite subgroup scheme of SL2 ; see [2]. Next, we consider the decomposition of e Mt . Since each MCM R-module Mt appears in f R for sufficiently large f  0 as a direct summand, e Mt also decomposes as t ⊕d t ⊕d t e ∼ R ⊕d0,e Mt = ⊕ M 1,e ⊕ · · · ⊕ Mn n,e . (2.1) 1

t / p ed , and call it the generalized Then we define the limit s(Mt , Ms )  lime→∞ ds,e F-signature of Ms with respect to Mt . The next corollary immediately follows from Theorem 6 and [9, 3.3.1 and 3.3.2].

Corollary 8 (Hashimoto–Nakajima [3]) Suppose an MCM R-module Mt decomposes as (2.1). Then, for all s, t = 0, . . . , n, we obtain s(Mt , Ms ) = (rank R Mt ) · s(R, Ms ) =

(dimk Vt ) · (dimk Vs ) (rank R Mt ) · (rank R Ms ) = . |G| |G|

Observation 9 In dimension two, it is known that an invariant subring R is of finite representation type, that is, it has only finitely many non-isomorphic indecomposable MCM R-modules {R, M1 , . . . , Mn }. From Corollary 8, every indecomposable MCM R-module appears in e Mt as a direct summand for sufficiently large e. Thus, the additive closure add R (e Mt ) coincides with the category of MCM R-modules CM(R). So we use several results so-called Auslander–Reiten theory (cf., [5, 16]) to add R (e Mt ). We will use this idea in the next section.

3 Dual F-Signature of Special Cohen–Macaulay Modules In the previous section, we could understand the structure of e R. By using such results, we can investigate some numerical invariants in positive characteristic. In this note we focus on the notion of the dual F-signature. As the name shows, this is also a kind of generalization of the F-signature.

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Definition 10 (Sannai [8]) Let A be a d-dimensional reduced F-finite Noetherian local ring with char A = p > 0. For a finitely generated A-module N and e ∈ N, we set be (N )  max {n | ∃ϕ : e N  N ⊕n }. Then we call the limit s(N )  lime→∞ be (N )/ p ed the dual F-signature of N , if it exists. Remark 11 Since the morphism e A  A⊕be (A) splits, if N is isomorphic to the basering A, then the dual F-signature of A in the sense of Definition 10 coincides with the F-signature of A. Thus, we use the same notation unless it causes confusion. We remark that the dual F-signature also characterizes some singularities. Theorem 12 (Sannai [8]) Let A be a d-dimensional reduced F-finite Cohen– Macaulay local ring with char A = p > 0. Then, (i) A is F-rational if and only if s(ω A ) > 0; (ii) s(A) ≤ s(ω A ); (iii) s(A) = s(ω A ) if and only if A is Gorenstein. In the rest of the paper we consider the property of this numerical invariant for the case where quotient surface singularities. Let G be a finite subgroup of GL(2, k) which contains no pseudo-reflections and S  k[[x, y]] be a power series ring. We assume that the order of G is coprime to p = chark. We denote the invariant subring of S under the action of G by R  S G . Suppose that M is an MCM R-module and we will consider the property of the dual F-signature of M. By the definition of the dual F-signature, we should understand the following topics: (1) The structure of e M, namely · What kind of MCM appears in e M as a direct summand? · The asymptotic behavior of e M on the order of p 2e . (2) How do we construct a surjection e M  M ⊕be efficiently? We can understand the first one by Corollary 8 and the second one by Observation 9. By paying attention to a certain MCM R-module so-called a special CM module, we have Theorem 14. So, we recall the definition of special CM modules. Definition 13  [12, 13] For an MCM R-module M, we say that M is special if (M ⊗ R ω R ) tor is also an MCM R-module. Special CM modules are compatible with the geometry as the special McKay correspondence. Namely, we can construct a one-to-one correspondence between non-free indecomposable special CM R-modules and irreducible exceptional curves on the minimal resolution of quotient surface singularity [12, 13]. So, we can state the following

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Theorem 14 (Nakajima [7]) Let R be a quotient surface singularity. If M is an indecomposable special CM R-module, then s(M) ≤ s(τ (M)). Also, R is Gorenstein if and only if s(M) = s(τ (M)), where τ (M) stands for the Auslander–Reiten translation of M. Remark 15 Since τ (R) ∼ = ω R in our situation, this theorem is an analogue of Theorem 12 (ii)–(iii). But it says that this characterization is obtained by not only the comparison between R and ω R but also the comparison between a special CM module and the AR translation of it. Also, by using the above ideas, we can compute the value of the dual F-signature for some MCM R-modules explicitly. For more about computations, see [6, 7].

References 1. I. Aberbach, G. Leuschke, The F-signature and strongly F-regularity. Math. Res. Lett. 10, 51–56 (2003) 2. N. Hara, T. Sawada, Splitting of Frobenius sandwiches. RIMS Kôkyûroku Bessatsu B24, 121– 141 (2011) 3. M. Hashimoto, Y. Nakajima, Generalized F-signature of invariant subrings. arXiv:1311.5963 4. C. Huneke, G. Leuschke, Two theorems about maximal Cohen-Macaulay modules. Math. Ann. 324(2), 391–404 (2002) 5. G. Leuschke, R. Wiegand, Cohen–Macaulay representations, in Mathematical Surveys and Monographs, vol. 181 (American Mathematical Society, 2012) 6. Y. Nakajima, Dual F-signature of Cohen–Macaulay modules over rational double points, in Algebras and Representation Theory. arXiv:1407.5230 7. Y. Nakajima, Dual F-signature of special Cohen–Macaulay modules over cyclic quotient surface singularities. arXiv:1311.5967 8. A. Sannai, On dual F-signature. Int. Math. Res. Not. IMRN 1, 197–211 (2015) 9. K.E. Smith, M. Van den Bergh, Simplicity of rings of differential operators in prime characteristic, in Proceedings of London Mathematical Society (3), vol. 75, no. 1 (1997), pp. 32–62 10. K. Tucker, F-signature exists. Invent. Math. 190(3), 743–765 (2012) 11. K. Watanabe, K. Yoshida, Minimal relative Hilbert-Kunz multiplicity. Illinois J. Math. 48(1), 273–294 (2004) 12. J. Wunram, Reflexive modules on cyclic quotient surface singularities. in Lecture Notes in Mathematics, vol. 1273 (Springer, 1987), pp. 221–231 13. J. Wunram, Reflexive modules on quotient surface singularities. Math. Ann. 279(4), 583–598 (1988) 14. Y. Yao, Modules with finite F-representation type. J. Lond. Math. Soc. (2) 72(2), 53–72 (2005) 15. Y. Yao, Observations on the F-signature of local rings of characteristic p. J. Algebra 299(1), 198–218 (2006) 16. Y. Yoshino, Cohen–Macaulay modules over Cohen–Macaulay rings, in London Mathematical Society Lecture Note Series, vol. 146 (Cambridge University Press, Cambridge, 1990)

Strong Generation of Some Derived Categories of Schemes Amnon Neeman

Abstract We survey some recent results on strong generation in Dperf (X ) and Dbcoh (X ). MSC 2000: Primary 18E30 · Secondary 18G20

1 Definitions Let T be a triangulated category. We begin by reminding the reader of some old definitions; see [2, 1.3.9] and [3, Sect. 2.2]: (i) if A and B are two subcategories of T , then A  B is the full subcategory of all objects y for which there exists a triangle x −→ y −→ z −→ with x ∈ A and z ∈ B; (ii) if A is a subcategory of T , then add(A) is the full subcategory containing all finite coproducts of objects in A; (iii) if A is a subcategory of T , and T is closed under coproducts, then Add(A) is the full subcategory containing all (set-indexed) coproducts of objects in A; (iv) if A is a full subcategory of T , then smd(A) is the full subcategory of all direct summands of objects in A;  G1 = smd add{ i G, i ∈ (v) if G  ∈ T is an object, then G1 is defined to be i Z} ; that is, let A be a category with  objects  { G, i ∈ Z}, then form add(A) as in (ii), then form G1 = smd add(A) as in (iv); (vi) if G ∈ T is an object and  T is closed under  small coproducts, let us define G1 to be G1 = smd Add{ i G, i ∈ Z} ; in other words, the recipe is as in (v), but we allow arbitrary coproducts as in (iii) instead of only finite ones as in (ii);

A. Neeman (B) Centre for Mathematics and Its Applications, Mathematical Sciences Institute, The Australian National University, Canberra, ACT 2601, Australia e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_21

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 (vii) for n > 1, define Gn and Gn inductively by the rules Gn+1 = smd G1     Gn and Gn+1 = smd G1  Gn . In this report we will consider only three kinds of triangulated categories. We will assume X to be a sufficiently nice scheme, and Dqc (X ) will be the unbounded derived category of sheaves of O X –modules with quasicoherent cohomology. The subcategories Dperf (X ) ⊂ Dbcoh (X ) ⊂ Dqc (X ) will be, respectively, the subcategory of perfect complexes and the subcategory of complexes with bounded, coherent cohomology. Observation 1 Let X be a quasicompact, quasiseparated scheme. Then the following holds: (i) if G ∈ Dperf (X ) and n ≥ 1 are such that Gn = Dqc (X ), then Gn = Dperf (X ); (ii) assume X is Noetherian; if G is an object of Dbcoh (X ) and if n ≥ 1 is an integer with Gn = Dqc (X ), then Gn = Dbcoh (X ). We will say that Dqc (X ) is strongly compactly generated if there is an object G ∈ Dperf (X ) and an integer n ≥ 1 as in Observation 1(i), and that Dqc (X ) is strongly boundedly generated if there is an object G ∈ Dbcoh (X ) and an integer n ≥ 1 as in Observation 1(ii).

2 A Survey of the Known Results Kelly [9] was probably the first to observe that some natural triangulated categories are strongly generated. One way to say it is that he studies strong compact generation of Dqc (X ) in the special case where X = Spec(R) is affine. As it happens the result does not assume R to be commutative, and the argument is so simple that we recall the proof. Theorem 2 (Kelly [9]) Let R be a ring. The category D(R) is strongly compactly generated if and only if R is of finite global dimension. Proof We will prove only one direction, namely, if R is of finite global dimension then D(R) is strongly compactly generated. There is the functor H ∗ : D(R) −→ Gr(R– Mod), which takes a complex X to the graded R–module {H i (X ), i ∈ Z}. In general, this functor is neither full nor faithful, i.e., the map   θ(X,Y ) HomD(R) (X, Y ) −−−−→ HomGr(R–Mod) H ∗ (X ), H ∗ (Y ) need not be either injective or surjective. But when X happens to be the complex 0

0

0

0

· · · −−−−→ X i−1 −−−−→ X i −−−−→ X i+1 −−−−→ · · · ,

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where all the X i are projective R–modules, then the map θ (X, Y ) is easily seen to be an isomorphism for every Y . We claim that if Y is an object of D(R), and H i (Y ) is projective for every i, then Y ∈ R1 . In fact, let X be the complex 0

0

0

0

· · · −−−−→ H i−1 (Y ) −−−−→ H i (Y ) −−−−→ H i+1 (Y ) −−−−→ · · · This is a complex of projective modules with zero differential, and hence the identity map H ∗ (X ) −→ H ∗ (Y ) lifts to a morphism in the derived category f : X −→ Y . But f is a homology isomorphism, hence f is an isomorphism in D(R). Thus Y is isomorphic to the complex 0

0

0

0

· · · −−−−→ H i−1 (Y ) −−−−→ H i (Y ) −−−−→ H i+1 (Y ) −−−−→ · · · , which is a direct summand of a complex F of the form 0

0

0

0

· · · −−−−→ F i−1 −−−−→ F i −−−−→ F i+1 −−−−→ · · · i with each F i free. The complex F belongs to Add{  R, i ∈ Z}, and its direct sumi mand Y belongs to R1 = smd Add{ R, i ∈ Z} . Now we claim that if Y is an object of D(R), and H i (Y ) is of projective dimension less than or equal to n for each i ∈ Z, then Y ∈ Rn+1 . In fact, we have already proved the case n = 0 in the previous paragraph. Assume therefore that we know the result for n − 1 ≥ 0 and we will prove it for n. Let Y be a complex with H i (Y ) of projective dimension less than or equal to n for all i. For each such i, choose a free module F i and a surjection F i −→ H i (Y ). If F is the complex 0

0

0

0

· · · −−−−→ F i−1 −−−−→ F i −−−−→ F i+1 −−−−→ · · · we have produced an epimorphism H i (F) −→ H i (Y ), which lifts to a morphism F −→ Y in D(R). Completing to a triangle F −→ Y −→ Q −→, we see that each H i (Q) is of projective dimension less than or equal to n − 1. Induction gives that Q ∈ Rn , while F ∈ R1 is obvious. Hence Y ∈ R1  Rn ⊂ Rn+1 . Finally, the object R ∈ D(R) is compact and, if R is of global dimension less than  or equal to n, we obtain from the claim above that Rn+1 = D(R). This theorem laid largely forgotten for 30 years. It was rediscovered by Christensen [4, Corollary 8.4]. Bondal–Van den Bergh [3, Theorem 3.1.4] proved the first global version: if X is a smooth, separated scheme over a field k, then the category Dqc (X ) is strongly compactly generated. Using a refinement of the argument, Rouquier [11, Theorem 7.38] shows that Dqc (X ) is strongly boundedly generated whenever X is a separated scheme of finite type over a perfect field k. Keller–Van den Bergh [8, Proposition 5.1.2] generalized it to separated schemes of finite type over arbitrary fields. Aihara–Takahashi [1] and Iyengar–Takahashi [7] studied Rouquier’s theorem further in the special case where X = Spec(R), where the union of the

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∼ D(R) is strongly boundedly generated if known results seems to be that Dqc (X ) = R is either an equicharacteristic excellent local ring, or essentially of finite type over a field. Orlov [10] obtained the best available version of the Bondal–Van den Bergh theorem: Dqc (X ) is strongly compactly generated if X is a regular scheme over a field k, and both X and X ×k X are Noetherian.

3 New Theorems The new results we wish to announce are the following. Theorem 3 Let X be a quasicompact, separated scheme. The category Dqc (X ) is strongly compactly generated if and only if X has a cover by open affine subschemes X i = Spec(Ri ), with each Ri of finite global dimension. The next theorem follows by using regular alterations; we remind the reader that a regular alteration is a generalization of a resolution of singularities, where we relax the assumption that the map must be birational. Theorem 4 Let X be a finite-dimensional, Noetherian, separated scheme and assume every closed subscheme of X admits a regular alteration. Then Dqc (X ) is strongly boundedly generated. Below we will sketch the proof. Note that, with the stronger assumption that every closed subscheme of X has a resolution of singularities, then Theorem 4 follows easily from Theorem 3. One has to work a little harder to handle more general alterations. But unlike resolutions of singularities, which are rarely known to exist, one has theorems of de Jong [5, 6] saying (among other facts) that any scheme X , separated and of finite type over an excellent scheme S of dimension less than or equal to two, has a regular alteration. Proof of Theorem 4 (special case) Assume that every closed subscheme of X has a resolution of singularities. By Noetherian induction, we may assume that every closed proper subscheme of X satisfies Theorem 4. Furthermore, an easy devissage allows us to assume that X is reduced. Let π : X −→ X be a resolution of singularities, consider the map f : O X −→ Rπ∗ O X and complete it to a triangle f

Q −−−−→ O X −−−−→ Rπ∗ O X −−−−→ . Since π is birational, the map f is an isomorphism on a dense open subset U ⊂ X and, since π is proper, the complex Rπ∗ O X belongs to Dbcoh (X ). The triangle tells us that Q belongs to Dbcoh (X ), and vanishes on the dense open subset U . Therefore, there is an inclusion of a closed subscheme i : Z −→ X , with image contained in X − U , so that Q = i ∗ P = Ri ∗ P for some P ∈ Dbcoh (Z ). Theorem 3 allows us to choose a G ∈ Dperf (X ) and an integer N with G N = Dqc (X ), and the fact that Z

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is a closed proper subscheme of X gives a G ∈ Dbcoh (Z ) and an integer M so that G  M = Dqc (Z ). If F is an arbitrary object of Dqc (X ), we may tensor the triangle above with F to obtain F ⊗LX Ri ∗ P

/F

/ F ⊗LX Rπ∗ O X

  Ri ∗ Li ∗ F ⊗LZ P

/F

/ Rπ∗ Lπ ∗ F

/

/

where the vertical isomorphisms formula. The bottom row shows  are  by the projection   that F belongs to Ri ∗ Dqc (Z )  Rπ∗ Dqc (X ) ⊂ Ri ∗ G  M  Rπ∗ G N , which is  contained in Rπ∗ G ⊕ Ri ∗ G  M+N , with Rπ∗ G ⊕ Ri ∗ G ∈ Dbcoh (X ). Note that, like Kelly’s theorem but unlike any of the modern versions, Theorems 3 and 4 do not assume X to be equicharacteristic. Perhaps we should say something about the proof of Theorem 3. If X = Spec(R) is affine, the result amounts to Kelly’s old theorem and the idea is to reduce ourselves N Ui exhibits X as the union of N open affine subsets, to this case. Suppose X = i=1 ˘ and so the Cech complex allows us to construct every object F ∈ Dqc (X ) in at most N steps, from objects of the form ji∗ ji∗ F , where ji : Ui −→ X is the inclusion and F = j∗ j ∗ F for the inclusion j : V −→ X with V being a finite intersection of the Ui ’s. By Kelly’s theorem ji∗ F ∈ OUi n+1 , where n is the global dimension of the ring Ri = O X (Ui ). Hence, ji∗ ji∗ F ∈  ji∗ OUi n+1 and it suffices to find a compact object G ∈ Dqc (X ) and an integer M ≥ 1 with ji∗ OUi ∈ G M , for the finitely many open affines Ui ⊂ X . Therefore, Theorem 3 follows from

Theorem 5 Let X be a quasicompact, separated scheme. If j : U −→ X is the inclusion of a quasicompact open subset and G ∈ Dqc (X ) is a compact generator, then there exists an integer M with R j∗ OU ∈ G M . Note the generality of Theorem 5: we do not, for example, need to assume that X is locally of finite projective dimension. Somehow, the assertion says that objects of the form R j∗ OU are special, and can always be constructed in a finite number of steps. The key is therefore to understand what is special about the objects R j∗ OU . Definition 6 Let X be a quasicompact, quasiseparated scheme. The length of an object F ∈ Dqc (X ) is the smallest integer n ≥ 0 for which there exists an integer m so that, on every open affine subset Spec(R) = U ⊂ X , F is quasiisomorphic in Dqc (U ) ∼ = D(R) to a complex ···

/0

/

Fm

/

F m+1

/ ···

/

F m+n−1

/

F m+n

/0

/ ··· ,

where all the F i are projective R–modules. Now we are ready to state the next theorem, from which Theorem 5 follows.

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Theorem 7 Let X be a scheme separated and of finite type over a Noetherian ring R, let G be a compact generator of Dqc (X ), and let n ≥ 0 be an integer. There exists an integer N > 0 so that every object in Dqc (X ) of length at most n belongs to G N . The idea, for the proof of Theorem 5, is to show that objects of the form R j∗ OU are all of finite length. But given the restrictions that Theorem 7 places on the scheme X , one needs to use Noetherian approximation to deduce Theorem 5 from Theorem 7. It requires a little effort to show, in full generality, that all the R j∗ OU are of finite length. Let us treat the easy special case where X is Noetherian and regular, and U ⊂ X is affine (for the proof of Theorem 4 this suffices). Then the complement of U is a divisor, that is there exists a line bundle L and a global section f : O X −→ L which vanishes exactly on X − U . Form the direct system OX

f

/L

f

/ L2

f

/ L3

f

/ ···

The colimit of the sequence is j∗ OU = R j∗ OU , and we deduce a triangle ∞ i=0

Li −−−−→ R j∗ OU −−−−→ 

∞

Li −−−−→ .

i=0

This triangle clearly shows that this particular R j∗ OU is of length less than or equal 1. Acknowledgements The author would like to thank the CRM in Barcelona, where much of the work was done, for its hospitality and congenial work environment. The author would also like to thank Ryo Takahashi, Ilya Tyomkin and Amnon Yekutieli for their questions during talks on which this survey is based, which helped the exposition. The research was partly supported by the Australian Research Council and by the CRM in Barcelona.

References 1. T. Aihara, R. Takahashi, Generators and dimensions of derived categories. Comm. Alg. (to appear). arXiv:1106.0205v3 2. A.A. Beilinson, J. Bernstein, P. Deligne, Analyse et topologie sur les éspaces singuliers. Astérisque 100, Soc. Math. France (French) (1982) 3. A.I. Bondal, M. Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3(1), 1–36 (2003) 4. J.D. Christensen, Ideals in triangulated categories: phantoms, ghosts and skeleta. Adv. Math. 136, 284–339 (1998) 5. A.J. de Jong, Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. 83, 51–93 (1996) 6. A.J. de Jong, Families of curves and alterations. Ann. Inst. Fourier 47(2), 599–621 (1997) 7. S. Iyengar, R. Takahashi, Annihilation of cohomology and strong generation of module categories. Int. Math. Res. Notices (to appear)

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8. B. Keller, M. Van den Bergh, On two examples by Iyama and Yoshino. arXiv:0803.0720v1 9. G.M. Kelly, Chain maps inducing zero homology maps. Proc. Camb. Philos. Soc. 61, 847–854 (1965) 10. D.O. Orlov, Smooth and proper noncommutative schemes and gluing of DG categories (2014). arXiv:1402.7364v2 11. R. Rouquier, Dimensions of triangulated categories. J. K-Theory 1(2), 193–256 (2008)

Regularity of Products over Quadratic Hypersurfaces Hop D. Nguyen and Thanh Vu

Abstract We study the free resolution of products of linear forms over a quadratic hypersurface. Our results support the conjecture that such free resolutions are linear.

Let k be a field and R a commutative standard graded k-algebra. In other words, R is N-graded and is generated by finitely many elements of degree 1 as an algebra over k. Let M be a finitely generated graded R-module. Then we define the (Castelnuovo– Mumford) regularity of M as an R-module by the formula reg R M = sup{ j − i : ToriR (k, M) j = 0}. The study of (Castelnuovo–Mumford) regularity is intimately connected with the study of minimal free resolutions. The later has its origin in invariant theory and is still having various connections with representation theory as well as methods from homotopy theory. The regularity is one of the most important complexity measures for graded modules. For recent surveys on the topic, we refer the reader to [3, 4]. Usually, the regularity of a given module is not easy to compute explicitly, unless for modules and ideals with nice combinatorial or algebraic structures. One of the interesting classes of ideals with known regularity is the class of ideals which are products of linear ideals. Recall that an ideal over R is called a linear ideal if it is generated by linear forms. A finitely generated graded R-module M is said to have a d-linear resolution (where d ∈ Z) if ToriR (k, M) j = 0 for all i, j with j − i = d. We say that M has a linear resolution if it has a d-linear resolution for some d. In 2003, Conca–Herzog [6, Theorem 3.1] proved that, in a polynomial ring over k, products of linear ideals have linear resolution. This result was extended by Derksen–Sidman [9, Theorem 4.4] to the effect that ideals obtainable from linear ideals from taking H.D. Nguyen (B) Fachbereich Mathematik/Informatik, Institut für Mathematik, Universität Osnabrück, Albrectstr. 28a, 49069 Osnabrück, Germany e-mail: [email protected] T. Vu Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_22

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intersection, sum and product admit sharp bounds on the Castelnuovo–Mumford regularity. In a recent survey on Koszul algebras, Conca–de Negri–Rossi [5] asked whether Conca-Herzog’s theorem can be extended to products of linear ideals in a Koszul filtration; see also [8]. Question 1 ([5, Question 5.8]) Let R be a standard graded k-algebra with a Koszul filtration F. Let I1 , . . . , Id be ideals in F, where d ≥ 1 is an integer. Is it true that if I1 I2 · · · Id = 0 then the equality reg R (I1 I2 · · · Id ) = d holds? The following example shows that this question has a negative answer. Example 2 Let R = k[a1 , . . . , a7 ]/I , where I = (a22 − a1 a3 , a2 a3 − a1 a5 , a32 − a2 a5 , a2 a4 − a1 a6 , a3 a4 − a2 a6 , a42 − a2 a7 , a4 a5 − a3 a6 , a4 a6 − a3 a7 , a62 − a5 a7 ) + (a4 a5 ). By abuse of notation, we use ai to denote its equivalence class in R, for i = 1, . . . , 7. Then R has the following Koszul filtration F = {(0), (a4 ), (a4 , a1 ), (a4 , a6 ), (a4 , a1 , a2 ), (a4 , a1 , a2 , a3 ), (a4 , a1 , a2 , a3 , a6 ),

(a4 , a1 , a2 , a3 , a6 , a7 ), (a4 , a1 , a2 , a3 , a6 , a5 ) = (a4 , a1 , a2 , a3 , a6 , a7 , a5 ), (a5 ), (a5 , a1 ), (a5 , a1 , a2 ), (a5 , a1 , a2 , a3 ), (a5 , a1 , a2 , a3 , a4 ), (a5 , a1 , a2 , a3 , a4 , a6 )} .

On the other hand, taking I = J = (a4 ), then I J = (a42 ) does not have 2-linear resolution over R, since 0 : a42 = (a5 , a2 a6 ). Moreover, we can find an algebra R with a Gröbner flag F (in the sense of Conca–Rossi–Valla [7]; see also Blum [2]) such that products of ideals in F need not have linear resolution. Therefore one of the most natural ways to generalize Conca–Herzog’s theorem is suggested by the following open question. Conjecture 3 Let f = 0 be a quadratic form in the polynomial ring R = k[x1 , . . . , xn ] (where n ≥ 1). Denote by x the residue class of x in S = R/( f ). Then, for any linear ideals I1 , . . . , Id of R (where d ≥ 1), it holds that reg S (I1 · I2 · · · Id ) = d. From the paper of Conca, De Negri and Rossi, this is true if S is of dimension at most 2 and if S = k[x, y, z, t]/(x y − zt); see also [5, Theorem 5.11]. It seems to be difficult to work with the regularity over S. One of the fundamental differences with the polynomial case considered in [6] is that resolutions over S are usually infinite. However, studying the regularity over S via the regularity over the polynomial ring R is possible and turns out to be fruitful. Indeed, on the one hand, it is known from Avramov–Eisenbud [1] that, for any finitely generated graded S-module M, (1) reg S M ≤ reg R M.

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On the other hand, since S is a hypersurface, reg R M ≤ reg R S + reg S M = reg S M + 1.

(2)

Hence, bounding above the regularity of M over R already gives a good bound for reg S M. One of the main results of this note is the following. Theorem 4 For any quadratic form 0 = f ∈ k[x1 , . . . , xn ] and linear ideals I1 , . . . , Id , we have reg(( f ) + I1 I2 · · · Id ) ≤ d + 1. If, moreover, f is contained in each of the ideals I1 , . . . , Id , then reg(( f ) + I1 I2 · · · Id )) = d. The proof is an application of Derksen–Sidman’s approximation theory for the regularity [9]. The main problem is to control the colon ideal I1 I2 · · · Id : f . Some consequences of Theorem 4 to Conjecture 3 are as follows. Firstly, using inequalities (1) and (2), we get Theorem 5 Any product of d linear ideals over a quadratic hypersurface S has regularity at most d + 1 as an S-module. Moreover, for any linear ideal I of S, all the powers of I have linear resolutions. Theorem 6 Let S be a quadratic hypersurface of Krull dimension at most 3. Then Conjecture 3 is true for S. Without the restriction on the Krull dimension, we have the following partial answer for products of two linear ideals. Proposition 7 Let S be a quadratic hypersurface, and I1 , I2 linear ideals of S. Then if the product I1 I2 is non-trivial, it has 2-linear resolution over S. The main results of this paper support the validity of Conjecture 3. We hope that they would also inspire further research on regularity over non-polynomial base rings. The proofs of the theorems in this extended abstract together with more detailed discussions can be found in [10].

References 1. L.L. Avramov, D. Eisenbud, Regularity of modules over a Koszul algebra. J. Algebra 153, 85–90 (1992) 2. S. Blum, Initially Koszul algebras. Beiträge Algebra Geom. 41, 455–467 (2000) 3. M. Brodmann, C.H. Linh, M.H. Seiler, Castelnuovo–Mumford regularity of annihilators, Ext and Tor modules, in Commutative Algebra: Expository Papers Dedicated to David Eisenbud on the Occasion of His 65th Birthday, ed. by I. Peeva (Springer, 2013), pp. 285–315 4. M. Chardin, Some results and questions on Castelnuovo-Mumford regularity, in Syzygies and Hilbert Functions, Lecture Notes in Pure and Applied Mathematics, vol. 254 (2007), pp. 1–40 5. A. Conca, E. de Negri, M.E. Rossi, Koszul algebra and regularity, in Commutative Algebra: Expository Papers Dedicated to David Eisenbud on the Occasion of His 65th Birthday, ed. by I. Peeva (Springer, 2013), pp. 285–315

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6. A. Conca, J. Herzog, Castelnuovo-Mumford regularity of products of ideals. Collect. Math. 54(2), 137–152 (2003) 7. A. Conca, M. Rossi, G. Valla, Gröbner flags and Gorenstein algebras. Compos. Math. 129, 95–121 (2001) 8. A. Conca, N.V. Trung, G. Valla, Koszul property for points in projective space. Math. Scand. 89, 201–216 (2001) 9. H. Derksen, J. Sidman, Castelnuovo-Mumford regularity by approximation. Adv. Math. 188, 104–123 (2004) 10. H.D. Nguyen, T. Vu, Regularity of products over quadratic hypersurfaces

Phantom Maps and Representability Oriol Raventós

Abstract Let T be a well generated triangulated category and let F be the ideal of phantom  maps with respect to the α-compact objects. In [5] we have proved that the ideal n≥1 F n has always square zero. We will show how this can be used to prove representability of cohomological and homological functors.

1 Phantom Maps A triangulated category T is well generated if it has coproducts and a set of αcompact generators S. We briefly recall some basic properties about well generated triangulated categories. We refer to [6] for a complete treatment. Let T α ⊂ T be the full subcategory of α-compact objects and let Modα −T α be the abelian category of contravariant functors from T α to the category of abelian groups Ab, preserving coproducts of less than α objects. The category Modα −T α is locally presentable and representable functors form a set of α-presentable projective generators. The functor / Modα −T α Sα : T defined by Sα (X ) = T (−, X )|T α is called a restricted Yoneda functor. It preserves products and coproducts, takes exact triangles to exact sequences, and reflects isomorphisms. However, in general, there is no reason for Sα to be neither essentially surjective, nor full, nor faithful. If the essential image of Sα coincides with the full subcategory of cohomological functors, i.e., the functors sending triangles to short exact sequences, we say that T satisfies α-Adams representability for objects. If Sα is full we say that T satisfies α-Adams representability for morphisms. To describe the kernel of Sα we make the following definition. Definition 1 A map f in T is phantom if T (s, f ) = 0 for every s ∈ T α . O. Raventós (B) Universität Regensburg, Regensburg, Germany e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_23

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The class of phantom maps forms an ideal in T by composition; we denote it by F. Example 2 Let T be the homotopy category of spectra. It is ℵ0 -compactly generated and a set of generators is given by the suspensions of the sphere spectrum { i S}i∈Z . Adams proved in [1] that T satisfies ℵ0 -Adams representability for objects and morphisms, and that this implied the representability of homology theories. Using this fact, it can be seen that a map f is phantom if and only if E ∗ ( f ) = 0 for every additive homology theory E. Theorem 3 (Neeman [7, Lemma 4.1]) Let T be an ℵ0 -compactly generated triangulated category that satisfies ℵ0 -Adams representability for morphisms. Then F 2 = 0, i.e., the composition of two phantom maps is always zero. The previous result does not extend to every well generated triangulated category. Example 4 Let T be the derived category of a ring R. It is ℵ0 -compactly generated and a set of generators is given by the suspensions of the ring considered as a chain concentrated in degree zero { i R}i∈Z . By [7], if R is a countable ring, then T satisfies ℵ0 -Adams representability for objects and morphisms. However, in general, T might not satisfy neither of the two ℵ0 -Adams representability statements, as explained in [3]. In [5] we show that there are well generated triangulated categories that do not satisfy α-Adams representability for objects nor for morphisms for any cardinal α. Nevertheless, we can still say something about the vanishing of the iterate powers of the ideal of phantom maps.  Theorem 5 Let T be a well generated triangulated category. Then n≥1 F n is a square zero ideal.

2 Projective and Injective Classes Given a class of objects P in a triangulated category T we write P-null = { f : X → Y in T | ∀g : p → X with p ∈ P, f ◦ g = 0} and P-conull = { f : X → Y in T | ∀h : Y → p with p ∈ P, h ◦ f = 0}. Notice that P-null and P-conull are always ideals. Given a class of morphisms F in T we write F-proj = { p in T | ∀g : p → X and f : X → Y in F, f ◦ g = 0}

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and F-inj = { p in T | ∀h : Y → p and f : X → Y in F, h ◦ f = 0}. Notice that F-proj is closed under coproducts and retracts and that F-inj is closed under products and retracts. Definition 6 Let T be a triangulated category. A pair (P, F) is a projective class if F = P-null and P = F-proj and, for every object X in T , there exists a triangle A

f

/X

g

/B

/ A

(1)

with A in P and g in F. Dually, a pair (F, P) is an injective class if F = P-conull and P = F-inj and, for every object X in T , there exists a triangle like (1) with B in P and f in F. If P1 and P2 are two classes of objects in a triangulated category T we say that an object X in T belongs to the class P1 ∗ P2 if there is a triangle like (1) with B ∈ P1 and A ∈ P2 . For a class of objects P we define recursively P∗n = P ∗ P∗(n−1) for all n > 1. We also define Pn∗ = P(n−1)∗ ∗ P for all n > 1. Theorem 7 (Christensen [2, Theorem 3.5]) Let T be a triangulated category with coproducts. Let F be a class of morphisms and P a class of objects in T , both closed by suspension and desuspension, and such that the pair (P, F) is a projective  n ω n class. Then (P  ∗n , F ) is a projectiveωclass for all n > 1. If we write F = n≥1 F and P∗ω = n≥1 P∗n , then (P∗ω , F ) is a projective class. Dually, if (F, P) is an injective class in a triangulated category with products, then (F n , Pn∗ ), for n > 1, and (F ω , Pω∗ ) are injective classes.

3 Brown Representability We say that a triangulated category with coproducts satisfies Brown representability / Ab sending coproducts in T to prodif every cohomological functor H : T op ucts in Ab is representable, i.e., there exists an object h such that H (X ) ∼ = T (X, h) for every X in T . Theorem 8 (Neeman [8, Theorem 1.11]) Let T be a triangulated category with coproducts. Let S be a set of objects in T closed by suspension and desuspension, and let P be the closure of S by coproducts and retracts. If P∗κ = T for an ordinal κ, then T satisfies Brown representability. Theorem 9 (Neeman [8, Theorem 1.12]) Let T be a triangulated category with products. Let S be a set of objects in T closed by suspension and desuspension, and let P be the closure of S by products and retracts. If Pκ∗ = T for an ordinal κ, then T op satisfies Brown representability.

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Next, we give a new proof of the following result by Neeman [6, Theorem 8.3.3]. Corollary 10 If T is a well generated triangulated category, then it satisfies Brown representability. Proof Let S be a generating set of α-compact objects and P the closure of S by coproducts and retracts, then (P, F) is a projective class with F the ideal of phantom maps of T . By Theorem 5, (F ω )2 = 0. Since (T , 0) is always a projective class, we deduce that P∗2ω = T . Hence, by Theorem 8, T satisfies Brown representability.  We can also recover the following result by Krause [4, Theorem B]. Corollary 11 Let T be an ℵ0 -compactly generated triangulated category. Then, T op satisfies Brown representability. Proof Let S be a set of compact generators for T . For every s ∈ S, let Hs : T op → Ab be defined by Hs (X ) = HomZ (T (s, X ), Q/Z). Since s is compact, Hs sends coproducts to products and, since Q/Z is an injective cogenerator of Ab, Hs is cohomological. Using Brown representability for T , Hs (X ) ∼ = T (X, s ∗ ) for an object ∗ ∗ ∗ s in T . Let S be the set of objects s for all s ∈ S. The set S ∗ cogenerates T , although its elements need not be α-compact in T op for any cardinal α. Notice that, for every morphism f in T , T (s, f ) = 0 if and only if T ( f, s ∗ ) = 0. Hence, if we let P be the closure of S ∗ by products and retracts, (F, P) defines an injective class in T with F the ideal of phantom maps. Since (F ω )2 = 0, P2ω∗ = T  by Theorem 5. Hence, by Theorem 9, T op satisfies Brown representability. Acknowledgments The author is supported by the grant SFB 1085 “Higher invariants” funded by the German Research Foundation, and by the grant MTM2013-42178-P funded by the Spanish Ministry of Science and Innovation.

References 1. J.F. Adams, A variant of E.H. Brown’s representability theorem. Topology 10, 185–198 (1971) 2. J.D. Christensen, Ideals in triangulated categories: phantoms, ghosts and skeleta. Adv. Math. 136(2), 284–339 (1998) 3. J.D. Christensen, B. Keller, A. Neeman, Failure of Brown representability in derived categories. Topology 40(6), 1339–1361 (2001) 4. H. Krause, A Brown representability theorem via coherent functors. Topology 41(4), 853–861 (2002) 5. F. Muro, O. Raventós, Transfinite Adams representability. Preprint arXiv:1304.3599 6. A. Neeman, Triangulated categories. Ann. Math. Stud. 148 (2001). Princeton University Press, Princeton, NJ 7. A. Neeman, On a theorem of Brown and Adams. Topology 36(3), 619–645 (2002) 8. A. Neeman, Brown representability follows from Rosický’s theorem. J. Topol. 2(2), 262–276 (2009)

Six Operations on dg Enhancements of Derived Categories of Sheaves and Applications Olaf M. Schnürer

Abstract We lift Grothendieck’s six functor formalism for derived categories of sheaves on ringed spaces over a field to differential graded enhancements. Two applications concerning homological smoothness of derived categories of schemes are given.

1 Motivation Grothendieck’s six functor formalism in the topological setting concerns the six functors ⊗L , RHom, Lα ∗ , Rα∗ , Rα! , and α ! between derived categories of sheaves on ringed spaces and their relations. It takes place in the 2-multicategory TRCATk of triangulated k-categories (the prefix “multi” takes care of functors with several inputs like ⊗L and RHom). The relevant objects are the derived categories D(X ) of sheaves of O-modules on ringed spaces (X, O). The six functors ⊗L , RHom, Lα ∗ , Rα∗ , Rα! , and α ! are 1-morphisms between these objects. The relations between these functors are encoded in two ways: first, by 2-morphisms like id → Rα∗ Lα ∗ ∼ and 2-isomorphisms like Rα∗ RHom(−, α ! (−)) − → RHom(Rα! (−), −); second, by commutative diagrams constructed from these 2-morphisms: they encode for example that (Lα ∗ , Rα∗ ) is a pair of adjoint functors or that (D(X ), ⊗L ) is a symmetric monoidal category. Nowadays, triangulated categories are often replaced by suitable differential graded (dg) enhancements because some useful constructions can be performed with dg categories but not with triangulated categories. Therefore, is natural to ask whether Grothendieck’s six functor formalism lifts to the level of dg enhancements. We give an affirmative answer to this question if we fix a field k and work with k-ringed spaces, i.e., pairs (X, O) consisting of a topological space X and a sheaf O of k-algebras on X . O.M. Schnürer (B) Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_24

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Theorem 1 Let k be a field. Then Grothendieck’s six functor formalism for k-ringed spaces lifts to dg k-enhancements. The precise meaning of Theorem 1 is explained in the rest of this note. For proofs and more details we refer the reader to [4]. The symbol k always denotes a field.

2 User’s Guide to the Enhanced Six Functor Formalism 2.1 Enhancements Considered Let C(X ) denote the dg k-category of complexes of sheaves on the k-ringed space (X, O). The dg k-subcategory I(X ) of h-injective complexes of injective sheaves is the dg k-enhancement of D(X ) we consider. It is a pretriangulated dg k-category with translation and the obvious functor from its homotopy category [I(X )] to D(X ) is an equivalence of triangulated k-categories.

2.2 Key Ingredient: dg k-Enriched Resolutions Theorem 2 (Existence of dg k-enriched h-injective resolutions) There is a dg k functor I : C(X ) → I(X ) together with a dg k-natural transformation id → I whose evaluation E → IE at each object E is a quasi-isomorphism. The proof of Theorem 2 uses enriched model category theory. The assumption that k is a field is crucial. Theorem 2 does not hold for k replaced by the integers. The proof that it fails for the ringed space (pt, Z) boils down to the following lemma. Lemma 3 There is no additive functor Mod(Z) → Mod(Z)[1] mapping an abelian group A to a monomorphism A → I A into an injective abelian group I A . Here, Mod(Z)[1] denotes the arrow category of abelian groups. Proof Consider T = Z/2Z. Then id T is mapped to the identity id T →IT = (id T , id IT ) of T → IT . Additivity and 0 = 2id T imply 0 = 2id IT . But IT is divisible and we  obtain the contradiction IT = 0. Fix a dg k-functor I as in Theorem 2. The induced functor [I] : [C(X )] → [I(X )] on homotopy categories factors to an equivalence ∼

[I] : D(X ) − → [I(X )] of triangulated k-categories.

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2.3 Lifts of the Six Functors Given a morphism α : Y → X of k-ringed spaces, we define the dg k-functor α ∗ as the composition α∗

I

→ C(X ) − → I(X ). α ∗ : I(Y ) − This dg k-functor lifts the derived functor Rα∗ in the sense that the diagram D(Y )

Rα∗

[I] ∼

 [I(Y )]

/ D(X )

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[I] ∼

[α ∗ ]

 / [I(X )]

of triangulated k-categories commutes up to a canonical 2-isomorphism. Using similar techniques, we lift the functors Lα ∗ , ⊗L , RHom, Rα! , and α ! to dg k-functors α ∗ , ⊗, Hom, α ! , and α ! , respectively. The definitions of α ∗ and ⊗ use dg k-enriched h-flat resolutions.

2.4 Key Definition: The 2-Multicategory ENHk Let DGCATk denote the 2-multicategory of dg k-categories. The objects of the 2-multicategory ENHk we want to define are the dg k-categories I(X ) introduced above. Given objects I(X ) and I(Y ) let τ : F  → F be a morphism in DGCATk (I(X ), I(Y )), F KS v τ I(Y ) h I(X ). F

We say that τ is an objectwise homotopy equivalence if τ I : F  (I ) → F(I ) is a homotopy equivalence (or, equivalently, a quasi-isomorphism) for all I ∈ I(X ). An equivalent condition is that the induced morphism [τ ] : [F  ] → [F] in TRCATk ([I(X )], [I(Y )]) is an isomorphism. Now we complete the definition of ENHk : the morphism category ENHk (I(X ), I(Y )) is defined as the localization of DGCATk (I(X ), I(Y )) with respect to the class of all objectwise homotopy equivalences. A similar definition applies if several source objects are involved. Obviously, mapping a dg k-category to its homotopy category induces a functor [−] : ENHk → TRCATk . This functor reflects 2-isomorphisms.

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It is possible to go back from ENHk to DGCATk : any 2-morphism ρ : F → G τ σ − F − → G of 2-morphisms in DGCATk , in ENHk can be represented by a roof F ← where τ is an objectwise homotopy equivalence. Moreover, ρ is a 2-isomorphism if and only if, in any such representing roof, σ is an objectwise homotopy equivalence.

2.5 Lifts of the Relations Explicit zig-zags of dg k-natural transformations define a 2-morphism id → α ∗ α ∗ ∼ and a 2-isomorphism α ∗ Hom(−, α ! (−)) − → Hom(α ! (−), −) in ENHk whose images under (3) coincide modulo the equivalences (1) and the canonical 2-isomorphisms (cf., (2)) with the 2-morphism id → Rα∗ Lα ∗ and the 2-isomorphism ∼

Rα∗ RHom(−, α ! (−)) − → RHom(Rα! (−), −). Similarly, we lift all standard 2-(iso)morphisms between compositions of the six functors to 2-(iso)morphisms in ENHk . Moreover, we show that (α ∗ , α ∗ ) is a pair of adjoint 1-morphisms in ENHk and that (I(X ), ⊗) is a symmetric monoidal object of ENHk . Other relations encoded by commutative diagrams lift similarly.

2.6 Relation to the Homotopy Category of dg k-Categories Let X and Y be k-ringed spaces. Results by Toën in [5], [6, 4.1, Prop. 1] show that the obvious map defines a bijection ∼

Isom ENHk (I(X ), I(Y )) − → HomHo(dgcatk ) (I(X ), I(Y )),

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where the left hand side denotes isomorphism classes of objects in ENHk (I(X ), I(Y )), and Ho(dgcat k ) denotes the homotopy category of dg k-categories, i.e., the localization of the category dgcat k of dg k-categories with respect to the class of quasiequivalences. This bijection says that the 2-multicategory ENHk contains finer information than the homotopy category of dg k-categories.

2.7 Some Remarks The formalism involving the four functors ⊗L , RHom, Lα ∗ , and Rα∗ lifts more generally to k-ringed topoi. Our techniques should apply to many other contexts, e.g. dg modules over dg kcategories, quasi-coherent sheaves on k-schemes, D-modules on k-schemes, sheaves

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of modules over other non-commutative structure sheaves of k-algebras, Q -sheaves on pro-étale sites, etc. Results on some of these topics will be explained in future work.

3 Applications The following results will appear in [3]. They are based on previous work from Lunts [1] and joint results in [2]. Recall that a dg k-category A is homologically k-smooth if the (right) dg (A ⊗k Aop )-module A (the “diagonal bimodule”) is a compact object of the derived category of dg (A ⊗k Aop )-modules. Theorem 4 Let X be a separated scheme of finite type over a field k. Then X is smooth over k (in the sense of algebraic geometry) if and only if Dperf (X ) is homologically k-smooth (which, by definition, means that its dg k-enhancement Iperf (X ) is homologically k-smooth). We hope to extend this result to suitable Deligne–Mumford stacks over a field. Theorem 5 Let X be a separated scheme of finite type over a perfect field k. Then Dbcoh (X ) is homologically k-smooth (which, by definition, means that its dg k-enhancement Ibcoh (X ) is homologically k-smooth).

References 1. V.A. Lunts, Categorical resolution of singularities. J. Algebra 323(10), 2977–3003 (2010) 2. V.A. Lunts, O.M. Schnürer, New enhancements of derived categories of coherent sheaves and applications. Preprint arXiv:1406.7559 3. O.M. Schnürer, Homological and geometric smoothness for schemes and Deligne–Mumford stacks. In preparation 4. O.M. Schnürer, Six operations on dg enhancements of derived categories of sheaves. In preparation 5. B. Toën, The homotopy theory of dg-categories and derived Morita theory. Invent. Math. 167(3), 615–667 (2007) 6. B. Toën, Lectures on DG-categories. Topics in algebraic and topological K -theory, in Lecture Notes in Mathematics, vol. 2011 (Springer, Berlin, 2008), pp. 243–30

Tensor Product of Dualizing Complexes over a Field Liran Shaul

Abstract Given a field k, Noetherian k-schemes (resp., k-formal schemes) X, Y , and dualizing complexes R X , RY over X, Y , we show that R X k RY (resp., its derived completion) is a dualizing complex over X ×k Y if and only if X ×k Y is Noetherian of finite Krull dimension.

1 A Basic Homological Question Fix a field k, and let A, B be commutative Noetherian k-algebras. We are interested in the following basic homological question: Question 1 Given M ∈ Dbf (Mod A), and N ∈ Dbf (Mod B), if M has finite injective dimension over A, and N has finite injective dimension over B, does M ⊗k N have finite injective dimension over A ⊗k B? We will approach this question using dualizing complexes. Let us recall the definition. Definition 2 Let A be a commutative Noetherian ring. A complex R ∈ Dbf (Mod A) is called a dualizing complex if it has finite injective dimension over A, and the canonical map R → R Hom A (R, R) is an isomorphism in D(Mod A). Any Noetherian ring which has a dualizing is of finite Krull dimension; see [3, Corollary V.7.2]. Thus, by using dualizing complexes, we will prove the following. Theorem 3 Let k be a field, and let A, B be commutative Noetherian k-algebras with dualizing complexes. Assume A ⊗k B is Noetherian of finite Krull dimension. Then for any M ∈ Dbf (Mod A), and any N ∈ Dbf (Mod B), if M has finite injective dimension over A, and N has finite injective dimension over B, then M ⊗k N has finite injective dimension over A ⊗k B. L. Shaul (B) Departement Wiskunde-Informatica, Universiteit Antwerpen, Middelheim Campus, Middelheimlaan 1, 2020 Antwerp, Belgium e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_25

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2 Equivalence of Question 1 to a Question on Dualizing Complexes Lemma 4 Let k be a field, and let A and B be Noetherian k-algebras with dualizing complexes R and S respectively, such that A ⊗k B is a Noetherian ring. Then the canonical map A ⊗k B → R Hom A⊗k B (R ⊗k S, R ⊗k S) is an isomorphism in D(Mod A ⊗k B). Proof See [7, Lemma 1.2]. (Notice that there is a typo in the proof given there, this natural map goes in the other direction, but it is still an isomorphism by [9, Lemma 8.4], so the proof there holds.)  Proposition 5 Let A be a Noetherian ring with a dualizing complex R. A complex M ∈ Dbf (Mod A) has finite injective dimension over A if and only if R Hom A (M, R) has finite projective dimension over A. Proof This is [3, Proposition V.2.6] combined with the fact that, for bounded complexes with finitely generated cohomology, the flat dimension is equal to the projective dimension.  With these two facts in mind, we reduce Question 1 to a problem about dualizing complexes: Proposition 6 Let k be a field, and let A, B be Noetherian k-algebras with dualizing complexes R, S respectively. Assume A ⊗k B is Noetherian of finite Krull dimension. Then, Question 1 has affirmative answer if and only if R ⊗k S is a dualizing complex over A ⊗k B. Proof In view of Lemma 4, it is clear that if − ⊗k − preserves finite injective dimension, then R ⊗k S is a dualizing complex over A ⊗k B. To see the converse, note that by [9, Lemma 8.4] there is an isomorphism R Hom A (M, R) ⊗k R Hom B (N , S) ∼ = R Hom A⊗k B (M ⊗k N , R ⊗k S) so, the result follows from Proposition 5 and the fact that − ⊗k − preserves finite projective dimension. 

3 Tensor Product of Dualizing Complexes over a Field In this section we show that the two equivalent conclusions of Proposition 6 hold. Proposition 7 Let A be a commutative Noetherian ring with a dualizing complex. Then there is a finite type A-algebra A which is Gorenstein of finite Krull dimension, such that there is a surjection A  A.

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Proof This follows from the proof of [4, Corollary 1.4]. See [7, Proposition 1.1] for details.  Proposition 8 Let k be a field, and let A, B be Noetherian Gorenstein k-algebras, such that A ⊗k B is Noetherian. Then A ⊗k B is Gorenstein.  Proof See [8, Theorem 6(a)]. Lemma 9 Let k be a field, and let A , B  be Noetherian k-algebras with dualizing complexes R  , S  , respectively, such that A ⊗k B  is Noetherian and R  ⊗k S  is a dualizing complex over it. Let A be a finite A -algebra, and let B be a finite B  algebra. Then, for any dualizing complex R over A and any dualizing complex S over B, the complex R ⊗k S is a dualizing complex over A ⊗k B. Proof This follows from [7, Lemmas 1.4 and 1.5].



Theorem 10 Let k be a field, and let A, B be Noetherian k-algebras with dualizing complexes R, S, respectively. Assume A ⊗k B is Noetherian of finite Krull dimension. Then R ⊗k S is a dualizing complex over A ⊗k B. Proof By Proposition 7 there are Gorenstein k-algebras A , B  and surjections A  A and B   B and, moreover, A ⊗k B  is also Noetherian of finite Krull dimension. By Proposition 8, A ⊗k B  is Gorenstein, so finiteness of its Krull dimension implies that it is a dualizing complex over itself. It follows that (A, A , B, B  ) satisfy the conditions of Lemma 9, so the result follows.  Theorem 3 now follows from this result and Proposition 6.

4 Tensor Product of Dualizing Complexes over Adic Rings Adic rings are the algebraic version of affine formal schemes. The above results will rarely work in this category, because tensor products of adic rings are rarely Noetherian. For example, if k is a field of characteristic zero, and A = k[[t]], then A ⊗k A is non-Noetherian (and of infinite Krull dimension). The reason for this failure is that the fiber product in the category of affine formal schemes over k is not represented by the tensor product, but rather by the complete tensor product. Takk A ∼ ing complete tensor product in the above example, we have that A⊗ = k[[t1 , t2 ]] which is a Noetherian ring of finite Krull dimension. We now explain how to generalize Theorem 10 to the formal setting. Definition 11 Let A be a commutative ring, and let a ⊆ A be a finitely generated  := lim A/an . (A, a) is called ideal. The a-adic completion of A is the A-algebra A ← −  The completion functors associated to a pair (A, a) are the functors adic if A ∼ = A. a : Mod A → Mod A,

 a : Mod A → Mod A 

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a (M) := lim M ⊗ A A/an , where in defined by a (M) := lim M ⊗ A A/an and  ← − ← −  a (M) is considered as a A-module. the latter case,  These functors are, in general, neither left exact nor right exact. However, they preserve surjections, so the zero-th cohomology of their left derived functors La : D(Mod A) → D(Mod A),

 a : D(Mod A) → D(Mod A) L

is equal to these functors themselves. Over a Noetherian ring, the functor La enjoys good properties. It has finite cohomological dimension, and there is an explicit formula to calculate it (see [5] and its references for details). However, below, we will have to use this functor over the tensor product of two adic rings, which, as explained above, is usually non-Noetherian. Luckily, it turns out that this good behavior also holds over certain non-Noetherian rings. Recall that if A is a Noetherian ring, and a ⊆ A is an ideal, then local cohomology of an A-module M with respect to a (that is, the right derived functor of the a-torsion functor a (M) := lim Hom A (A/an , M)) may be calculated by tensoring M with − → the Cech complex associated to a finite sequence that generates a; see [2, Theorem 5.1.19]. Given a (not necessarily Noetherian) ring A, and a finitely generated ideal a ⊆ A, we say that a is weakly proregular if, as in the Noetherian case, local cohomology with respect to a coincides with tensoring with the Cech complex associated to some finite sequence generating a. As shown in [1, 5, 6], weak proregularity is exactly the condition needed to ensure that the good properties the La functor has over Noetherian rings hold also over non-Noetherian rings. The above definition (which is somewhat non-standard, but equivalent to the usual one) shows that over Noetherian rings any ideal is weakly proregular. In the non-Noetherian case, the next fact is very useful. Proposition 12 Let k be a field, and let A, B be Noetherian k-algebras. Let a ⊆ A and b ⊆ B be ideals. Then the ideal a ⊗k B + A ⊗k b ⊆ A ⊗k B is weakly proregular. Proof See [5, Example 4.35].



Using this fundamental fact, and some additional homological and homotopical tools, we have the following formal generalization of Theorem 10. Theorem 13 Let k be a field, and let (A, a) and (B, b) be Noetherian adic k-algebras. Set I = a ⊗k B + A ⊗k b ⊆ A ⊗k B. Let R, S be dualizing complexes over A and B, respectively. Assume that the ring A ⊗k B :=  I (A ⊗k B) is  I (R ⊗k S) is a dualizing complex Noetherian of finite Krull dimension. Then L over A ⊗k B. Proof See [7, Theorem 2.9]. As an immediate corollary, we obtain an adic generalization of Proposition 8.



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Corollary 14 Let k be a field, and let (A, a), (B, b) be Noetherian adic Gorenstein k-algebras of finite Krull dimension, such that A ⊗k B is Noetherian of finite Krull dimension. Then A ⊗k B is Gorenstein. Acknowledgements The author acknowledges the support of the European Union for the ERC grant No 257004-HHNcdMir.

References 1. L. Alonso, A. Jeremias, J. Lipman, Local homology and cohomology on schemes. Annales Scientifiques de l’École Normale Supérieure 30(1), 1–39 (1997) 2. M.P. Brodman, R.Y. Sharp, Local cohomology. Camb. Stud. Adv. Math. 60 (1998) 3. R. Hartshorne, Residues and duality, in Lecture Notes in Mathematics, vol. 20 (1966). (Lecture notes of a Seminar on the Work of A. Grothendieck, given at Harvard 1963/64, with an appendix by P. Deligne) 4. T. Kawasaki, On arithmetic Macaulayfication of Noetherian rings. Trans. Am. Math. Soc. 354(1), 123–149 (2002) 5. M. Porta, L. Shaul, A. Yekutieli, On the homology of completion and torsion. Algebras Represent. Theory 17(1), 31–67 (2014) 6. P. Schenzel, Proregular sequences, local cohomology, and completion. Math. Scand. 92, 161– 180 (2003) 7. L. Shaul, Tensor product of dualizing complexes over a field. Preprint arXiv:1412.3759v1 8. M. Tousi, S. Yassemi, Tensor products of some special rings. J. Algebra 268(2), 672–676 (2003) 9. A. Yekutieli, J.J. Zhang, Dualizing complexes and perverse modules over differential algebras. Compositio Mathematica 141(03), 620–654 (2005)

Strong Generators in Tensor Triangulated Categories Johan Steen

Abstract Let T be an essentially small rigid tensor triangulated category. In Balmer (J Reine Angew Math 588:149–168, 2005, [1]), Balmer associates to T a topological space Spc T whose points are the proper prime ideals of T. We show that if Spc T is connected, then T has no nonzero and proper tensor ideals admitting a strong generator.

1 Generators and Representability This paper is based on joint work with Greg Stevenson [7]. Triangulated categories occur naturally in both algebra and topology, and give a unifying framework in which we can do homological algebra via cohomological functors, i.e., functors to abelian categories which turn distinguished triangles to long exact sequences. Common to most naturally occurring triangulated categories is that they come equipped with a set of generators. The existence of generators is immensely helpful, as they allow us to show that a property that is stable under coproducts and the formation of cones automatically holds for the entire triangulated category, once it holds on the generators. In particular, the existence of generators allows us to show representability results for certain cohomological functors (depending on the nature of the generators). Furthermore, in the presence of representability results, one can show the existence of adjoint functors for exact functors between triangulated categories. Let us be more precise by considering an example. Let T be a compactly generated triangulated category, i.e., T is a triangulated category admitting set-indexed coproducts and a set of compact objects U such that the smallest localizing subcategory

J. Steen (B) Institutt for Matematiske Fag, NTNU, 7491 Trondheim, Norway e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_26

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of T containing the objects U coincides with T. The famous Brown representability theorem for compactly generated triangulated categories, due to Neeman [5], then states that a cohomological functor H : Top −→ Ab is representable, that is, of the form HomT (−, X ) for some X ∈ T, if and only if H sends coproducts in T to products in Ab. From this, it is a consequence that an exact functor T −→ T admits a right adjoint provided it commutes with coproducts. Neeman’s proof goes by building a tower of objects in T X 0 −→ X 1 −→ X 2 −→ · · · whose homotopy colimit is the representing object X . When T is not cocomplete (as is the case for the categories we study), this construction does not make sense in general as each of the approximating objects X i , as well as the homotopy colimit construction, involve infinite coproducts. Instead, one imposes conditions on the generators to ensure that each X i can be built inside of T and that the object X is reached after a finite number of steps. Definition 1 A triangulated subcategory of T is said to be thick if it is closed under taking direct summands in T. An object g ∈ T is a (classical) generator for T if the smallest thick subcategory containing g, denoted by thick(g), coincides with T. Put g1 = add{ k g | k ∈ Z} and, for n ≥ 1, gn+1 = add{cone( f ) | f : x −→ y, x ∈ g1 , y ∈ gn }. The object g is a strong generator if there is an N such that g N = T, i.e., g builds every object of T in a bounded number of steps.  Note that n gn = thick(g), and if T has a strong generator, then any generator is strong. Rouquier provides a representation theorem for triangulated categories admitting a strong generator. This theorem simultaneously generalizes Neeman’s representability theorem as well as that of Bondal–Van den Bergh [3], and is crucial to the proof of our main result. Theorem 2 (Rouquier, [6, Corollary 4.17]) Let T be an idempotent complete triangulated category admitting a strong generator, and let H : Top −→ Ab be a cohomological functor. Then H is representable if and only if it is locally finitely presented. The functor H is locally finitely generated if for any object x there is an object a and a natural transformation HomT (−, a) −→ H which is an epimorphism when evaluated on  k x, for all integers k. It is locally finitely presented if it is locally finitely generated and for all objects b the kernel of any natural transformation HomT (−, b) −→ H is locally finitely generated as well.

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2 Rigid Tensor Triangulated Categories Let (T, ⊗, 1) be a tensor triangulated category, i.e., both triangulated and symmetric monoidal such that ⊗ is biexact with respect to the triangulated structure. A thick subcategory S ⊆ T is a tensor ideal if s ⊗ x ∈ S for all s ∈ S and all x ∈ T. A proper tensor ideal P  T is prime if x ⊗ y ∈ P implies that x ∈ P or y ∈ P. If T is essentially small, the prime tensor ideals form a set, and the Balmer spectrum is Spc T = {P  T prime}. This is topologized by choosing the subsets supp x = {P ∈ Spc T | x ∈ / P} for all objects x in T to be a closed basis. As shown by Balmer [1, Theorem 4.10], this topological space classifies the (radical) tensor ideals. We study tensor triangulated categories that additionally have internal homobjects, meaning that the monoidal structure is closed, i.e., for all objects x, − ⊗ x admits a right adjoint, hom(x, −). Moreover, we say that the category is rigid if the canonical morphism hom(x, 1) ⊗ y −→ hom(x, y) is a natural isomorphism for all x and y. This ensures that one has well-behaved dual objects x ∨ = hom(x, 1).

3 The Main Theorem We are now ready to state our main result. Theorem 3 (Steen–Stevenson, [7, Theorem 4.1]) Let T be an essentially small rigid tensor triangulated category whose Balmer spectrum Spc T is connected as a topological space. If S ⊆ T is a strongly generated tensor ideal, then either S = 0 or S = T. The following example shows that the distinction between thick subcategories and tensor ideals is necessary. Example 4 Consider the bounded derived category of coherent sheaves on the projective line over a field k, Db (coh P1k )  Dperf (P1k ). The tensor product over O = OP1k makes this an essentially small tensor triangulated category. It is also rigid, and has connected spectrum Spc Db (coh P1k ) ≈ P1k ;

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see [1, Corollary 5.6]. The proper thick subcategory thick(O) is strongly generated, but not a tensor ideal, since the smallest tensor ideal containing the tensor unit is the whole category. We remark that every thick subcategory is automatically a tensor ideal provided thick(1) = T. Example 5 A rigidly compactly generated tensor triangulated category (T, ⊗, 1) is a compactly generated triangulated category, whose thick subcategory of compact objects Tc ⊆ T contains 1 and is a rigid tensor triangulated category with the structure inherited from T. In particular, it is essentially small. All of the following examples are of this form and so, our theorem applies to Tc : (i) Let R be a commutative ring. The unbounded derived category (D(R), ⊗LR , R) is a rigidly compactly generated tensor triangulated category. The compact objects D(R)c are the perfect complexes Dperf (R), i.e., bounded complexes of finitely generated projective modules. The internal hom-objects are given by RHom(−, −). As pointed out in [2, Example 4.4], there is a homeomorphism of spectra Spc Dperf (R) ≈ Spec(R). We conclude that, for a commutative ring R with no non-trivial idempotents, there are no strongly generated proper thick subcategories of Dperf (R) except 0. (ii) Let k be a field of characteristic p > 0 and G a finite group whose order is divisible by p. Then (StMod kG, ⊗k , k), the stable category of kG-modules, is a tensor triangulated category. Here, ⊗k denotes the tensor product over k with the diagonal group action. The subcategory of compact objects is precisely stmod kG, the stable category of finitely generated kG-modules. There is a homeomorphism Spc(stmod kG) ≈ Proj H • (G; k); see [2, Proposition 8.5]. Consequently, there are no proper tensor ideals of stmod kG, except 0, admitting a strong generator. This is true for all nonzero proper thick subcategories provided G is a p-group, since thick(k) = stmod kG in this case. (iii) The stable homotopy category (SH, ∧, S 0 ) is a tensor triangulated category. The subcategory of finite spectra SHfin is precisely the subcategory of compact objects in SH, and is moreover the thick subcategory generated by S 0 . Balmer has shown in [2, Corollary 9.5], using the classification of thick subcategories given by Hopkins–Smith [4], that Spc(SHfin ) is a connected space. This category is moreover rigid, its dual objects being given by the Spanier–Whitehead duals. Thus our main theorem applies, and we can additionally show that the category SHfin has no strong generators either; see [7, Theorem 5.2]. Thus, the only thick subcategory of SHfin admitting a strong generator is the zero subcategory.

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3.1 Sketch of the Proof We sketch the proof of Theorem 3. Our aim is to show that if S ⊆ T is a tensor ideal admitting a strong generator then Spc T ≈ Spc S  Spc S⊥ , where S⊥ = {x ∈ T | HomT (S, x) = 0}. Thus if S is both nonzero and proper, the space Spc T decomposes. The proof proceeds in two steps. First, we show that the inclusion functor i ∗ : S −→ T admits a right adjoint. This uses Rouquier’s representability result, and is the most technical part. For any t ∈ T, the functor   Ht = HomT i ∗ (−), t : Sop −→ Ab is locally finitely presented. This can be reduced to checking only on suspensions of the generator, and relies heavily on both the ⊗-hom-adjunction and rigidity. Since the Balmer spectrum is invariant under idempotent completion, [1, Corollary 3.14], we can assume that T is idempotent complete. Since S is assumed to be strongly generated, we invoke Rouquier’s result and obtain an isomorphism Ht ∼ = HomS (−, i ! t). This extends to a functor i ! : T −→ S which is right adjoint to i ∗ . The second part of the proof goes by noting that rigidity ensures that S⊥ is again a tensor ideal and, moreover, that (−)∨ : T −→ T is a duality such that x ∈ S implies x ∨ ∈ S. Consequently, S⊥ ∼ = S ⊕ S⊥ . Thus the spectrum decomposes = ⊥ S and T ∼ as Spc T ≈ Spc S  Spc S⊥ .

References 1. P. Balmer, The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math. 588, 149–168 (2005) 2. P. Balmer, Spectra, spectra, spectra–tensor triangular spectra versus Zariski spectra of endomorphism rings. Algebr. Geom. Topol. 10(3), 1521–1563 (2010) 3. A. Bondal, M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3(1), 1–36 (2003) 4. M.J. Hopkins, J.H. Smith, Nilpotence and stable homotopy theory, II. Ann. Math. (2) 148(1), 1–49 (1998) 5. A. Neeman, The chromatic tower for D(R). Topology, 31(3), 519–532 (1992). (with an appendix by Marcel Bökstedt) 6. R. Rouquier, Dimensions of triangulated categories. J. K-Theory 1(2), 193–256 (2008) 7. J. Steen, G. Stevenson, Strong generators in tensor triangulated categories. Bull. Lond. Math. Soc. 47(4), 607–616 (2015)

Abelian Model Structures and Applications Jan Šˇtovíˇcek

Abstract We give an overview on the recent development and applications of model structures in algebra (more precisely in representation theory of rings and algebras, and in studying quasi-coherent sheave over suitable schemes).

1 Motivation Although the theory of triangulated categories is an extremely successful one and plays an important rôle in representation theory and algebraic geometry (see e.g., [1]), it is well known to have certain imperfections and limitations. Various methods have been developed to fix these, and here we focus on a classical one: Quillen structures and categories. The main point here is that we bring into life Hovey’s observation from [5] that there is a very close relation between model structures and cotorsion pairs. This makes construction of model structures tailored to one’s need practical in the context of representation theory. This short note highlights relevant results from papers [8, 10] and references therein. Another application which is not discussed here can be found in [9].

2 Model Structures on Abelian Categories In order to explain the main idea, let us recall a primitive of model structures. Definition 1 Given two morphisms f : A → B and g : X → Y in a category C, we write f  g if for each solid commutative square the dotted arrow can be (not necessarily uniquely) filled in so that both triangles commute.

J. Šˇtovíˇcek (B) Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_27

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/X >

A f

 B

h

g

 / Y.

A weak factorization system in C is a pair (L, R) of classes of morphisms such that: (i) L and R are closed under existing retracts in C; (ii) f  g whenever f ∈ L and g ∈ R; (iii) each morphism h in C factors as h = g f with f ∈ L and g ∈ R. Algebraic counterparts of weak factorization systems are so-called complete cotorsion pairs. From now on, A will stand for an abelian category. Definition 2 A pair (C, F) of classes of objects of A is a complete cotorsion pair if (i) L and R are closed under retracts; (ii) Ext 1A (C, F) = 0 for each C ∈ C and F ∈ F; (iii) given an object X ∈ A, there exist (typically non-unique) short exact sequences 0 → X → F1 → C1 → 0 and 0 → F2 → C2 → X → 0. The word “complete” appears for historical reasons. Cotorsion pairs had been originally defined only as pairs of classes of objects mutually orthogonal to Ext 1A . The important of completeness, that is part (3) of the definition, was only recognized later. The relation between the two above definitions has been implicitly formalized by Hovey [5]. Theorem 3 (Hovey [5, Proposition 5.4]) Let (C, F) be a cotorsion pair in A and put Mono C = { f : A → B | f is mono and Coker f ∈ C}, and Epi F = {g : X → Y | g is epi and Ker g ∈ F}. Then, (Mono C, Epi F) is a weak factorization system in A. The latter theorem can be adapted to give a similar correspondence for model structures on abelian categories. To make the correspondence completely transparent, recall that Definition 4 A model structure on a category C is a triple of classes of morphisms (Cof, W, Fib) such that: (i) W is closed under retracts, and if u, v is a pair of composable morphisms such that two of u, v, v ◦ u are in W, so is the third; (ii) (Cof ∩ W, Fib) and (Cof, W ∩ Fib) are weak factorization systems. Then the main result of [5] is the following (a detailed treatment can also be found in [2, 8]).

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Theorem 5 (Hovey [5]) Suppose that A is an abelian category and let (C, T , F) be a triple of classes of objects such that (i) T is closed under retracts, and if 0 → T1 → T → T2 → 0 is exact and two of T1 , T2 , T belong to T , so does the third; (ii) (C ∩ T , F) and (C, T ∩ F) are complete cotorsion pairs. Then, there is a unique model structure on A such that Cof = Mono C and Fib = Epi F. The class W of weak equivalences depends only on T and not on the particular choice of C and F. It is needless to say that T will become the class of trivial objects, i.e., those weakly equivalent to the zero object. Of course, we do not get all model structures from cotorsion pairs. We precisely get those compatible with the abelian structure in the sense that cofibrations are exactly monomorphisms with cofibrant cokernel and fibrations are precisely epimorphisms with fibrant kernel. Example 6 (Šˇt ovíˇcek, [8, Sect. 8.5]) Let X be a quasi-compact separated scheme. Then there exists a monoidal model structure on the closed monoidal category (C(Qcoh X ), ⊗, O X ) of complexes of quasi-coherent sheaves over X . The homotopy category is the ordinary derived category D(Qcoh X ). The cofibrant objects are best described as K -flat complexes of sheaves.

3 Approximation Theory The reader may wonder how to actually construct pairs of complete cotorsion pairs needed in Theorem 5. To start with, one may even wonder how to construct just one single complete cotorsion pair. In the context of modules over rings, cotorsion pairs have always existed in abundance in tilting theory [1] and approximation theory [4]. The latter has the goal to approximate arbitrary modules by special ones as in Definition 2 (iii). Such approximations are then conceptually very close to cofibrant and fibrant replacements in homotopy theory. At this point, we will be more ambitious and present an improvement of the standard approximation theory. A usual method to construct weak factorization systems in homotopy theory is Quillen’s small object argument and, as noted already by Hovey [5], the principal method to construct cotorsion pairs in [4] actually is a form of Quillen’s argument. This observation evolved into Theorem 7 (Saorín–Šˇt ovíˇcek, [7, 2.15], and Šˇt ovíˇcek, [8, 5.16]) Let A be a Grothendieck abelian category and S ⊆ A be a small generating set of objects. Then there is a complete cotorsion pair (C, F) where F = {F ∈ A | Ext 1A (S, F) = 0 ∀S ∈ S} and C consists of transfinite extensions of objects of S.

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Here, a transfinite extension of objects of S stands for a colimit of a well ordered chain of inclusions 0 = E 0  E 1  E 2  · · ·  E α  E α+1  · · · such that all the factors E α+1 /E α are isomorphic to objects of S, and E γ = colim α 2) gives no additional structure on the object resulting from reindexing k-times. This and similar observations led to the following considerations.

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2.1 Stabilization Hypothesis If k ≥ n + 2 then any k-tuply monoidal weak n-category is a (k + 1)-tuply monoidal weak n-category. This is a hypothesis rather than a conjecture, because in 1995 there was not a good definition of a weak n-category. Carlos Simpson proved the hypothesis holds for Tamsamani’s notion of a weak n-category [8], and we seek to prove it holds for Rezk’s notion and to also prove a more general version of stabilization. Batanin has reduced this question to a simpler question in a series of papers over the past several years. First, the structure of a k-tuply monoidal n-category can be encoded by something called a k-operad in nCat (a weak version of this theory would ask it to be encoded specifically by an E k -operad). Indeed, k-operads were primarily introduced to study this phenomenon, as [5] makes clear. Let tr n M denote the n-truncation of M (i.e., for all X,Y the space map(X, Y ) is n-truncated in sSet), and let O pk (tr n M) denote the model category of k-operads in tr n M; see [4]. Batanin defined a suspension funcloc (tr n M) → O pkloc (tr n M) on so-called n-locally constant k-operads tor S ∗ : O pk+1 which captures the suspension studied by Baez and Dolan. This allows for a proof of the Baez–Dolan stabilization hypothesis via the following result. Theorem 4 If k ≥ n + 2 then S ∗ is a right Quillen equivalence. It is interesting to highlight the following easier to state result, which is implied by Theorem 4 and implies the classical Baez–Dolan stabilization hypothesis: Theorem 5 (Batanin [3]) Suppose the unit of (M, ⊗, I ) is cofibrant. Let 1k denote the canonical k-operad 1k (T ) = I , T ∈ Or d(k), and let G n,k denote the ntruncation of its cofibrant replacement. If k ≥ n + 2 then AlgG n,k (tr n M) is Quillen equivalent to AlgG n+1,k (tr n M). This result can be proven by mapping each of AlgG n,k (tr n M) and AlgG n,k+1 (tr n M) to Alg E∞ (tr n M), where E ∞ is a cofibrant replacement of Com taken in S O p(M). As a consequence of our work, both of these statements will be proven when tr n M is Rezk’s model of weak n-categories, i.e., the truncation tr≤n (n -spaces).

3 Tools from Abstract Homotopy Theory The proof of Theorem 2 involves a careful analysis of the free-algebra extensions coming from the free-forgetful adjunctions S : MN  S O p(M) : U and  O pn (M). For simplicity, we focus our exposition on the former. F : Colln (M) N = n∈N M has the product model structure and this adjunction passes Here, M through n∈N Mn with the projective model structure. As always, when attempting to transfer a model structure across an adjunction, what must be checked is that, for any trivial cofibration f : K → L in MN and for any map S(K ) → O in S O p(M),

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 the pushout map O → P := O S(K ) S(L) is a weak equivalence. If this is only true for cofibrant O then the resulting structure is that of a semi-model category, which means the lifting of a trivial cofibration f against a fibration, and the factorization of a map g into a trivial cofibration followed by a fibration only hold for maps f, g with cofibrant domain. The analysis in [4, Sect. 9.4] demonstrates that the monad S can be represented by a -cofibrant colored operad. Then, [12, Theorem 6.3.1] provides the semi-model structure, which Batanin has used to prove Theorem 5 in [3]. In order  to promote this to a left proper model structure we view K = (K n )n∈N as (K 0 ) K r ed , where K r ed := (K n )n≥1 and we factor the pushout as two pushouts S(K )

/ S(L)



 / Pz  Pr ed

O

 O

O

 / P,

  where Pz = S(L 0 ) S(K 0 ) O and Pr ed = S(L r ed ) S(Kr ed ) O. This reduces us to studying the maps O → Pz and O → Pr ed separately. The commutative monoid axiom of [10], there used to prove that commutative monoids in M inherit a model structure, can be generalized to this setting to provide control over the former map. The hypotheses that M be strongly h-monoidal and compactly generated allow for control over the latter map, as described in [4]. However, since a semi-model structure suffices, we prefer to say here some few words about Theorem 3, which will appear in [9]. The main tool to prove Theorem 3 is a semi-model category version of Jeff Smith’s theorem for creating combinatorial model structures (presented in [2], where Theorem 3 is conjectured). By carefully proving semi-model categorical versions of several results from [7], we are able to prove that the set of generating trivial cofibrations JC produced by Smith’s theorem does in fact provide a Bousfield localization semi-model structure which satisfies the universal property of localization, and which is Quillen equivalent (as a semi-model category) to the model category localization L C (M) should it exist. This theorem is applied to O pk (tr n (M)) and completes the proof of Theorem 4.

References 1. J.C. Baez, J. Dolan, Higher-dimensional algebra and topological quantum field theory. J. Math. Phys. 36(11), 6073–6105 (1995) 2. C. Barwick, On left and right model categories and left and right Bousfield localizations. Homology, Homotopy Appl. 12(2), 245–320 (2010)

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3. M. Batanin, An operadic proof of the Breen-Baez–Dolan stabilisation hypothesis. In preparation 4. M. Batanin, C. Berger, Homotopy theory for algebras over polynomial monads. Preprint. http:// arxiv.org/abs/1305.0086 (2014) 5. M. Batanin, The Eckmann-Hilton argument and higher operads. Adv. Math. 217(1), 334–385 (2008) 6. P. Hackney, M. Robertson, D. Yau, Relative left properness of colored operads (2014). http:// arxiv.org/abs/1411.4668v1 7. P.S. Hirschhorn, Model categories and their localizations. Mathematical Surveys and Monographs (American Mathematical Society, Providence, RI, 2003) 8. C. Simpson, On the Breen-Baez–Dolan stabilization hypothesis for Tamsamanis weak ncategories (1998). arxiv:math/9810058 9. D. White, Bousfield localization without left properness. In preparation 10. D. White, Model structures on commutative monoids in general model categories (2014). arxiv:1403.6759 11. D. White, Monoidal Bousfield localizations and algebras over operads (2014). arxiv:1404.5197 12. D. White, D. Yau, Bousfield localizations and algebras over colored operads (2015). arxiv:1503.06720

Vanishing of Tor Olgur Celikbas and Roger Wiegand

Abstract Let M and N be finitely generated modules over a local ring (R, m). We seek conditions implying that ToriR (M, N ) = 0 for all i > 0. In addition, we examine consequences of the vanishing of Tor.

1 The Inspiration Throughout, R is assumed to be a local Noetherian ring with maximal ideal m; modules are always assumed to be finitely generated. The theme of this note is that taking tensor products tends to make things worse. To put a positive spin on it, if M ⊗ R N is nice (e.g., torsion-free, maximal Cohen–Macaulay, . . .), then both M and N should be nice. Here is an easy example: Exercise 1 If M ⊗ R N is a non-zero free module, then both M and N are free. (Hint: Write M = F ⊕ M  , where F is free and M  has no non-zero free summand; then show M  = 0.) Our inspiration is the following result due to Auslander [1] (Lichtenbaum [13] in the ramified case): Theorem 2 Assume R is a regular local ring and M ⊗ R N is non-zero and torsionfree. Then both M and N are torsion-free. Moreover, ToriR (M, N ) = 0 for all i > 0, and (1) pd R M + pd R N = pd R (M ⊗ R N ) .

O. Celikbas (B) University of Connecticut, Storrs, CT 06269, USA e-mail: [email protected] R. Wiegand University of Nebraska, Lincoln, NE 68588-0130, USA e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_32

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The technical key to Auslander’s proof is Tor-rigidity. A pair (M, N ) of Rmodules is Tor-rigid provided the vanishing of Tor Rj (M, N ) for some j ≥ 1 forces ToriR (M, N ) = 0 for all i ≥ j; the module M is rigid provided the pair (M, N ) is rigid for every N . Notation 3 We let M denote the torsion submodule of M, that is, the kernel of the natural map M → Q(R) ⊗ R M, where Q(R) denotes the total quotient ring (obtained by inverting the non-zero divisors of R). We let ⊥M denote the torsionfree module M/M. A slight rearrangement of Auslander’s proof yields the following. Theorem 4 Let M and N be nonzero modules over a reduced local ring R. Assume that either ⊥M or ⊥N is rigid and that M ⊗ R N is torsion-free. Then both M and N are torsion-free, and ToriR (M, N ) = 0 for all i ≥ 1.

2 The Wish In view of the Auslander–Buchsbaum formula [2], Eq. (1) can be rewritten in a way that makes sense even for modules of infinite projective dimension: depth M + depth N = depth R + depth M ⊗ R N .

(2)

This formula is known as the depth formula, and it is useful in determining depth properties (e.g., Serre’s conditions) of tensor products. For example, if the depth formula holds for modules M and N over a Cohen–Macaulay ring, one knows that M ⊗ R N is maximal Cohen–Macaulay (MCM) if and only if both M and N are MCM. Auslander [1] showed that the depth formula follows whenever M and N are Tor-independent (that is, ToriR (M, N ) = 0 for all i > 0), and either M or N has finite projective dimension. Huneke–Wiegand [12] showed that the depth formula follows from Tor-independence if R is a complete intersection (defined in the next section). Christensen–Jorgensen [5] verified the depth formula for Tor-independent modules over an AB ring. This is a Gorenstein ring R for which there is a bound b, depending only on the ring, such that ToriR (M, N ) = 0 for all i > b whenever ToriR (M, N ) = 0 for all i  0. AB-rings were originally defined, in [10], in terms of vanishing of Ext, but it is shown in [10, §3] that the condition is equivalent to the one given above. One might wish (somewhat short of conjecture) that the depth formula holds for Tor-independent modules over every local ring. At this point no counterexamples are known.

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3 Tor-Independence Particularly over complete intersections and AB-rings, it is worthwhile to find criteria for Tor-independence, for one has the depth formula for Tor-independent modules. Definition 5 A local ring (R, m) is a complete intersection provided the completion  is isomorphic to S/( f 1 , . . . , f c ), where (S, n) is a regular local ring and (f) = R ( f 1 , . . . , f c ) is a regular sequence in n. The number c is the relative codimension  in S. (If (f) ⊆ n2 , then c is the (absolute) codimension codim R, that is the of R difference embdim R − dim R, where embdim R, the embedding dimension, is the minimal number of generators for the maximal ideal m.) One would like to have results for complete intersections that emulate Auslander’s theorem for regular rings: M ⊗ R N torsion-free implies M and N are Torindependent. Here is such a result for hypersurfaces (complete intersections of codimension one): Theorem 6 (Huneke–Wiegand, [12]) Let M and N be modules over a hypersurface (R, m), and assume that M has rank (that is, Q(R) ⊗ R M is Q(R)-free). If M ⊗ R N is reflexive, then M and N are Tor-independent. The assumption that M has rank rules out trivial counterexamples such as R = k[[x, y]]/(x y), M = R/(x). Here, M ⊗ R M = M, which is MCM and hence reflexive, yet ToriR (M, M) = 0 for odd positive i. Consideration of Serre’s conditions suggests a conjecture, or at least another wish. Following Evans–Griffith [8], we say that a module M satisfies Serre’s condition (Sn ) provided depth Rp ≥ min{n, dim Rp } for every prime ideal p of R. (Warning: some sources use dim Mp instead of dim Rp in the definition, but that formulation does not work well for our purposes.) Noting that “torsion-free” = (S1 ) and “reflexive” = (S2 ) over complete intersections, one is led naturally to the following: Conjecture Let M and N be modules over a complete intersection R of codimension c, and assume that M has rank. If M ⊗ R N satisfies Serre’s condition (Sc+1 ), then M and N are Tor-independent. Huneke–Jorgensen–Wiegand [11] made some progress on this conjecture in codimensions two and three and also introduced the notion of “quasi-liftings”, which permit arguments using induction on codimension. One can always assume the rings in question are complete. If, now, M is a torsion-free module over a complete intersection R of codimension c > 0, one has R = S/( f ), where S is a complete intersection of codimension c − 1 and f is a non-zero divisor in the square of the maximal ideal of S. The quasi-lifting E of M to S is the first syzygy over S of the first cosyzygy of M over R. More recently, Dao [6, 7] introduced the “η-pairing”, which generalizes the θ -pairing used by Hochster [9] in 1980 in his study of the Direct Summand Conjecture. Assume that there is some non-negative integer m such that ToriR (M, N ) has finite length for all i ≥ m. Then,

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O. Celikbas and R. Wiegand n 1  (−1)i length(ToriR (M, N )) . n→∞ n e i=m

ηeR (M, N ) = lim

Here, e can be any positive integer so that ηeR (M, N ) can be infinite. However, if (R, m, k) is a complete intersection of codimension c, then ηcR (M, N ) < ∞, and  is a complete ηeR (M, N ) = 0 if e > c; see [7, Theorem 4.3]. Furthermore, if R intersection of relative codimension c in an unramified regular local ring, and if ηcR (M, N ) = 0, then the pair (M, N ) is c-rigid, that is, the vanishing of c consecutive Tor Rj (M, N ) forces the vanishing of all subsequent ToriR (M, N ). In [4, §3, §4], the authors exploit vanishing of the η-pairing to obtain new results on vanishing of Tor, including the following: Theorem 7 (Celikbas–Iyengar–Piepmeyer–Wiegand [4, Corollary 3.14]) Let (R, m) be a complete intersection of codimension c, and let M and N be finitely generated R-modules. Assume that M and N satisfy (Sc ), M ⊗ R N satisfies (Sc+1 ), and Mp is a free Rp -module for all prime ideals p of R of height at most c. Then, ToriR (M, N ) = 0 for all i ≥ 1, and hence the depth formula holds. We refer the reader to the survey [3] for many more results on rigidity and vanishing of Tor. Acknowledgements R. Wiegand thanks the Simons Foundation for support through a Collaboration Grant.

References 1. M. Auslander, Modules over unramified regular local rings. Illinois J. Math. 5, 631–647 (1961) 2. M. Auslander, D.A. Buchsbaum, Homological dimension in local rings. Trans. Am. Math. Soc. 85, 390–405 (1957) 3. O. Celikbas, R. Wiegand, Vanishing of tor, and why we care about it. J. Pure Appl. Algebra 219, 429–448 (2015) 4. O. Celikbas, S.B. Iyengar, G. Piepmeyer, R. Wiegand, Criteria for vanishing of Tor over complete intersections. Pacific J. Math. 276, 93–115 (2015) 5. L.W. Christensen, D.A. Jorgensen, Vanishing of Tate homology and depth formulas over local rings. J. Pure Appl. Algebra 219, 464–481 (2015) 6. H. Dao, Decent intersection and Tor-rigidity for modules over local hypersurfaces. Trans. Am. Math. Soc. 365, 2803–2821 (2013) 7. H. Dao, Asymptotic behaviour of Tor over complete intersections and applications. Preprint available at arXiv:07105818 8. E.G. Evans, P. Griffith, Syzygies. London Mathematical Society Lecture Note Series, vol. 106 (Cambridge University Press, Cambridge, 1985) 9. M. Hochster, The dimension of an intersection in an ambient hypersurface, In Algebraic geometry (Chicago, Ill., 1980). Lecture Notes in Mathematics, vol. 862 (Springer, Berlin, 1981), pp. 93–106 10. C. Huneke, D.A. Jorgensen, Symmetry in the vanishing of Ext over Gorenstein rings. Math. Scand. 93, 161–184 (2003)

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11. C. Huneke, D.A. Jorgensen, R. Wiegand, Vanishing theorems for complete intersections. J. Algebra 238, 684–702 (2001) 12. C. Huneke, R. Wiegand, Tensor products of modules and the rigidity of Tor. Math Ann 299, 449–476 (Correction at: Math. Ann. 338(2007), 291–293 (1994)) 13. S. Lichtenbaum, On the vanishing of Tor in regular local rings. Illinois J. Math. 10, 220–226 (1996)

Prime Ideals in Noetherian Rings Sylvia Wiegand

Abstract We discuss some questions and some results concerning partially ordered sets of prime ideals in Noetherian rings. Our focus is two-dimensional integral domains of polynomials and power series.

1 Introduction and Background Let R be a commutative Noetherian ring, and let Spec(R) denote the prime spectrum, i.e., the set of prime ideals of R as a partially ordered set, or poset, under inclusion. Our work on this topic is motivated by the following question raised by Kaplansky about 1950: Question 1 Which partially ordered sets occur as Spec(R) for some Noetherian ring R? This difficult question remains unanswered, although there have been many related results, such as: (1) Hochster’s characterization of the prime spectrum of a commutative ring as a topological space (see [4]); (2) Lewis’ result that every finite poset is the prime spectrum of a commutative ring (see [5]); (3) the identification of some properties of prime spectra of Noetherian rings (see [13]); (4) examples of Noetherian rings that do not have certain other properties (see [8–10]); (5) characterizations of prime spectra of specific classes of Noetherian rings or of particular Noetherian rings (see [3, 11]); etc. This work is discussed in [12], along with other results and references. Theorem 2 is a generalization of examples due to Nagata–Heitmann–McAdam: prime ideals in a Noetherian ring can exhibit any finite amount of “misbehavior”. Theorem 2 (Wiegand [14, Th. 1]) Let F be an arbitrary finite poset. There exist a Noetherian ring R and a saturated order-embedding ϕ : F → Spec(R) such that ϕ preserves minimal upper bound sets and maximal lower bound sets. In detail, for u, v ∈ F, we have S. Wiegand (B) Department of Mathematics, University of Nebraska, Lincoln, NE 68502-0130, USA e-mail: [email protected] © Springer International Publishing Switzerland 2016 D. Herbera et al. (eds.), Extended Abstracts Spring 2015, Trends in Mathematics 5, DOI 10.1007/978-3-319-45441-2_33

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u < v if and only if ϕ(u) < ϕ(v); v covers u if and only if ϕ(v) covers ϕ(u); ϕ(mub F (u, v)) = mub(ϕ(u), ϕ(v)); and ϕ(Mlb F (u, v)) = Mlb(ϕ(u), ϕ(v)).

Here v covers u means v > u and no element is between them; the minimal upper bound set of u and v is mub F (u, v) = min({w ∈ F | w ≥ u, w ≥ v}); and the maximal lower bound set of u and v is Mlb F (u, v) = max({w ∈ F | w ≤ u, w ≤ v}). Theorem 3, due to Steve McAdam, guarantees that noncatenary misbehavior cannot be too widespread in the prime spectrum of a Noetherian ring: Theorem 3 (McAdam, [7]) Let P be a prime ideal of height n in a Noetherian ring. Then all but finitely many covers of P have height n + 1. One might conjecture from Theorem 3 that, if a finite “bad” subset is removed, the remaining prime ideals of Noetherian rings behave well, like those in excellent rings.1 Corollary 4 follows from Theorem 2 and is related to our focus on dimension two. Corollary 4 (Wiegand, [14, Th. 2])] Let U be a countable poset of dimension two. Assume that U has a unique minimal element and max(U ) is finite. Then U ∼ = Spec(R) for some countable Noetherian domain R if and only if, for each element u with ht(u) = 2, the set L e (u) := {v ∈ U | v < u and v is not less than any other element} is infinite. Our current and recent investigations show that certain finite subsets help determine prime spectra for power series rings; see Theorems 7 and 8.

2 Spectra of 2-Dim Polynomial-Power Series Domains Let x, y, z be indeterminates over a one-dimensional Noetherian domain R or a field k. Let Z be the ring of integers, Q the field of rational numbers, C the field of complex numbers, and R the field of real numbers. Let Q be the algebraic closure of Q in C. Question 5 Which pairs of the following prime spectra are order-isomorphic? What properties do the prime spectra have? Which, if any, are shown in the diagrams below? (1) Spec(Z[y]) (4) Spec(Q[x, y]) (7) Spec(Z[[x]])

(2) Spec((Z/2Z)[y]) (5) Spec(Q[x, y]) (8) Spec(Q[y][[x]])

(3) Spec(Z(2) [y]) (6) Spec(C[x, y]) (9) Spec(Q[[x]][y])

1 For the definition of “excellent ring”, see [6, p. 260]. Basically, “excellence” means the ring is catenary and has other nice properties that homomorphic images of polynomial rings over a field possess.

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(a)

|R|

•

•

•

•

|R|

|R|

•

189

··· ···

(b)

•

•

•

•

|R|

|R|

··· ···

•

Diagram 1 Two possible prime spectra for certain of the rings of (1)–(13)

(10) Spec(C[x, y, z]/(x 4 + y 4 + z 4 − 1)) (11) Spec(C[[x, y, z]]/(x 4 + y 4 + z − 1)) or (13) Spec(Z[y][[x]]/(x − 2y 2 − (12) Spec(Z[[x]][y]/(x − 2y 2 − 2)), 2)). In each diagram there is a unique height-one element with more than one cover; |L e (u)| = |R| for every height-two u; and the number of height-two elements is ℵ0 . 4

Remark 6 Some answers: (i) The spectra of the rings in items (1)–(5) are countable and thus are not orderisomorphic to the others, which are uncountable. (ii) Diagram 1b shows Spec(R[[x]]), for every one-dimensional countable Noetherian domain R with countably many maximal ideals; see [2]. Thus, Spec(Z[[x]]) ∼ = Spec(Q[y][[x]]). Diagram 1a shows Spec(Q[[x]][y]) and so, Spec(Q[[x]][y])  Spec(Q[y][[x]]); see [12]. (iii) Roger Wiegand gives five axioms that characterize Spec(Z[y]) in [11]. His axioms imply that, if F is a field, then Spec(Z[y]) ∼ = Spec(F[x, y]) ⇐⇒ F is contained in the algebraic closure of a finite field; see [12]. Thus Spec(Z[y]) ∼ = Spec((Z/2Z)[x, y]), but Spec(Z[y])  Spec(Q[x, y]). Translated to algebraic geometry, the crucial axiom for Spec(Z[y]) is “For every curve C and finite set of points P1 , . . . , Pn on C, there exists an irreducible curve that meets C in exactly the points P1 , . . . , Pn .” In Q[x, y], the point P = (1, 1) is on the curve C of x 3 − y 2 − 1 = 0, but every curve that meets C at P meets C in a second point. In ring theory language, this is to say that the maximal ideal M = (x − 1, y − 1)Q[x, y] contains the prime ideal P = (x 3 − y 2 − 1)Q[x, y]. But every height-one prime ideal Q ⊆ M with Q = P is such that P + Q is contained in another maximal ideal N = M of Spec(Q[x, y]). The three spectra Spec(Q[x, y]), Spec(Q[x, y]) and Spec(C[x, y]) are unknown. It is not known if Spec(Q[x, y]) ∼ = Spec(Q[x, y]) or if Spec(Q[x, y]) ∼ = Spec(Q[x, y]) \ {u}, for u = (0). (iv) Diagram 1.a is a subset of but not the same as the spectrum in (3); it contains other height-ones below exactly n height-two elements, for every n ∈ N; see [3].

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(v) The prime spectrum of item (10) might appear to be the same as Spec(C[x, y]). But in Spec(C[x, y]) every maximal ideal is uniquely determined by a pair of height-one prime ideals; e.g., the maximal ideal (x − 2, y − 3)C[x, y] is unique for (x − 2) and (y − 3). In algebraic geometric terms “every point is the set-theoretic intersection of two curves.” Roger Wiegand has shown this property fails for spectrum (10); see [11, Cor 3]. (vi) The spectrum of item (11), given in [1], is not like the others because the ring is local. (vii) We discuss spectra (12) and (13) briefly in Sect. 3.

3 Prime Spectra for Quotients of Mixed Polynomial-Power Series Rings We give a sample of results from joint work with Ela Celibas and Christina EubanksTurner on prime spectra of quotients of mixed polynomial-power series rings over a Noetherian one-dimensional domain R or a field k. For example, we show: Theorem 7 (Celikbas–Eubanks-Turner–Wiegand [1]) Let R be a countable Noetherian one-dimensional domain with infinitely many maximal ideals, and let x and y be indeterminates over R. Let Q be a height-one prime ideal of R[y][[x]] such that x ∈ / Q and dim(R[y][[x]]/Q) = 2. Then, there exists a finite partially ordered set F of dimension at most one and an order-embedding ϕ : F → U such that U = Spec(R[y][[x]]/Q) is determined by ϕ(F) and the following properties: (i) U is a two-dimensional partially ordered set with a unique minimal element; (ii) ht(ϕ( f )) = 1 + ht( f ), for every f ∈ F, and ϕ preserves minimal upper bounds, that is, for every pair f, g ∈ F, ϕ(mub F ( f, g)) = mubU (ϕ( f ), ϕ(g)); (iii) for every u ∈ U of height two, there exists an f ∈ min F such that ϕ( f ) < u, and |L e (u)| = |R| (|L e (u)| as in Corollary 4); (iv) for every u ∈ min ϕ(F), |{w ∈ U | u < w}| = ℵ0 ; (v) if u ∈ U \ ϕ(F) and ht u = 1, then |{w ∈ U | u < w}| = 1. Diagram 2 of Spec(Z[y][[x]]/(x − 2y 2 + 2)) illustrates Theorem 7, except that we cannot show the boxes of cardinality |R| below every height-two element. In Diagram 2, the partially ordered set F from Theorem 7 can be taken to be the finite subset {(x, y − 1), (x, 2), (x, y + 1), (x, y − 1, 2), (x, y + 1, 2)} of Spec(Z[y][[x]]).

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ℵ0 ; (x, y − 1, 5) ∈ (x, 2, y − 1) ℵ0 (x, y − 1)

|R|

(x, 2)

(x, 2, y + 1) |R|

ℵ0 ; (x, y + 1, 3) ∈ (x, y + 1)

(0) = (x − 2(y − 1)(y + 1)) Diagram 2 Spec(Z[y][[x]]/(x − 2(y − 1)(y + 1))

Theorem 8 (Celikbas–Eubanks-Turner–Wiegand, [1]) Let  E be a finite partially ordered set of dimension at most one. Let F = min E ∪ u=v,u,v∈min E mub(u, v). Then there exists a height-one prime ideal Q of Z[y][[x]] and an embedding ϕ : F → U , such that U = Spec(Z[y][[x]]/Q) satisfies properties (i)-(v) of Theorem 7 for F and ϕ. Results similar to Theorems 7 and 8 hold for prime spectra U = Spec(R[[x]][y]/Q) over a countable Noetherian one-dimensional domain R with infinitely many maximal ideals, where Q = (x) is a height-one prime ideal of R[[x]][y], except that there may be height-one maximal elements in U . Also, we have results concerning prime spectra for two-dimensional images of mixed polynomial-power series over semilocal Noetherian one-dimensional domains, for two-dimensional images k[[x]][y, z]/Q of polynomial-power series over a field k, where again Q = (x) and Q is a height-one prime ideal of k[[x]][y, z], and for rings or fields that are uncountable. The prime spectra of one-dimensional domains are rather trivial; we provide conditions that yield the one-dimensional case, and we give the cardinalities that occur in that case [1]. Acknowledgements We thank the organizers of the semester-long program and the workshop, and thank the Universitat Autònoma de Barcelona for providing facilities for this project. We had an enjoyable and mathematically stimulating time.

References 1. E. Celikbas, C. Eubanks-Turner, S. Wiegand, Prime ideals in quotients of mixed polynomialpower series rings. Work in progress 2. W. Heinzer, C. Rotthaus, S. Wiegand, Mixed polynomial-power series rings and relations among their spectra (Springer, New York, Multiplicative ideal theory in commutative algebra, 2006), pp. 227–242 3. W. Heinzer, S. Wiegand, Prime ideals in two-dimensional polynomial rings. Proc. Am. Math. Soc. 107(3), 577–586 (1989) 4. M. Hochster, Prime ideal structure in commutative rings. Trans. Am. Math. Soc. 142, 43–60 (1969) 5. W. Lewis, The spectrum of a ring as a partially ordered set. J. Algebra 25, 419–434 (1973)

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6. H. Matsumura, Commutative ring theory, in Cambridge Studies in Advanced Mathematics, vol. 8, 2nd edn. (Cambridge University Press, Cambridge, 1989). Translated from the Japanese by M. Reid 7. S. McAdam, Saturated chains in Noetherian rings. Indiana Univ. Math. J. 23, 719–728 (1973/74) 8. S. McAdam, Intersections of height 2 primes. J. Algebra 49(2), 315–321 (1977) 9. M. Nagata, On the chain problem of prime ideals. Nagoya Math. J. 10, 51–64 (1956) 10. L.J. Ratliff, Chain Conjectures in Ring Theory. Lecture Notes in Mathematics, vol. 647 (Springer, Berlin, 1978) 11. R. Wiegand, The prime spectrum of a two-dimensional affine domain. J. Pure Appl. Algebra 40(2), 209–214 (1986) 12. R. Wiegand, S. Wiegand, Prime ideals in Noetherian rings: a survey in ring and module theory. Trends in Mathematics vol. 175 (Birkhäuser/Springer Basel AG, Basel, 2010) 13. S. Wiegand, R. Wiegand, The maximal ideal space of a Noetherian ring. J. Pure Appl. Algebra 8, 129–141 (1976) 14. S. Wiegand, Intersections of prime ideals in Noetherian rings. Commun. Algebra 11, 1853– 1876 (1983)