**Contemporary Mathematics**

Volume: 436;
2007;
334 pp;
Softcover

MSC: Primary 13; 18; 55;
Secondary 20; 14

Print ISBN: 978-0-8218-3814-3

Product Code: CONM/436

List Price: $105.00

AMS Member Price: $84.00

MAA member Price: $94.50

**Electronic ISBN: 978-0-8218-8115-6
Product Code: CONM/436.E**

List Price: $105.00

AMS Member Price: $84.00

MAA member Price: $94.50

# Interactions between Homotopy Theory and Algebra

Share this page *Edited by *
*Luchezar L. Avramov; J. Daniel Christensen; William G. Dwyer; Michael A. Mandell; Brooke E. Shipley*

This book is based on talks presented at the Summer School on Interactions between Homotopy theory and Algebra held at the University of Chicago in the summer of 2004. The goal of this book is to create a resource for background and for current directions of research related to deep connections between homotopy theory and algebra, including algebraic geometry, commutative algebra, and representation theory. The articles in this book are aimed at the audience of beginning researchers with varied mathematical backgrounds and have been written with both the quality of exposition and the accessibility to novices in mind.

#### Readership

Graduate students and research mathematicians interested in algebraic topology, commutative algebra, algebraic geometry and homological algebra.

# Table of Contents

## Interactions between Homotopy Theory and Algebra

- Contents v6 free
- Preface vii8 free
- List of Participants xiii14 free
- Introductory Lecture Series 116 free
- Model Categories and Simplicial Methods 318
- Lectures on Local Cohomology 5166
- 1. Introduction 5267
- 2. Local Cohomology 5469
- 3. Injective Modules over Noetherian Rings and Matlis Duality 6176
- 4. Cohen-Macaulay and Gorenstein rings 6883
- 5. Vanishing Theorems and the Structure of Hdm(R) 7590
- 6. Vanishing Theorems II 8095
- 7. Appendix 1: Using local cohomology to prove a result of Kalkbrenner and Sturmfels 87102
- 8. Appendix 2: Bass numbers and Gorenstein Rings 93108
- References 98113

- Derived categories, resolutions, and Brown representability 101116
- Introduction 102117
- 1. Derived categories 103118
- 2. Triangulated categories 108123
- 3. Localization of triangulated categories 112127
- 4. Brown representability 116131
- 5. Resolutions 123138
- 6. Differential graded algebras and categories 125140
- 7. Algebraic triangulated categories 130145
- 7.1. Exact categories 130145
- 7.2. Frobenius categories 131146
- 7.3. The derived category of an exact category 131146
- 7.4. The stable category of a Frobenius category 132147
- 7.5. Algebraic triangulated categories 132147
- 7.6. The stable homotopy category is not algebraic 134149
- 7. 7. The differential graded category of an exact category 135150
- 7.8. Notes 136151

- Appendix A. The octahedral axiom 136151
- References 138153

- Exercises on derived categories, resolutions, and Brown representability 141156

- Topics Lecture Series 147162
- Spectra for commutative algebraists 149164
- 0. Introduction. 149164
- 1. Motivation via the derived category. 150165
- 2. Why consider spectra? 152167
- 3. How to construct spectra (Step 1). 154169
- 4. The smash product (Step 2). 159174
- 5. Brave new rings. 165180
- 6. Some algebraic uses of ring spectra. 167182
- 7. Local ring spectra. 169184
- References 172187

- Rational Homotopy Theory: A Brief Introduction 175190
- André-Quillen homology of commutative algebras 203218
- 1. Introduction 203218
- 2. Kähler differentials 205220
- 3. Simplicial algebras 208223
- 4. Simplicial resolutions 211226
- 5. The cotangent complex 217232
- 6. Basic properties 221236
- 7. André-Quillen homology and the Tor functor 224239
- 8. Locally complete intersection homomorphisms 225240
- 9. Regular homomorphisms 230245
- References 233248

- Local cohomology in commutative algebra, homotopy theory, and group cohomology 235250
- First steps in brave new commutative algebra 239254
- 1. Introduction 239254
- Part 1. Localization and completion for ideals. 240255
- 2. Algebraic definitions: Local and Cech cohomology and homology 241256
- 3. Homotopical analogues of the algebraic definitions 248263
- 4. Completion at ideals and Bousfield localization 251266
- 5. Localization away from ideals and Bousfield localization 253268
- 6. Chromatic filtrations for MU-modules. 254269
- 7. Completion theorems and their duals. 257272

- Part 2. Morita equivalences and Gorenstein rings. 259274
- References 273288

- Cotorsion pairs and model categories 277292
- Coherent sheaves on an elliptic curve 297312
- Lectures on the Cohomology of Finite Groups 317332