Interactions between homotopy and algebra : Summer School on Interactions between Homotopy Theory and Algebra, University of Chicago, July 26-August 6, 2004, Chicago, Illinois

Contemporary Mathematics 436......Page 1
Interactions between Homotopy Theory and Algebra......Page 3
Contents......Page 5
Preface......Page 6
List of Participants......Page 12
Introductory Lecture Series......Page 14
Contents......Page 15
1.1. Chain complexes......Page 16
1.2. Model Categories......Page 17
1.3. Chain complexes form a model category......Page 19
1.5. Simplicial Sets.......Page 21
2.1. Quillen Functors and Quillen Equivalence......Page 26
2.2. Homotopies and the homotopy category......Page 27
2.3. Total derived functors......Page 30
3.1. Cofibrantly Generated Model Categories......Page 31
3.2. Promoting model category structures......Page 34
4.1. The Dold-Kan theorem and simplicial resolutions......Page 35
4.2. Simplicial model categories......Page 39
4.3. Simplicial algebras......Page 42
4.4. Homology and Cohomology......Page 44
5.1. Skeletons and the skeletal decomposition......Page 50
5.2. Reedy model categories; the invariance of realization......Page 53
5.3. Resolution model categories......Page 56
References......Page 60
Contents......Page 62
1. Introduction......Page 63
2. Local Cohomology......Page 65
2.1. Two Other Important Ways to Think about Local Cohomology......Page 68
2.2. The Graded Case......Page 70
3. Injective Modules over Noetherian Rings and Matlis Duality......Page 72
3.1. Exercises......Page 78
4. Cohen-Macaulay and Gorenstein rings......Page 79
4.1. Exercises......Page 85
5. Vanishing Theorems and Structure of Hdm(R)......Page 86
5.1. The Graded Case Revisited......Page 89
5.3. Exercises......Page 90
6. Vanishing Theorems II......Page 91
6.1. Applications to Intersections of Algebraic Varieties......Page 94
6.2. The multi-graded case......Page 95
6.4. Exercises......Page 96
7.1. Introduction......Page 98
7.2. Quasi-Homogenization......Page 99
7.3. The Main Theorem......Page 102
8. Appendix 2: Bass numbers and Gorenstein Rings......Page 104
References......Page 109
Contents......Page 111
Introduction......Page 112
1.1. Additive and abelian categories......Page 113
1.3. Localization......Page 114
1.4. An alternative definition......Page 115
1.5. Extension groups......Page 116
1.6. Hereditary categories......Page 117
2.1. The axioms......Page 118
2.2. The octahedral axiom......Page 119
2.3. Cohomological functors......Page 120
2.5. K(A) is triangulated......Page 121
3.1. Quasi-isomorphisms......Page 122
3.2. D(A) is triangulated......Page 123
3.3. Triangulated and thick subcategories......Page 124
3.5. Verdier localization......Page 125
4.1. Coherent functors......Page 126
4.3. The idempotent completion of a triangulated category......Page 128
4.5. Brown representability......Page 129
4.6. Notes......Page 132
5.1. Injective resolutions......Page 133
5.2. Projective Resolutions......Page 134
6.1. Differential graded algebras and modules......Page 135
6.2. Differential graded categories......Page 136
6.4. Injective and projective resolutions......Page 138
6.5. Compact objects and perfect complexes......Page 139
7.1. Exact categories......Page 140
7.3. The derived category of an exact category......Page 141
7.5. Algebraic triangulated categories......Page 142
7.6. The stable homotopy category is not algebraic......Page 144
7.7. The differential graded category of an exact category......Page 145
Appendix A. The octahedral axiom......Page 146
References......Page 148
Exercises on derived categories, resolutions and Brown representability......Page 150
Topics Lecture Series......Page 155
0. Introduction......Page 156
1.A. Why consider the derived category?......Page 157
1.B. How to construct the derived category......Page 158
2. Why consider spectra?......Page 159
2.B. Second answer......Page 160
3. How to construct spectra (Step 1)......Page 161
4.A. Method 1: EKMM spectra......Page 166
4.B. Method 2: symmetric spectra......Page 169
4.C. Method 3: orthogonal spectra......Page 170
5. Brave new rings......Page 172
6.A. Topological Hochschild homology and cohomology......Page 174
6.C. Topological equivalence......Page 175
7.A. Regularity......Page 176
7.C. Orientability......Page 177
7.E. The local cohomology theorem......Page 178
References......Page 179
Introduction......Page 181
1.1. Rationalization and rational homotopy type......Page 182
1.2. The passage to commutative cochain algebras......Page 185
2.1. Examples and elementary construction......Page 191
2.2. Models of fiber squares......Page 194
2.3. Lusternik-Schnirelmann category......Page 197
2.4. Dichotomy......Page 201
3. Commutative algebra and rational homotopy theory......Page 204
References......Page 207
1. Introduction......Page 209
2. Kähler differentials......Page 211
3. Simplicial algebras......Page 214
4. Simplicial resolutions......Page 217
5. The cotangent complex......Page 223
6. Basic properties......Page 227
7. André-Quillen homology and the Tor functor......Page 230
8. Locally complete intersection homomorphisms......Page 231
9. Regular homomorphisms......Page 236
References......Page 239
Local Cohomology as Approximation......Page 241
Gorenstein rings and their generalizations......Page 242
Group rings and group cohomology......Page 243
1. Introduction......Page 245
Part 1. Localization and completion for ideals......Page 246
2.A. The functors......Page 247
2.B. The stable Koszul complex......Page 248
2.C. Invariance statements......Page 249
2.D. Local homology and cohomology......Page 250
2.E. Derived functors......Page 251
2.F. The shape of local cohomology......Page 252
2.G. Čech homology and cohomology......Page 253
3. Homotopical analogues of the algebraic definitions......Page 254
3.C. The Čech spectra......Page 255
3.D. Basic properties......Page 256
4.B. Bousfield\'s terminology......Page 257
4.C. Homotopical completion is a Bousfield localization......Page 258
5.A. The Čech complex as a Bousfield localization......Page 259
6.A. Cobordism notation......Page 260
6.B. Chromatic filtration and the Cousin complex......Page 261
6.C. Chromatic completions of MU-modules and S-modules......Page 262
7. Completion theorems and their duals......Page 263
8. The context, and some examples......Page 265
8.C. The principle examples......Page 266
9.A. First variant......Page 268
9.C. Complete modules and torsion modules......Page 270
10.A. The definition......Page 272
10.C. Brown-Comenetz duality and Matlis lifts......Page 273
10.D. Local duality and the dualizing complex......Page 274
11. The Gorenstein condition......Page 275
11.A. Orientability......Page 276
11.B. The local cohomology theorem......Page 277
11.C. The Gorenstein condition and the dualizing complex......Page 278
References......Page 279
Contents......Page 282
1. Cotorsion pairs......Page 283
2. Relation between cotorsion pairs and model categories......Page 285
2.1. Abelian model categories......Page 286
2.2. From cotorsion pairs to an abelian model category......Page 287
3. Cofibrant generation......Page 289
4. Monoidal structure......Page 291
5. Standard examples......Page 292
6. Gorenstein rings......Page 293
7.1. The general approach......Page 294
7.2. Making the theorem concrete......Page 295
7.3. Sheaves and schemes......Page 299
References......Page 300
1. Introduction......Page 302
2. Modules over Dedekind domains......Page 304
3. Coherent sheaves on projective varieties......Page 305
4. Coherent sheaves on an elliptic curve......Page 311
References......Page 319
1. Introduction......Page 321
2. Preliminaries......Page 322
3. Minimal resolutions......Page 326
4. Computations and further structure......Page 328
5. Cohomology and actions of finite groups......Page 332
References......Page 337