Triangulated Categories

Cover......Page 1
Title Page......Page 5
Contents......Page 7
Preface......Page 9
Triangulated categories: definitions, properties, and examples......Page 11
1. Additive categories......Page 12
1.1. The category of complexes......Page 13
1.2. The homotopy category of complexes......Page 15
2. Abelian categories......Page 16
3. Definition of triangulated categories......Page 21
4. Some formal properties of triangulated categories......Page 24
5. Abelian categories vs. triangulated categories......Page 26
6. The homotopy category of complexes is triangulated......Page 27
7.1. Homology and quasi-isomorphisms......Page 34
7.2. Localisation of categories......Page 37
7.3. Morphisms in the derived category......Page 39
7.4. Derived categories are triangulated......Page 47
7.5. Comparing morphisms......Page 48
7.6.1. Mapping cylinders......Page 50
8. Frobenius categories and stable categories......Page 53
References......Page 60
Introduction......Page 62
1. Levels......Page 63
2. Complexities......Page 65
3. Composition products......Page 69
4. Differential Graded Hopf algebras......Page 70
5. Exterior algebras I.......Page 75
6. Complete intersection local rings I.......Page 76
8. Exterior algebras II.......Page 80
9. Complete intersection local rings II.......Page 83
References......Page 84
1.1. Context......Page 86
1.2. Contents......Page 87
Acknowledgments......Page 88
2.2. The knitting algorithm......Page 89
3.1. Quivers......Page 93
3.2. Seeds and mutations......Page 94
3.3. Examples of seed and quiver mutations......Page 95
3.4. Definition of cluster algebras......Page 98
3.6. The example A3......Page 99
3.7. Cluster algebras with finitely many cluster variables......Page 100
4.1. Definition......Page 102
4.2. Example: SL(2,C)......Page 103
4.3. Example: Planes in affine space......Page 104
4.4. Example: The big cell of the Grassmannian......Page 105
4.5. Compatible Poisson structures......Page 106
4.6. Example: The maximal unipotent subgroup of SL(n + 1,C)......Page 108
5.1. Quiver representations and Gabriel’s theorem......Page 109
5.2. Tame and wild quivers......Page 115
5.3. The Caldero-Chapoton formula......Page 116
5.4. The derived category......Page 118
5.5. Presentation of the derived category of a Dynkin quiver......Page 121
5.6. Caldero-Chapoton’s proof......Page 124
5.7. The cluster category......Page 125
5.8. From cluster categories to cluster algebras......Page 126
5.10. Mutation of cluster-tilting sets......Page 129
6.1. Categorification......Page 132
6.2. Two applications......Page 134
6.3. Cluster categories and singularities......Page 135
7.1. Definition and main examples......Page 136
7.3. Mutation......Page 139
7.4. Simple mutations, reachable cluster-tilting objects......Page 141
7.5. Combinatorial invariants......Page 142
7.6. More mutants categorified......Page 144
7.7. 2-CY categories from algebras of global dimension 2......Page 147
8. Application: The periodicity conjecture......Page 148
9.1. A reminder on reflection functors......Page 153
9.2. Mutation of quivers with potentials......Page 155
9.3. Derived equivalence of Ginzburg dg algebras......Page 157
9.4. A geometric illustration......Page 161
9.5. Ginzburg algebras from algebras of global dimension 2......Page 162
9.6. Cluster-tilting objects, spherical collections, decorated representations......Page 163
References......Page 164
1. Introduction......Page 171
Acknowledgement......Page 173
2.2. Categories of fractions......Page 174
2.3. Adjoint functors......Page 175
2.4. Localization functors......Page 176
2.5. Local objects......Page 177
2.7. Localization functors preserving coproducts......Page 179
2.9. Example: Localization of modules......Page 180
2.10. Example: Localization of spectra......Page 181
3.1. Calculus of fractions......Page 182
3.2. Calculus of fractions and adjoint functors......Page 184
3.3. A criterion for the fractions to form a small set......Page 185
3.5. Calculus of fractions and coproducts......Page 186
4.1. Triangulated categories......Page 188
4.3. Multiplicative systems......Page 190
4.4. Cohomological functors......Page 191
4.6. Verdier localization......Page 192
4.7. Localization of subcategories......Page 193
4.8. Orthogonal subcategories......Page 194
4.9. Bousfield localization......Page 195
4.11. A functorial triangle......Page 197
4.12. Localization versus colocalization......Page 198
4.13. Recollements......Page 199
4.14. Example: The derived category of a module category......Page 200
4.16. Example: The recollement induced by an idempotent......Page 201
5.1. Brown representatbility......Page 202
5.2. Localization functors via Brown representability......Page 203
5.3. Compactly generated triangulated categories......Page 204
5.5. Localization functors preserving coproducts......Page 205
5.6. Finite localization......Page 207
5.7. Cohomological localization via localization of graded modules......Page 208
5.8. Example: Resolutions of chain complexes......Page 211
5.9. Example: Homological epimorphisms......Page 212
6.3. Well generated triangulated categories......Page 213
6.4. Filtered categories......Page 214
6.6. The comma category of an exact triangle......Page 215
6.7. A Kan extension......Page 217
6.8. A criterion for well generatedness......Page 219
6.9. Cohomological functors via filtered colimits......Page 220
6.10. A universal property......Page 221
6.11. Notes......Page 222
7.1. Cohomological localization......Page 223
7.2. Localization with respect to a small set of objects......Page 225
7.3. Functors between well generated categories......Page 228
7.4. The kernel of a functor between well generated categories......Page 229
7.5. The kernel of a cohomological functor on a well generated category......Page 230
7.6. Localization of well generated categories versus abelian localization......Page 231
7.7. Example: The derived category of an abelian Grothendieck category......Page 232
7.8. Notes......Page 233
8. Epilogue: Beyond well generatedness......Page 234
Appendix A. The abelianization of a triangulated category......Page 235
Appendix B. Locally presentable abelian categories......Page 237
References......Page 243
1. Introduction......Page 246
2.1. Generalities about ideals in additive categories......Page 249
2.2. Examples of ideals......Page 250
2.3. What is a triangulated category?......Page 253
2.4. The universal homological functor......Page 257
2.5. Homological ideals in triangulated categories......Page 258
3. From homological ideals to derived functors......Page 259
3.1. Basic notions......Page 260
3.2. Exact chain complexes......Page 264
3.2.1. More homological algebra with chain complexes......Page 268
3.3. Projective objects......Page 270
3.4. Projective resolutions......Page 272
3.5. Derived functors......Page 273
3.6. Projective objects via adjointness......Page 277
3.7. The universal exact homological functor......Page 280
3.8. Derived functors in homological algebra......Page 283
4. The plain Universal Coefficient Theorem......Page 285
4.1. Universal Coefficient Theorem in the hereditary case......Page 287
5. Crossed products for compact quantum groups......Page 290
5.1. The Pimsner–Voiculescu exact sequence......Page 296
References......Page 297
Derived categories and Grothendieck duality......Page 300
0. Introduction......Page 301
1. Historical overview......Page 304
2. Background on RHom complexes......Page 311
3. Dualizing complexes......Page 323
4. When R f∗ respects compacts......Page 335
5. Where we can prove Conjecture 4.16......Page 347
6. Dualizing complexes and f !......Page 352
Appendix A. A fact concerning strongly dualizable objects......Page 354
References......Page 356
1. Introduction......Page 361
3.1. Abelian case......Page 362
3.2.2. Open subvarieties and quotients......Page 364
3.2.3. Perfect complexes......Page 365
3.2.4. Extensions of perfect complexes......Page 366
3.2.5. Applications to K-theory......Page 369
4.1.1. Classification of Serre subcategories......Page 370
4.1.2. Centres......Page 371
4.2.1. Classification of thick subcategories......Page 372
4.2.2. Centres......Page 373
4.2.3. Remarks on centres......Page 374
4.2.5. (Anti-)ample canonical bundles......Page 376
References......Page 379
Introduction......Page 381
0.1 Notations......Page 382
1. Generalized functions......Page 383
2. D-modules......Page 385
3. Microsupport......Page 388
4. Microlocal analysis......Page 390
5. The use of Grothendieck topologies......Page 394
References......Page 396
Algebraic versus topological triangulated categories......Page 399
References......Page 416
1. Introduction......Page 418
2.1. Derived category of coherent sheaves......Page 419
2.2. Fourier-Mukai transforms......Page 420
2.3. Fourier-Mukai partners......Page 422
2.4. Several categorical invariants......Page 424
2.5. The group of autoequivalences......Page 426
3.1. Semiorthogonal decomposition......Page 428
3.2. Blow-up formula of derived categories......Page 429
3.3. Derived equivalence under birational transformations......Page 430
3.4. Perverse t-structures and flops......Page 432
3.5. D-equivalence and K-equivalence......Page 434
4.2. Exceptional collections......Page 435
4.3. McKay correspondence......Page 437
4.4. Non-commutative crepant resolutions......Page 438
5.2. Bridgeland’s stability conditions......Page 440
5.3. The space of stability conditions......Page 442
5.4. Group actions......Page 443
5.5. Background from string theory......Page 444
5.7. Stability conditions on K3 surfaces......Page 447
5.8. Stability conditions and birational geometry......Page 450
5.9. Stability conditions and 3-fold flops......Page 451
5.10. Holomorphic generating functions on the space of stability conditions......Page 455
References......Page 458
0. Introduction......Page 462
1. Dualizing Complexes: Overview......Page 463
2. Rigid Complexes and DG Algebras......Page 464
3. Properties of Rigid Complexes......Page 466
4. Rigid Dualizing Complexes......Page 468
5. Rigid Complexes and CM Homomorphisms......Page 469
References......Page 472