Handbook of tilting theory

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents......Page 6
1 Introduction page......Page 10
2 Basic results of classical tilting theory......Page 18
References......Page 21
1 Introduction......Page 24
2 Notation......Page 25
3 Representation-finite algebras......Page 27
4 Critical algebras......Page 33
5 Tame algebras......Page 35
References......Page 37
1 Introduction......Page 40
2 Tilting modules......Page 41
3 Tilting functors, spectral sequences and filtrations......Page 44
4 Applications......Page 52
5 Edge effects, and the case t ..2......Page 55
References......Page 56
1 Introduction......Page 58
2 Derived categories......Page 60
3 Derived functors......Page 72
4 Tilting and derived equivalences......Page 75
5 Triangulated categories......Page 81
6 Morita theory for derived categories......Page 87
7 Comparison of t-structures, spectral sequences......Page 92
8 Algebraic triangulated categories and dg algebras......Page 99
References......Page 106
6 Hereditary categories......Page 114
1 Fundamental concepts......Page 115
2 Examples of hereditary categories......Page 117
3 Repetitive shape of the derived category......Page 121
4 Perpendicular categories......Page 123
5 Exceptional objects......Page 124
6 Piecewise hereditary algebras and Happel’s theorem......Page 126
8 Modules over hereditary algebras......Page 130
9 Spectral properties of hereditary categories......Page 133
10 Weighted projective lines......Page 134
11 Quasitilted algebras......Page 151
References......Page 152
1 Some background......Page 156
3 Basics on Fourier-Mukai transforms......Page 158
4 The reconstruction theorem......Page 164
5 Curves and surfaces......Page 168
6 Threefolds and higher dimensional varieties......Page 175
7 Non-commutative rings in algebraic geometry......Page 179
References......Page 182
8 Tilting theory and homologically finite subcategories with applications to quasihereditary algebras......Page 188
1 The Basic Ingredients......Page 190
2 The Correspondence Theorem......Page 200
3 Quasihereditary algebras and their generalizations......Page 209
4 Generalizations......Page 216
References......Page 220
9 Tilting modules for algebraic groups and finite dimensional algebras......Page 224
1 Quasi-hereditary algebras......Page 226
2 Coalgebras and Comodules......Page 229
3 Linear Algebraic Groups......Page 234
4 Reductive Groups......Page 237
5 Infinitesimal Methods......Page 242
6 Some support for tilting modules......Page 247
7 Invariant theory......Page 248
8 General Linear Groups......Page 250
9 Connections with symmetric groups and Hecke algebras......Page 253
10 Some recent applications to Hecke algebras......Page 256
References......Page 263
1 Introduction......Page 268
2 The partial order of tilting modules......Page 269
3 The quiver of tilting modules......Page 270
4 The simplicial complex of tilting modules......Page 279
References......Page 284
11 Infinite dimensional tilting modules and cotorsion pairs......Page 288
1 Cotorsion pairs and approximations of modules......Page 290
2 Tilting cotorsion pairs......Page 301
3 Cotilting cotorsion pairs......Page 307
4 Finite type, duality, and some examples......Page 313
5 Tilting modules and the finitistic dimension conjectures......Page 321
References......Page 325
12 Infinite dimensional tilting modules over finite dimensional algebras......Page 332
1 Definitions and preliminaries......Page 333
2 The subcategory correspondence......Page 336
3 The finitistic dimension conjectures......Page 341
4 Complements of tilting and cotilting modules......Page 345
5 Classification of all cotilting modules......Page 349
References......Page 350
13 Cotilting dualities......Page 354
1 Generalized Morita Duality and Finitistic Cotilting Modules......Page 357
2 Cotilting Modules and Bimodules......Page 359
3 Weak Morita Duality......Page 362
4 Pure Injectivity of Cotilting Modules and Reflexivity......Page 364
References......Page 365
1 A brief introduction to modular representation theory......Page 368
2 The abelian defect group conjecture......Page 369
3 Symmetric algebras......Page 370
4 Characters and derived equivalence......Page 375
5 Splendid equivalences......Page 379
6 Derived equivalence and stable equivalence......Page 382
7 Lifting stable equivalences......Page 384
8 Clifford theory......Page 385
9 Cases for which the Abelian Defect Group Conjecture has been verified......Page 387
10 Nonabelian defect groups......Page 392
References......Page 393
1 Introduction......Page 402
2 Spectral Algebra......Page 405
3 Quillen model categories......Page 408
4 Differential graded algebras......Page 412
5 Two topologically equivalent DGAs......Page 415
References......Page 418
Appendix Some remarks concerning tilting modules and tilted algebras. Origin. Relevance. Future......Page 422
1 Basic Setting......Page 423
2 Connections......Page 432
3 The new cluster tilting approach......Page 455
References......Page 479