Handbook of tilting theory
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| Format: | Buch |
| Sprache: | English |
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Cambridge [u.a.]
Cambridge Univ. Press
2007
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| Ausgabe: | 1. publ. |
| Schriftenreihe: | London Mathematical Society lecture notes series
332 |
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| 100 | 1 | |a Angeleri Hügel, Lidia |d 1960- |e Verfasser |0 (DE-588)138988234 |4 aut | |
| 245 | 1 | 0 | |a Handbook of tilting theory |c ed. by Lidia Angeleri Hügel ... |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2007 | |
| 300 | |a VIII, 472 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a London Mathematical Society lecture notes series |v 332 | |
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| 650 | 4 | |a Modules (Algebra) | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-015481986 | ||
Datensatz im Suchindex
| DE-BY-TUM_call_number | 0102/MAT 001z 2001 A 985-332 |
|---|---|
| DE-BY-TUM_katkey | 1583765 |
| DE-BY-TUM_media_number | 040010251659 |
| _version_ | 1816712678807175168 |
| adam_text | Contents
1 Introduction page 1
2 Basic results of classical tilting theory
L. Angeleri Hiigel, D. Happel, and H. Krause 9
References 12
3 Classification of representation finite algebras and their
modules
T. Brustle 15
1 Introduction 15
2 Notation 16
3 Representation finite algebras 18
4 Critical algebras 24
5 Tame algebras 26
References 28
4 A spectral sequence analysis of classical tilting func¬
tors
S. Brenner and M. C. R. Butler 31
1 Introduction 31
2 Tilting modules 32
3 Tilting functors, spectral sequences and nitrations 35
4 Applications 43
5 Edge effects, and the case t = 2 46
References 47
5 Derived categories and tilting
B. Keller 49
1 Introduction 49
2 Derived categories 51
v
vi Contents
3 Derived functors 63
4 Tilting and derived equivalences 66
5 Triangulated categories 72
6 Morita theory for derived categories 78
7 Comparison of t structures, spectral sequences 83
8 Algebraic triangulated categories and dg algebras 90
References 97
6 Hereditary categories
H. Lenzing 105
1 Fundamental concepts 106
2 Examples of hereditary categories 108
3 Repetitive shape of the derived category 112
4 Perpendicular categories 114
5 Exceptional objects 115
6 Piecewise hereditary algebras and HappePs theorem 117
7 Derived equivalence of hereditary categories 121
8 Modules over hereditary algebras 121
9 Spectral properties of hereditary categories 124
10 Weighted projective lines 125
11 Quasitilted algebras 142
References 143
7 Fourier Mukai transforms
L. Hille and M. Van den Bergh 147
1 Some background 147
2 Notations and conventions 149
3 Basics on Fourier Mukai transforms 149
4 The reconstruction theorem 155
5 Curves and surfaces 159
6 Threefolds and higher dimensional varieties 166
7 Non commutative rings in algebraic geometry 170
References 173
8 Tilting theory and homologically finite subcategories
with applications to quasihereditary algebras
I. Reiten 179
1 The Basic Ingredients 181
2 The Correspondence Theorem 191
3 Quasihereditary algebras and their generalizations 200
4 Generalizations 207
References 211
Contents vii
9 Tilting modules for algebraic groups and finite dimen¬
sional algebras
S. Donkin 215
1 Quasi hereditary algebras 217
2 Coalgebras and Comodules 220
3 Linear Algebraic Groups 225
4 Reductive Groups 228
5 Infinitesimal Methods 233
6 Some support for tilting modules 238
7 Invariant theory 239
8 General Linear Groups 241
9 Connections with symmetric groups and Hecke algebras244
10 Some recent applications to Hecke algebras 247
References 254
10 Combinatorial aspects of the set of tilting modules
L. Unger 259
1 Introduction 259
2 The partial order of tilting modules 260
3 The quiver of tilting modules 261
4 The simplicial complex of tilting modules 270
References 275
11 Infinite dimensional tilting modules and cotorsion pairs
J. Trlifaj 279
1 Cotorsion pairs and approximations of modules 281
2 Tilting cotorsion pairs 292
3 Cotilting cotorsion pairs 298
4 Finite type, duality, and some examples 304
5 Tilting modules and the finitistic dimension conjectures312
References 316
12 Infinite dimensional tilting modules over finite dimen¬
sional algebras
0. Solberg 323
1 Definitions and preliminaries 324
2 The subcategory correspondence 327
3 The finitistic dimension conjectures 332
4 Complements of tilting and cotilting modules 336
5 Classification of all cotilting modules 340
References 341
viii Contents
13 Cotilting dualities
R. Colpi and K. R. Fuller 345
1 Generalized Morita Duality and Finitistic Cotilting
Modules 348
2 Cotilting Modules and Bimodules 350
3 Weak Morita Duality 353
4 Pure Injectivity of Cotilting Modules and Reflexivity 355
References 356
14 Representations of finite groups and tilting
J. Chuang and J. Richard 359
1 A brief introduction to modular representation theory 359
2 The abelian defect group conjecture 360
3 Symmetric algebras 361
4 Characters and derived equivalence 366
5 Splendid equivalences 370
6 Derived equivalence and stable equivalence 373
7 Lifting stable equivalences 375
8 Clifford theory 376
9 Cases for which the Abelian Defect Group Conjecture
has been verified 378
10 Nonabelian defect groups 383
References 384
15 Morita theory in stable homotopy theory
B. Shipley 393
1 Introduction 393
2 Spectral Algebra 396
3 Quillen model categories 399
4 Differential graded algebras 403
5 Two topologically equivalent DGAs 406
References 409
Appendix Some remarks concerning tilting modules and
tilted algebras. Origin. Relevance. Future.
C. M. Ringel 413
1 Basic Setting 414
2 Connections 423
3 The new cluster tilting approach 446
References 470
|
| adam_txt |
Contents
1 Introduction page 1
2 Basic results of classical tilting theory
L. Angeleri Hiigel, D. Happel, and H. Krause 9
References 12
3 Classification of representation finite algebras and their
modules
T. Brustle 15
1 Introduction 15
2 Notation 16
3 Representation finite algebras 18
4 Critical algebras 24
5 Tame algebras 26
References 28
4 A spectral sequence analysis of classical tilting func¬
tors
S. Brenner and M. C. R. Butler 31
1 Introduction 31
2 Tilting modules 32
3 Tilting functors, spectral sequences and nitrations 35
4 Applications 43
5 Edge effects, and the case t = 2 46
References 47
5 Derived categories and tilting
B. Keller 49
1 Introduction 49
2 Derived categories 51
v
vi Contents
3 Derived functors 63
4 Tilting and derived equivalences 66
5 Triangulated categories 72
6 Morita theory for derived categories 78
7 Comparison of t structures, spectral sequences 83
8 Algebraic triangulated categories and dg algebras 90
References 97
6 Hereditary categories
H. Lenzing 105
1 Fundamental concepts 106
2 Examples of hereditary categories 108
3 Repetitive shape of the derived category 112
4 Perpendicular categories 114
5 Exceptional objects 115
6 Piecewise hereditary algebras and HappePs theorem 117
7 Derived equivalence of hereditary categories 121
8 Modules over hereditary algebras 121
9 Spectral properties of hereditary categories 124
10 Weighted projective lines 125
11 Quasitilted algebras 142
References 143
7 Fourier Mukai transforms
L. Hille and M. Van den Bergh 147
1 Some background 147
2 Notations and conventions 149
3 Basics on Fourier Mukai transforms 149
4 The reconstruction theorem 155
5 Curves and surfaces 159
6 Threefolds and higher dimensional varieties 166
7 Non commutative rings in algebraic geometry 170
References 173
8 Tilting theory and homologically finite subcategories
with applications to quasihereditary algebras
I. Reiten 179
1 The Basic Ingredients 181
2 The Correspondence Theorem 191
3 Quasihereditary algebras and their generalizations 200
4 Generalizations 207
References 211
Contents vii
9 Tilting modules for algebraic groups and finite dimen¬
sional algebras
S. Donkin 215
1 Quasi hereditary algebras 217
2 Coalgebras and Comodules 220
3 Linear Algebraic Groups 225
4 Reductive Groups 228
5 Infinitesimal Methods 233
6 Some support for tilting modules 238
7 Invariant theory 239
8 General Linear Groups 241
9 Connections with symmetric groups and Hecke algebras244
10 Some recent applications to Hecke algebras 247
References 254
10 Combinatorial aspects of the set of tilting modules
L. Unger 259
1 Introduction 259
2 The partial order of tilting modules 260
3 The quiver of tilting modules 261
4 The simplicial complex of tilting modules 270
References 275
11 Infinite dimensional tilting modules and cotorsion pairs
J. Trlifaj 279
1 Cotorsion pairs and approximations of modules 281
2 Tilting cotorsion pairs 292
3 Cotilting cotorsion pairs 298
4 Finite type, duality, and some examples 304
5 Tilting modules and the finitistic dimension conjectures312
References 316
12 Infinite dimensional tilting modules over finite dimen¬
sional algebras
0. Solberg 323
1 Definitions and preliminaries 324
2 The subcategory correspondence 327
3 The finitistic dimension conjectures 332
4 Complements of tilting and cotilting modules 336
5 Classification of all cotilting modules 340
References 341
viii Contents
13 Cotilting dualities
R. Colpi and K. R. Fuller 345
1 Generalized Morita Duality and Finitistic Cotilting
Modules 348
2 Cotilting Modules and Bimodules 350
3 Weak Morita Duality 353
4 Pure Injectivity of Cotilting Modules and Reflexivity 355
References 356
14 Representations of finite groups and tilting
J. Chuang and J. Richard 359
1 A brief introduction to modular representation theory 359
2 The abelian defect group conjecture 360
3 Symmetric algebras 361
4 Characters and derived equivalence 366
5 Splendid equivalences 370
6 Derived equivalence and stable equivalence 373
7 Lifting stable equivalences 375
8 Clifford theory 376
9 Cases for which the Abelian Defect Group Conjecture
has been verified 378
10 Nonabelian defect groups 383
References 384
15 Morita theory in stable homotopy theory
B. Shipley 393
1 Introduction 393
2 Spectral Algebra 396
3 Quillen model categories 399
4 Differential graded algebras 403
5 Two topologically equivalent DGAs 406
References 409
Appendix Some remarks concerning tilting modules and
tilted algebras. Origin. Relevance. Future.
C. M. Ringel 413
1 Basic Setting 414
2 Connections 423
3 The new cluster tilting approach 446
References 470 |
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| index_date | 2024-07-02T16:46:16Z |
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| isbn | 9780521680455 052168045X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015481986 |
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| series | London Mathematical Society lecture notes series |
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| spellingShingle | Angeleri Hügel, Lidia 1960- Handbook of tilting theory London Mathematical Society lecture notes series Categories (Mathematics) Modules (Algebra) Kategorientheorie (DE-588)4120552-2 gnd |
| subject_GND | (DE-588)4120552-2 |
| title | Handbook of tilting theory |
| title_auth | Handbook of tilting theory |
| title_exact_search | Handbook of tilting theory |
| title_exact_search_txtP | Handbook of tilting theory |
| title_full | Handbook of tilting theory ed. by Lidia Angeleri Hügel ... |
| title_fullStr | Handbook of tilting theory ed. by Lidia Angeleri Hügel ... |
| title_full_unstemmed | Handbook of tilting theory ed. by Lidia Angeleri Hügel ... |
| title_short | Handbook of tilting theory |
| title_sort | handbook of tilting theory |
| topic | Categories (Mathematics) Modules (Algebra) Kategorientheorie (DE-588)4120552-2 gnd |
| topic_facet | Categories (Mathematics) Modules (Algebra) Kategorientheorie |
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