Derived equivalences for group rings:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kon
Springer
1998
|
| Series: | Lecture notes in mathematics
1685 |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008024033&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | X, 246 S. graph. Darst. |
| ISBN: | 3540643117 |
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MARC
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| 100 | 1 | |a König, Steffen |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Derived equivalences for group rings |c Steffen König ; Alexander Zimmermann. With contributions by Bernhard Keller ... |
| 264 | 1 | |a Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kon |b Springer |c 1998 | |
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| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Lecture notes in mathematics |v 1685 | |
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Record in the Search Index
| DE-BY-TUM_call_number | 0102 MAT 001z 2001 B 999-1685 |
|---|---|
| DE-BY-TUM_katkey | 942353 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040020227161 |
| _version_ | 1821931387241889793 |
| adam_text | Contents
1 Introduction (by Alexander Zimmermann) 1
2 Basic definitions and some examples (by Steffen Konig) 5
2.1 Notations 5
2.2 Complexes 5
2.3 Triangulated categories 8
2.4 Stable categories 13
2.5 Localization and derived categories 14
2.6 Examples 18
2.7 Derived functors 19
2.8 Double complexes 21
2.9 The Mittag Leffler condition 22
2.10 Invariants of derived categories 24
2.11 Six examples 24
2.11.1 25
2.11.2 26
2.11.3 26
2.11.4 27
2.11.5 28
2.11.6 30
3 Rickard s fundamental theorem (by Steffen Konig) 33
3.1 Motivation: History of tilting theory 33
3.2 Tilting complexes, Rickard s theorem and some examples 35
3.3 Strategy of the proof 39
3.4 Construction of F 40
3.5 F is a full embedding 44
3.6 Construction of G 45
3.7 Proof of theorem 3.2.1 48
3.7.1 Proof of (1) 48
3.7.2 Proof of (2) 48
3.7.3 Proof of (3) 49
4 Some modular and local representation theory (by Alexander Zimmer¬
mann) 51
4.1 Motivation 51
4.2 Elementary modular representation theory 52
4.2.1 Basic definitions and properties 53
4.2.2 The Cartan Brauer triangle 57
4.2.3 Green correspondence 61
4.2.4 Brauer correspondence 64
4.3 Brauer tree algebras 64
4.4 Green orders 69
4.4.1 The rational components; isotypic orders 73
4.4.2 Structure theorem for Green orders 76
5 Onesided tilting complexes for group rings (by Alexander Zimmermann) 81
5.1 A special class of tilting complexes 81
5.2 Examples 90
5.2.1 Derived equivalences; the cyclic defect group case 90
5.2.2 Derived equivalences; dihedral defect groups 101
6 Tilting with additional structure: twosided tilting complexes (by
Alexander Zimmermann) 105
6.1 Introduction 105
6.2 The theorem and its proof 105
6.3 Properties of derived equivalences 113
6.3.1 Properties following from the existence of onesided tilting com¬
plexes 113
6.3.2 Properties following from the existence of twosided tilting com¬
plexes 118
6.3.3 Characters and perfect isometries 130
6.4 A twosided tilting complex for Green orders 137
6.4.1 A first reduction 137
6.4.2 Setup for the actual construction 138
6.4.3 Constructing the complex 139
6.4.4 The proof of proposition 6.4.1 144
7 Historical remarks (by Alexander Zimmermann) 151
8 On the construction of triangle equivalences (by Bernhard Keller) 155
8.1 Unbounded derived categories 155
8.1.1 Unbounded resolutions 156
8.1.2 Unbounded derived categories 158
8.1.3 Infinite devissage 158
8.1.4 Derived equivalences 159
8.1.5 Triangulated categories with infinite sums and a set of compact
generators 161
8.2 Differential graded algebras and their derived categories 162
8.2.1 DG algebras 162
8.2.2 DG modules 163
8.2.3 The homotopy category. I63
8.2.4 Resolutions 164
8.2.5 The derived category. 164
8.2.6 Derived equivalences 165
8.3 Applications 166
8.3.1 Construction of bimodule complexes 166
8.3.2 Rickard s Morita theorem 167
8.3.3 Stable categories and DG algebras 168
8.3.4 Invariance of cyclic homology under derived equivalence 169
8.4 Appendix: Proof of theorem 8.1.1 172
9 Triangulated Categories in the Modular Representation Theory of
Finite Groups (by Jeremy Richard) 177
9.1 Introduction and notation 177
9.1.1 Introduction 177
9.1.2 Notation 178
9.2 Equivalences of derived categories 178
9.2.1 Some remarks on symmetric algebras 179
9.2.2 Derived equivalences between symmetric algebras 183
9.2.3 Derived equivalences between blocks: generalities 184
9.2.4 Derived equivalences between blocks: Broue s conjectures .... 188
9.2.5 Splendid equivalences 189
9.3 Bousfield localization in the stable module category 193
9.3.1 The stable module category and varieties for modules 193
9.3.2 Bousfield localization 195
9.3.3 Varieties for arbitrary modules 196
9.3.4 The classification of epaisse subcategories of stmod(fcP) 196
10 The derived category of blocks with cyclic defect groups (by Raphael
Rouquier) 199
10.1 Introduction 199
10.2 Miscellany : stable equivalences, Rickard equivalences and more .... 200
10.2.1 Notations 200
10.2.2 Stable category, stable equivalences and invariants 201
10.2.3 Derived category and Rickard equivalences 203
10.2.4 Some more lemmas 209
10.3 Blocks stably equivalent to OD xE 211
10.3.1 Exceptional characters 211
10.3.2 Decomposition numbers 212
10.3.3 The Brauer tree and its walk 213
10.3.4 Construction of the complex 214
10.4 Local study 215
10.4.1 OP{G) = 1 216
10.4.2 OP(G) jt l 217
10.5 An example : PSL2(p) 218
11 On stable equivalences of Morita type (by Afarkus Linckelmann) 221
11.1 Types of equivalences 221
11.2 Stable equivalences of Morita type 223
11.3 Stable equivalences and p groups 225
11.4 Formal invariants of stable equivalences of Morita type 227
11.5 A criterion for tilting complexes 231
Bibliography 233
Index 244
|
| any_adam_object | 1 |
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| id | DE-604.BV011875467 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T10:20:24Z |
| institution | BVB |
| isbn | 3540643117 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008024033 |
| oclc_num | 246274778 |
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| owner_facet | DE-91G DE-BY-TUM DE-824 DE-739 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-706 DE-83 DE-11 DE-188 |
| physical | X, 246 S. graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spellingShingle | König, Steffen Zimmermann, Alexander Derived equivalences for group rings Lecture notes in mathematics Algebra, Homological Group rings Homologische Algebra (DE-588)4160598-6 gnd Gruppenring (DE-588)4158469-7 gnd Abgeleitete Kategorie (DE-588)4504943-9 gnd |
| subject_GND | (DE-588)4160598-6 (DE-588)4158469-7 (DE-588)4504943-9 |
| title | Derived equivalences for group rings |
| title_auth | Derived equivalences for group rings |
| title_exact_search | Derived equivalences for group rings |
| title_full | Derived equivalences for group rings Steffen König ; Alexander Zimmermann. With contributions by Bernhard Keller ... |
| title_fullStr | Derived equivalences for group rings Steffen König ; Alexander Zimmermann. With contributions by Bernhard Keller ... |
| title_full_unstemmed | Derived equivalences for group rings Steffen König ; Alexander Zimmermann. With contributions by Bernhard Keller ... |
| title_short | Derived equivalences for group rings |
| title_sort | derived equivalences for group rings |
| topic | Algebra, Homological Group rings Homologische Algebra (DE-588)4160598-6 gnd Gruppenring (DE-588)4158469-7 gnd Abgeleitete Kategorie (DE-588)4504943-9 gnd |
| topic_facet | Algebra, Homological Group rings Homologische Algebra Gruppenring Abgeleitete Kategorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008024033&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT konigsteffen derivedequivalencesforgrouprings AT zimmermannalexander derivedequivalencesforgrouprings |
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