Brauer groups, Hopf algebras and Galois theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer
1998
|
| Series: | K-monographs in mathematics
4 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 488 S. |
| ISBN: | 079234829X |
Staff View
MARC
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| 245 | 1 | 0 | |a Brauer groups, Hopf algebras and Galois theory |c by Stefaan Caenepeel |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer |c 1998 | |
| 300 | |a XVI, 488 S. | ||
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| 650 | 4 | |a Galois theory | |
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Record in the Search Index
| _version_ | 1804126373090951168 |
|---|---|
| adam_text | Contents
Preface xi
I The Brauer group of a commutative ring 1
1 Morita theory for algebras without a unit 3
1.1 Morita contexts 3
1.2 Dual pairs and elementary algebras 10
1.3 Invertible modules 13
1.4 Left modules versus bimodules 15
2 Azumaya algebras and Taylor Azumaya algebras 19
2.1 Central algebras, the separator and the trace map 19
2.2 Taylor Azumaya algebras 23
2.3 The Rosenberg Zelinsky exact sequence 33
3 The Brauer group 35
3.1 Equivalent Taylor Azumaya algebra 35
3.2 The (big) Brauer group 39
3.3 The splitting theorem for Taylor Azumaya algebras 41
3.4 The determinant map for an Azumaya algebra 48
3.5 The splitting theorem for semilocal rings 49
4 Central separable algebras 55
4.1 Separable algebras 55
4.2 Central separable algebras 60
4.3 Flat Taylor Azumaya algebras 70
5 Amitsur cohomology and etale cohomology 77
5.1 Grothendieck topologies 77
viii Contents
5.2 Amitsur cohomology 79
5.3 The category of sheaves 81
5.4 Direct and inverse image sheaves and presheaves 87
5.5 Stalks in the etale topology 91
5.6 Etale cohomology 96
5.7 Flabby sheaves 102
6 Cohomological interpretation of the Brauer group 107
6.1 Cohomology with values in the category of invertible modules .... 107
6.2 The Brauer group versus the second cohomology group 113
6.3 Taylor s theorem 121
6.4 Verschoren s construction and Takeuchi s exact sequence 130
6.5 The Brauer group is torsion 134
6.6 The Mayer Vietoris exact sequence 136
6.7 Gabber s theorem 139
6.8 The Brauer group modulo a nilpotent ideal 142
6.9 The Brauer group of a regular ring 144
6.10 Further results and examples 149
6.11 The Brauer group of a scheme and further generalizations 158
II Hopf algebras and Galois theory 171
7 Hopf algebras 173
7.1 Algebras, coalgebras and Hopf algebras 173
7.2 Modules and comodules 189
8 Galois objects 197
8.1 Relative Hopf modules and Galois objects 197
8.2 Galois objects and graded ring theory 205
8.3 Galois objects and Morita theory 207
8.4 Galois extensions 210
8.5 Galois objects and classical Galois theory 212
8.6 Integrals 213
Contents jx
8.7 Galois coobjects 215
9 Cohomology over Hopf algebras 223
9.1 Sweedler cohomology 223
9.2 Harrison cohomology 226
10 The group of Galois (co)objects 235
10.1 Galois coobjects and Harrison cohomology 235
10.2 Galois coobjects with geometric normal basis 242
10.3 The group of Galois coobjects and Amitsur cohomology 247
10.4 The Picard group of a coalgebra 250
10.5 The group of Galois objects 261
10.6 The split part of the group of Galois objects 275
10.7 About the Picard invariant map 276
10.8 Pairings and noncommutative Galois objects 277
11 Some examples 283
11.1 Group algebras 283
11.2 Monogenic Larson orders 286
11.3 Examples in characteristic p 295
III The Brauer Long group of a commutative ring 303
12 /f Azumaya algebras 305
12.1 Dimodules and dimodule algebras 305
12.2 // Azumaya algebras 313
12.3 Separability conditions 317
12.4 Examples of // Azumaya algebras 327
13 The Brauer Long group of a commutative ring 339
13.1 The Brauer Long group and its subgroups 339
13.2 The Brauer group of // module Azumaya algebras 345
13.3 The Picard group of // dimodules 351
13.4 The cup product 355
x Contents
13.5 The split part of the Brauer Long group 359
13.6 A dimodule version of the Rosenberg Zelinsky exact sequence .... 366
13.7 A complex for the Brauer Long group 368
13.8 The Hopf algebra H = EomR(H,K) 372
13.9 A short exact sequence for the Brauer Long group 375
13.10Application to some particular cases 384
13.11 Computing 0(7?,^)^ 405
13.12The multiplication rules 412
13.13The Brauer Long group of a scheme 436
14 The Brauer group of Yetter Drinfel d module algebras 439
14.1 Yetter Drinfel d modules 439
14.2 // Azumaya algebras and the Brauer group 442
14.3 The subgroups of BQ(k,H) 444
A Abelian categories and homological algebra 451
A.I Abelian categories 451
A.2 Derived functors 453
B Faithfully flat descent 459
C Elementary algebraic K theory 467
Bibliography 473
Index 480
|
| any_adam_object | 1 |
| author | Caenepeel, Stefaan 1956- |
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| author_facet | Caenepeel, Stefaan 1956- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.55 |
| dewey-search | 512/.55 |
| dewey-sort | 3512 255 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV011829351 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:16:25Z |
| institution | BVB |
| isbn | 079234829X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007988005 |
| oclc_num | 37640417 |
| open_access_boolean | |
| owner | DE-739 DE-19 DE-BY-UBM DE-11 DE-29T |
| owner_facet | DE-739 DE-19 DE-BY-UBM DE-11 DE-29T |
| physical | XVI, 488 S. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Kluwer |
| record_format | marc |
| series | K-monographs in mathematics |
| series2 | K-monographs in mathematics |
| spelling | Caenepeel, Stefaan 1956- Verfasser (DE-588)123804132 aut Brauer groups, Hopf algebras and Galois theory by Stefaan Caenepeel Dordrecht [u.a.] Kluwer 1998 XVI, 488 S. txt rdacontent n rdamedia nc rdacarrier K-monographs in mathematics 4 Galois-theorie gtt Groepen (wiskunde) gtt Hopf-algebra's gtt Brauer groups Galois theory Hopf algebras K-monographs in mathematics 4 (DE-604)BV011222840 4 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007988005&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Caenepeel, Stefaan 1956- Brauer groups, Hopf algebras and Galois theory K-monographs in mathematics Galois-theorie gtt Groepen (wiskunde) gtt Hopf-algebra's gtt Brauer groups Galois theory Hopf algebras |
| title | Brauer groups, Hopf algebras and Galois theory |
| title_auth | Brauer groups, Hopf algebras and Galois theory |
| title_exact_search | Brauer groups, Hopf algebras and Galois theory |
| title_full | Brauer groups, Hopf algebras and Galois theory by Stefaan Caenepeel |
| title_fullStr | Brauer groups, Hopf algebras and Galois theory by Stefaan Caenepeel |
| title_full_unstemmed | Brauer groups, Hopf algebras and Galois theory by Stefaan Caenepeel |
| title_short | Brauer groups, Hopf algebras and Galois theory |
| title_sort | brauer groups hopf algebras and galois theory |
| topic | Galois-theorie gtt Groepen (wiskunde) gtt Hopf-algebra's gtt Brauer groups Galois theory Hopf algebras |
| topic_facet | Galois-theorie Groepen (wiskunde) Hopf-algebra's Brauer groups Galois theory Hopf algebras |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007988005&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011222840 |
| work_keys_str_mv | AT caenepeelstefaan brauergroupshopfalgebrasandgaloistheory |