Central simple algebras and Galois cohomology:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2017
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| Edition: | Second edition |
| Series: | Cambridge studies in advanced mathematics
165 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xi, 417 Seiten 23 cm |
| ISBN: | 9781107156371 9781316609880 |
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Record in the Search Index
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| adam_text | Titel: Central simple algebras and Galois cohomology
Autor: Gille, Philippe
Jahr: 2017
Central Simple Algebras and Galois Cohomology Second Edition PHILIPPE GILLE Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan, Lyon TAMÁS SZAMUELY Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest ¡gi Cambridge UNIVERSITY PRESS
Contents Preface page ix 1 Quaternion algebras 1 1.1 Basic properties 1 1.2 Splitting over a quadratic extension 4 1.3 The associated conic 7 1.4 A theorem of Witt 10 1.5 Tensor products of quaternion algebras 13 2 Central simple algebras and Galois descent 19 2.1 Wedderburn’s theorem 19 2.2 Splitting fields 22 2.3 Galois descent 27 2.4 The Brauer group 33 2.5 Cyclic algebras 36 2.6 Reduced norms and traces 42 2.7 A basic exact sequence and applications 47 2.8 Index and period 52 2.9 Central simple algebras over complete discretely valued fields 59 2.10 K of central simple algebras 61 3 Techniques from group cohomology 70 3.1 Definition of cohomology groups 70 3.2 Explicit resolutions 77 3.3 Relation to subgroups 82 3.4 Cup-products 92 4 The cohomological Brauer group 105 4.1 Profinite groups and Galois groups 105 4.2 Cohomology of profinite groups 111 v
vi Contents 4.3 The cohomology exact sequence 118 4.4 The Brauer group revisited 123 4.5 Another characterization of the index 127 4.6 The Galois symbol 130 4.7 Cyclic algebras and symbols 133 5 Severi-Brauer varieties 139 5.1 Basic properties 140 5.2 Classification by Galois cohomology 142 5.3 Geometric Brauer equivalence 147 5.4 Amitsur’s theorem 152 5.5 An application: making central simple algebras cyclic 159 6 Residue maps 163 6.1 Cohomological dimension 163 6.2 Ci-fields 170 6.3 Cohomology of complete discretely valued fields 177 6.4 Cohomology of function fields of curves 182 6.5 Application to class field theory 187 6.6 Application to the rationality problem: the method 190 6.7 Application to the rationality problem: the example 197 6.8 Residue maps with finite coefficients 202 6.9 The Faddeev sequence with finite coefficients 208 7 Milnor K-theory 215 7.1 The tame symbol 215 7.2 Milnor’s exact sequence and the Bass-Tate lemma 222 7.3 The norm map 228 7.4 Reciprocity laws 237 7.5 Applications to the Galois symbol 243 7.6 The Galois symbol over number fields 249 8 The Merkurjev-Suslin theorem 258 8.1 Gersten complexes in Milnor K-theory 259 8.2 Properties of Gersten complexes 263 8.3 A property of Severi-Brauer varieties 268 8.4 Hilbert’s Theorem 90 for K 2 275 8.5 The Merkurjev-Suslin theorem: a special case 283 8.6 The Merkurjev-Suslin theorem: the general case 288 8.7 Reduced norms and /^-symbols 296 8.8 A useful exact sequence 302 8.9 Reduced norms and cohomology 310
Contents vii 9 Symbols in positive characteristic 316 9.1 The theorems of Teichmiiller and Albert 317 9.2 Differential forms and p-torsion in the Brauer group 325 9.3 Logarithmic differentials and flat p-connections 329 9.4 Decomposition of the de Rham complex 336 9.5 The Bloch-Gabber-Kato theorem: statement and reductions 340 9.6 Surjectivity of the differential symbol 345 9.7 Injectivity of the differential symbol 351 9.8 Application to p-torsion in Milnor K-theory 359 Appendix: a breviary of algebraic geometry 367 Bibliography 394 Index 413
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| any_adam_object | 1 |
| author | Gille, Philippe 1968- Szamuely, Tamás |
| author_GND | (DE-588)17385382X (DE-588)134142314 |
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| ctrlnum | (OCoLC)993431305 (DE-599)BSZ491693915 |
| dewey-full | 512/.32 514.23 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra 514 - Topology |
| dewey-raw | 512/.32 514.23 |
| dewey-search | 512/.32 514.23 |
| dewey-sort | 3512 232 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Second edition |
| format | Book |
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| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:58:21Z |
| institution | BVB |
| isbn | 9781107156371 9781316609880 |
| language | English |
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| physical | xi, 417 Seiten 23 cm |
| publishDate | 2017 |
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| series | Cambridge studies in advanced mathematics |
| series2 | Cambridge studies in advanced mathematics |
| spelling | Gille, Philippe 1968- Verfasser (DE-588)17385382X aut Central simple algebras and Galois cohomology Philippe Gille, Centre Nationale de la Recherche Scientifique (CNRS), Institut Camille Jordan, Lyon, Tamás Szamuely, Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest Second edition Cambridge Cambridge University Press 2017 xi, 417 Seiten 23 cm txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 165 Algebra, Homological Galois cohomology Zentral einfache Algebra (DE-588)4620287-0 gnd rswk-swf Galois-Kohomologie (DE-588)4019172-2 gnd rswk-swf Zentral einfache Algebra (DE-588)4620287-0 s Galois-Kohomologie (DE-588)4019172-2 s DE-604 Szamuely, Tamás Verfasser (DE-588)134142314 aut Cambridge studies in advanced mathematics 165 (DE-604)BV000003678 165 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030051737&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gille, Philippe 1968- Szamuely, Tamás Central simple algebras and Galois cohomology Cambridge studies in advanced mathematics Algebra, Homological Galois cohomology Zentral einfache Algebra (DE-588)4620287-0 gnd Galois-Kohomologie (DE-588)4019172-2 gnd |
| subject_GND | (DE-588)4620287-0 (DE-588)4019172-2 |
| title | Central simple algebras and Galois cohomology |
| title_auth | Central simple algebras and Galois cohomology |
| title_exact_search | Central simple algebras and Galois cohomology |
| title_full | Central simple algebras and Galois cohomology Philippe Gille, Centre Nationale de la Recherche Scientifique (CNRS), Institut Camille Jordan, Lyon, Tamás Szamuely, Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest |
| title_fullStr | Central simple algebras and Galois cohomology Philippe Gille, Centre Nationale de la Recherche Scientifique (CNRS), Institut Camille Jordan, Lyon, Tamás Szamuely, Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest |
| title_full_unstemmed | Central simple algebras and Galois cohomology Philippe Gille, Centre Nationale de la Recherche Scientifique (CNRS), Institut Camille Jordan, Lyon, Tamás Szamuely, Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest |
| title_short | Central simple algebras and Galois cohomology |
| title_sort | central simple algebras and galois cohomology |
| topic | Algebra, Homological Galois cohomology Zentral einfache Algebra (DE-588)4620287-0 gnd Galois-Kohomologie (DE-588)4019172-2 gnd |
| topic_facet | Algebra, Homological Galois cohomology Zentral einfache Algebra Galois-Kohomologie |
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