Algebraic groups and their birational invariants
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| Format: | Buch |
| Sprache: | English |
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Providence, RI
American Math. Soc.
1998
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| Schriftenreihe: | Translations of mathematical monographs
179 |
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| 100 | 1 | |a Voskresenskij, Valentin E. |e Verfasser |4 aut | |
| 240 | 1 | 0 | |a Algebraičeskie gruppy i ich biracional'nye invarianty |
| 245 | 1 | 0 | |a Algebraic groups and their birational invariants |c V. E. Voskresenskii |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 1998 | |
| 300 | |a XIII, 218 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Translations of mathematical monographs |v 179 | |
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Datensatz im Suchindex
| _version_ | 1819577799916126208 |
|---|---|
| adam_text | Contents
Preface xi
Chapter 1. Forms and Galois Cohomology 1
§1 Group schemes and their cohomology
1.1 Group objects in a category 1
1.2 Group schemes 3
1.3 Affine groups, Hopf algebras 4
1.4 Group schemes over a field, algebraic groups 7
1.5 Probenius morphisms 7
1.6 Diagonal groups 10
1.7 Characters of group schemes 11
1.8 Bicharacters 13
1.9 Exactness of the functor D 14
1.10 Galois cohomology 16
1.11 Sheaves and cohomology in the etale topology 17
1.12 Cartier divisors and Weil divisors 19
§2 The Brauer group of a projective variety 20
2.1 The unramified Brauer group of a function field 20
2.2 The Kummer exact sequence 20
2.3 The Tate group, the Picard number, the Lefschetz number 21
§3 The theory of fc forms 23
3.1 Forms and one dimensional cohomology 23
3.2 Splitting fields of a fc form 24
3.3 Forms of group schemes 25
3.4 Groups of multiplicative type 25
3.5 Principal homogeneous spaces 27
3.6 Projective groups and associated fc forms 29
3.7 The Brauer group of a field 30
3.8 Chevalley groups 32
3.9 Semisimple groups 35
3.10 Inner and outer forms 36
3.11 Almost simple semisimple groups 37
3.12 The Weil restriction 37
Chapter 2. Birational Geometry of Algebraic Tori 41
§4 Birational invariants of linear algebraic groups 41
4.1 The variety of maximal tori of a reductive group 41
4.2 Structure of the generic torus of a semisimple group 42
vii
viii CONTENTS
4.3 The Picard group and the Brauer group of a linear algebraic
group 44
4.4 Criteria for birational equivalence of algebraic varieties 46
4.5 Projective models of linear algebraic groups 47
4.6 Flasque resolutions of a module 49
4.7 Stable equivalence 51
4.8 Chevalley modules 52
4.9 Tori of small dimension 57
4.10 Tori with a biquadratic splitting field 58
4.11 The semigroup of stable equivalence 59
§5 Tori with a cyclic splitting field 60
5.1 Devissage of a quasi split torus 60
5.2 Invertibility of the Picard class 62
5.3 The Chistov multiplication 62
§6 Stable rationality of varieties 65
6.1 Stably rational tori as orbit varieties 65
6.2 Covariants of linear represntations 67
6.3 Rationality of tori of type pq 69
6.4 Universal torsors 71
6.5 Counterexamples to Zariski s conjecture 73
Chapter 3. Invariants of Finite Transformation Groups 75
§7 Fields of invariants of finite transformation groups 75
7.1 Fields of invariants and their models 75
7.2 Invariants of finite abelian groups 76
7.3 The fields (k,pa), p 2 78
7.4 The fields (k, 2Q) 79
7.5 General case 79
7.6 Invariants of finite groups over an algebraically closed field 81
7.7 Invariants of finite linear groups 82
7.8 Invariants of finite groups acting on tori 85
7.9 Invariants of connected algebraic groups 87
§8 Invariant projective Demazure models 90
8.1 Cones and fans 90
8.2 Projective invariant fans 93
8.3 Birational invariants of tori without affect 97
8.4 The graded ring of a toric variety 99
Chapter 4. Arithmetic of Linear Algebraic Groups 103
§9 Tori over a finite field 103
9.1 Number of rational points 103
9.2 Zeta function 104
§10 Tori over local fields 106
10.1 Tori over reals 106
10.2 Tori over a nonarchimedean field 107
10.3 Integer structures in linear algebraic groups 107
10.4 Canonical integer form of a quasisplit torus 109
10.5 Canonical form of a norm torus 111
§11 Tori over global fields 111
CONTENTS ix
11.1 Adele groups 111
11.2 Canonical integer model of a torus over a number field 113
11.3 Cohomology of adele groups 114
11.4 Descent of the ground field 118
11.5 Approximation problems 119
11.6 Arithmetical meaning of the birational invariant H1(k,p(T)) 120
§12 Arithmetic of semisimple groups 122
12.1 Cohomology of semisimple groups 122
12.2 Weak approximation 124
12.3 The group H k, Pic X) 125
§13 Artin L functions 127
13.1 Partial Artin L functions 127
13.2 Theorems of Artin and Brauer 129
13.3 Global zeta function of a torus 131
Chapter 5. Tamagawa Numbers 133
§14 Haar measure on adele groups 133
14.1 Product of local measures 133
14.2 Computation of local volumes 134
14.3 Canonical convergence factors 136
14.4 The Tamagawa measure 137
14.5 Properties of Tamagawa numbers 142
14.6 Tamagawa numbers of algebraic tori 142
14.7 The group $ 147
14.8 Further development of the method 148
14.9 Chevalley group Z schemes 148
14.10 Gindikin Karpelevich integrals 149
14.11 Langlands method of computing Tamagawa numbers 153
14.12 Elementary computations of volumes of some classical quotients 160
§15 The Minkowski Siegel Tamagawa formula 163
15.1 Infinite products 163
15.2 The weight of a genus of an odd positive lattice 165
15.3 The weight of a genus of an even positive unimodular lattice 169
15.4 Sums of squares 169
15.5 Sum of two squares 172
15.6 Sum of four squares 173
15.7 Sum of six squares 173
15.8 Sum of eight squares 174
15.9 Sum of three squares 174
15.10 Sum of five squares 175
15.11 Sum of seven squares 176
Chapter 6. .R equivalence in Algebraic Groups 177
§16 The group of .R equivalence classes 177
16.1 First properties of /^ equivalence on varieties 177
16.2 Birational invariance of /^ equivalence in groups 179
§17 /^ equivalence on algebraic tori 180
17.1 Flasque resolution of a torus and /^ equivalence 180
17.2 Some special tori 182
x CONTENTS
17.3 The group T(k(t)) 184
§18 The unimodular group of a simple algebra 185
18.1 Reduction to the anisotropic kernel 185
18.2 The Whitehead group of a simple algebra 185
18.3 Platonov s examples 187
18.4 The Whitehead group of an isotropic group 188
18.5 inequivalence over special fields 188
§19 Algebras with involutions and groups of adjoint type 190
19.1 Algebras with involutions 190
19.2 Indecomposable algebras with involutions 190
19.3 Automorphisms of indecomposable algebras with involutions 192
19.4 Forms of algebras with involutions 192
19.5 The covering of Go 193
19.6 Merkurjev s theorems 194
Chapter 7. Index Formulas in Arithmetic of Algebraic Tori 197
§20 Arithmetic of the projective group of a field 197
20.1 Ratio of class numbers 197
20.2 Index formulas for quadratic extensions 200
20.3 The Hasse relations for an imaginary extension 201
§21 Arithmetic of a norm hypersurface 202
Bibliographical Remarks 207
Bibliography 211
|
| any_adam_object | 1 |
| author | Voskresenskij, Valentin E. |
| author_facet | Voskresenskij, Valentin E. |
| author_role | aut |
| author_sort | Voskresenskij, Valentin E. |
| author_variant | v e v ve vev |
| building | Verbundindex |
| bvnumber | BV013265014 |
| classification_rvk | SK 240 |
| ctrlnum | (OCoLC)245954687 (DE-599)BVBBV013265014 |
| dewey-full | 512.55 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.55 |
| dewey-search | 512.55 |
| dewey-sort | 3512.55 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV013265014 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-23T15:24:39Z |
| institution | BVB |
| isbn | 0821809059 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009042991 |
| oclc_num | 245954687 |
| open_access_boolean | |
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| owner_facet | DE-355 DE-BY-UBR DE-29T DE-11 DE-19 DE-BY-UBM |
| physical | XIII, 218 S. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | American Math. Soc. |
| record_format | marc |
| series | Translations of mathematical monographs |
| series2 | Translations of mathematical monographs |
| spellingShingle | Voskresenskij, Valentin E. Algebraic groups and their birational invariants Translations of mathematical monographs Algebraische Geometrie (DE-588)4001161-6 gnd Lineare algebraische Gruppe (DE-588)4295326-1 gnd |
| subject_GND | (DE-588)4001161-6 (DE-588)4295326-1 |
| title | Algebraic groups and their birational invariants |
| title_alt | Algebraičeskie gruppy i ich biracional'nye invarianty |
| title_auth | Algebraic groups and their birational invariants |
| title_exact_search | Algebraic groups and their birational invariants |
| title_full | Algebraic groups and their birational invariants V. E. Voskresenskii |
| title_fullStr | Algebraic groups and their birational invariants V. E. Voskresenskii |
| title_full_unstemmed | Algebraic groups and their birational invariants V. E. Voskresenskii |
| title_short | Algebraic groups and their birational invariants |
| title_sort | algebraic groups and their birational invariants |
| topic | Algebraische Geometrie (DE-588)4001161-6 gnd Lineare algebraische Gruppe (DE-588)4295326-1 gnd |
| topic_facet | Algebraische Geometrie Lineare algebraische Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009042991&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000002394 |
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