Algebraic groups: the theory of group schemes of finite type over a field
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| Main Author: | |
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| Format: | Book |
| Language: | English |
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Cambridge, United Kingdom
Cambridge University Press
2017
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| Series: | Cambridge studies in advanced mathematics
170 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | xvi, 644 Seiten |
| ISBN: | 9781107167483 |
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| 245 | 1 | 0 | |a Algebraic groups |b the theory of group schemes of finite type over a field |c J. S. Milne, University of Michigan, Ann Arbor |
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| adam_text | Contents
A
Preface............................................................ xv
Introduction...................................................... 1
Conventions and notation............................................ 3
1 Definitions and Basic Properties 6
a Definition........................................................ 6
b Basic properties of algebraic groups........................... 12
c Algebraic subgroups .......................................... 18
d Examples...................................................... 22
e Kernels and exact sequences................................... 23
f Group actions................................................. 26
g The homomorphism theorem for smooth groups.................... 28
h Closed subfunctors: definitions and statements................ 29
i Transporters.................................................. 30
j Normalizers................................................... 31
k Centralizers.................................................. 33
1 Closed subfunctors: proofs.................................... 35
Exercises ......................................................... 38
2 Examples and Basic Constructions 39
a Affine algebraic groups....................................... 39
b Étale group schemes .......................................... 44
c Anti-affine algebraic groups ................................. 45
d Homomorphisms of algebraic groups............................. 46
e Products...................................................... 49
f Semidirect products........................................... 30
g The group of connected components............................. 31
h The algebraic subgroup generated by a map..................... 53
i Restriction of scalars........................................ 37
j Torsors ...................................................... 60
Exercises ......................................................... 61
3 Affine Algebraic Groups and Hopf Algebras 64
a The comultiplication map...................................... 64
vii
viii Contents
b Hopf algebras.............................................. 65
c Hopf algebras and algebraic groups ............................. 66
d Hopf subalgebras........................................... 67
e Hopf subalgebras of 0(G) versus subgroups of G............. 68
f Subgroups of G(k) versus algebraic subgroups of G.......... 68
g Affine algebraic groups in characteristic zero are smooth .... 70
h Smoothness in characteristic p ^ 0......................... 72
i Faithful flatness for Hopf algebras........................ 73
j The homomorphism theorem for affine algebraic groups .... 74
k Forms of algebraic groups ...................................... 76
Exercises ...................................................... 81
4 Linear Representations of Algebraic Groups 83
a Representations and comodules.............................. 83
b Stabilizers................................................ 85
c Representations are unions of finite-dimensional representations 86
d Affine algebraic groups are linear......................... 86
e Constructing all finite-dimensional representations ............ 88
f Semisimple representations...................................... 90
g Characters and eigenspaces...................................... 92
h Chevalley’s theorem............................................. 94
i The subspace fixed by a group................................... 96
Exercises .......................................................... 97
5 Group Theory; the Isomorphism Theorems 98
a The isomorphism theorems for abstract groups.................... 98
b Quotient maps................................................... 99
c Existence of quotients..........................................102
d Monomorphisms of algebraic groups...............................106
e The homomorphism theorem........................................108
f The isomorphism theorem.........................................Ill
g The correspondence theorem......................................112
h The connected-etale exact sequence..............................114
i The category of commutative algebraic groups....................115
j Sheaves.........................................................116
k The isomorphism theorems for functors to groups.................118
1 The isomorphism theorems for sheaves of groups..................118
m The isomorphism theorems for algebraic groups...................119
n Some category theory............................................121
Exercises ..........................................................122
6 Subnormal Series; Solvable and Nilpotent Algebraic Groups 124
a Subnormal series................................................124
b Isogenies ......................................................126
c Composition series for algebraic groups.........................127
Contents
IX
d The derived groups and commutator groups.......................129
e Solvable algebraic groups......................................131
f Nilpotent algebraic groups.....................................133
g Existence of a largest algebraic subgroup with a given property . 134
h Semisimple and reductive groups................................135
i A standard example........................................... 136
7 Algebraic Groups Acting on Schemes 138
a Group actions..................................................138
b The fixed subscheme............................................138
c Orbits and isotropy groups.....................................139
d The functor defined by projective space........................141
e Quotients of affine algebraic groups...........................141
f Linear actions on schemes......................................145
g Flag varieties.................................................146
Exercises ..........................................................146
8 The Structure of General Algebraic Groups 148
a Summary........................................................148
b Normal affine algebraic subgroups..............................149
c Pseudo-abelian varieties ......................................149
d Local actions..................................................150
e Anti-affine algebraic groups and abelian varieties.............151
f Rosenlicht’s decomposition theorem.............................151
g Rosenlicht’s dichotomy.........................................153
h The Barsotti—Chevalley theorem.................................154
i Anti-affine groups.............................................156
j Extensions of abelian varieties by affine algebraic groups: a survey 159
k Homogeneous spaces are quasi-projective........................160
Exercises ..........................................................162
9 Tannaka Duality; Jordan Decompositions 163
a Recovering a group from its representations....................163
b Jordan decompositions..........................................166
c Characterizing categories of representations...................171
d Categories of comodules over a coalgebra.......................174
e Proof of Theorem 9.24..........................................178
f Tannakian categories...........................................183
g Properties of G versus those of Rep(G).........................184
10 The Lie Algebra of an Algebraic Group 186
a Definition.....................................................186
b The Lie algebra of an algebraic group..........................188
c Basic properties of the Lie algebra............................190
d The adjoint representation; definition of the bracket..........191
X
Contents
e Description of the Lie algebra in terms of derivations.........194
f Stabilizers..................................................196
g Centres......................................................197
h Centralizers.................................................197
i An example of Chevalley........................................198
j The universal enveloping algebra...............................199
k The universal enveloping p-algebra.............................204
1 The algebra of distributions (hyperalgebra) of an algebraic group 207
Exercises ........................................................208
11 Finite Group Schemes 209
a Generalities.................................................209
b Locally free finite group schemes over a base ring...........211
c Cartier duality................................................212
d Finite group schemes of order p................................215
e Derivations of Hopf algebras...................................216
f Structure of the underlying scheme of a finite group scheme . . 218
g Finite group schemes of order n are killed by n................220
h Finite group schemes of height at most one ....................222
i The Verschiebung morphism......................................224
j The Witt schemes Wn............................................226
k Commutative group schemes over a perfect field.................227
Exercises ..........................................................229
12 Groups of Multiplicative Type; Linearly Reductive Groups 230
a The characters of an algebraic group...........................230
b The algebraic group Z (M)......................................230
c Diagonalizable groups .........................................233
d Diagonalizable representations.................................234
e Tori...........................................................236
f Groups of multiplicative type..................................236
g Classification of groups of multiplicative type................239
h Representations of a group of multiplicative type..............241
i Density and rigidity...........................................242
j Central tori as almost-factors.................................245
k Maps to tori...................................................246
1 Linearly reductive groups......................................248
m Unirationality.................................................250
Exercises ..........................................................252
13 Tori Acting on Schemes 254
a The smoothness of the fixed subscheme..........................254
b Limits in schemes..............................................258
c The concentrator scheme in the affine case.....................260
d Limits in algebraic groups ....................................263
Contents
xi
e Luna maps.....................................................267
f The Bialynicki-Birula decomposition...........................272
g Proof of the Bialynicki-Birula decomposition..................276
Exercises .........................................................278
14 Unipotent Algebraic Groups 279
a Preliminaries from linear algebra.............................279
b Unipotent algebraic groups....................................280
c Unipotent elements in algebraic groups .......................286
d Unipotent algebraic groups in characteristic zero ............288
e Unipotent algebraic groups in nonzero characteristic .........292
f Algebraic groups isomorphic to ...............................298
g Split and wound unipotent groups .............................299
Exercises .........................................................301
15 Cohomology and Extensions 302
a Crossed homomorphisms.........................................302
b Hochschild cohomology ........................................304
c Hochschild extensions ........................................307
d The cohomology of linear representations......................310
e Linearly reductive groups.....................................311
f Applications to homomorphisms.................................312
g Applications to centralizers..................................313
h Calculation of some extensions................................313
Exercises .........................................................323
16 The Structure of Solvable Algebraic Groups 324
a Trigonalizable algebraic groups...............................324
b Commutative algebraic groups..................................327
c Structure of trigonalizable algebraic groups..................330
d Solvable algebraic groups.....................................334
e Connectedness ................................................338
f Nilpotent algebraic groups....................................340
g Split solvable groups.........................................343
h Complements on unipotent algebraic groups.....................346
i Tori acting on algebraic groups...............................347
Exercises .........................................................351
17 Borel Subgroups and Applications 352
a The Borel fixed point theorem.................................352
b Borel subgroups and maximal tori..............................353
c The density theorem ..........................................361
d Centralizers of tori..........................................363
e The normalizer of a Borel subgroup............................366
f The variety of Borel subgroups................................368
Contents
xii
g Chevalley’s description of the unipotent radical...............370
h Proof of Chevalley’s theorem.........................................373
i Borel and parabolic subgroups over an arbitrary base field .... 375
j Maximal tori and Cartan subgroups over an arbitrary base field . 376
k Algebraic groups over finite fields.........................382
1 Split algebraic groups......................................384
Exercises .............................................................385
18 The Geometry of Algebraic Groups 387
a Central and multiplicative isogenies..............................387
b The universal covering......................................388
c Line bundles and characters.................................389
d Existence of a universal covering.................................392
e Applications................................................393
f Proof of theorem 18.15......................................395
Exercises .............................................................396
19 Semisimple and Reductive Groups 397
a Semisimple groups...........................................397
b Reductive groups..................................................399
c The rank of a group variety.......................................401
d Deconstructing reductive groups...................................403
Exercises ....................................................... - 406
20 Algebraic Groups of Semisimple Rank One 407
a Group varieties of semisimple rank 0..............................407
b Homogeneous curves................................................408
c The automorphism group of the projective line.....................409
d A fixed point theorem for actions of tori.........................410
e Group varieties of semisimple rank 1..............................412
f Split reductive groups of semisimple rank 1.......................414
g Properties of SL2.................................................415
h Classification of the split reductive groups of semisimple rank 1 418
i The forms of SL2, GL2, and PGL2...................................419
j Classification of reductive groups of semisimple rank one .... 421
k Review of SL2 ....................................................422
Exercises ............................................................423
21 Split Reductive Groups 424
a Split reductive groups and their roots............................424
b Centres of reductive groups.......................................427
c The root datum of a split reductive group.........................428
d Borel subgroups; Weyl groups; Tits systems........................433
e Complements on semisimple groups..................................439
f Complements on reductive groups...................................442
Contents
xiii
g Unipotent subgroups normalized by T.............................444
h The Bruhat decomposition........................................446
i Parabolic subgroups.............................................452
j The root data of the classical semisimple groups................456
Exercises ...........................................................461
22 Representations of Reductive Groups 463
a The semisimple representations of a split reductive group .... 463
b Characters and Grothendieck groups..............................473
c Semisimplicity in characteristic zero...........................475
d Weyi’s character formula........................................479
e Relation to the representations ofLie(G)........................481
Exercises ...........................................................482
23 The Isogeny and Existence Theorems 483
a isogenies of groups and of root data............................483
b Proof of the isogeny theorem ...................................487
c Complements.....................................................492
d Pinnings........................................................495
e Automorphisms...................................................497
f Quasi-split forms...............................................499
g Statement of the existence theorem; applications................501
h Proof of the existence theorem..................................503
Exercises ...........................................................511
24 Construction of the Semisimple Groups 512
a Deconstructing semisimple algebraic groups......................512
b Generalities on forms of semisimple groups......................514
c The centres of semisimple groups ...............................516
d Semisimple algebras ............................................518
e Algebras with involution........................................520
f The geometrically almost-simple groups of type A ...............523
g The geometrically almost-simple groups of type C ...............526
h Clifford algebras ..............................................527
i The spin groups.................................................531
j The geometrically almost-simple group of types B and D . . . . 533
k The classical groups in terms of sesquilinear forms.............534
1 The exceptional groups..........................................538
m The trialitarian groups (groups of subtype 3D4 and 6D4) .... 542
Exercises ...........................................................542
25 Additional Topics 544
a Parabolic subgroups of reductive groups.........................544
b The small root system...............................-............548
c The Satake—Tits classification .................................551
XIV
Contents
d Representation theory.......................................553
e Pseudo-reductive groups.......................................557
f Nonreductive groups: Levi subgroups.........................558
g Galois cohomology.............................................559
Exercises ......................................................565
Appendix A Review of Algebraic Geometry 566
a Affine algebraic schemes......................................566
b Algebraic schemes.............................................569
c Subschemes....................................................571
d Algebraic schemes as functors.................................572
e Fibred products of algebraic schemes..........................574
f Algebraic varieties...........................................575
g The dimension of an algebraic scheme..........................576
h Tangent spaces; smooth points; regular points.................577
i Étale schemes.................................................579
j Galois descent for closed subschemes..........................580
k Flat and smooth morphisms.....................................581
1 The fibres of regular maps.................................. 582
m Complete schemes; proper maps.................................583
n The Picard group..............................................584
o Flat descent..................................................584
Appendix B Existence of Quotients of Algebraic Groups 586
a Equivalence relations.........................................586
b Existence of quotients in the finite affine case..............591
c Existence of quotients in the finite case.....................596
d Existence of quotients in the presence of quasi-sections......599
e Existence generically of a quotient...........................602
f Existence of quotients of algebraic groups....................604
Appendix C Root Data 607
a Preliminaries.................................................607
b Reflection groups.............................................608
c Root systems..................................................610
d Root data.....................................................613
e Duals of root data............................................615
f Deconstructing root data......................................620
g Classification of reduced root systems........................621
References 627
Index
637
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 170
Editorial Board
B. BOLLOBÁS, W. FULTON, F. KIRWAN,
P. SARNAK, B. SIMON, B. TOTARO
ALGEBRAIC GROUPS
Algebraic groups play much the same role for algebraists as Lie groups play for
analysts. This book is the first comprehensive introduction to the theory of algebraic
group schemes over fields that includes the structure theory of semisimple algebraic
groups and is written in the language of modem algebraic geometry.
The first eight chapters study general algebraic group schemes over a field and
culminate in a proof of the Barsotti—Chevalley theorem realizing every algebraic
group as an extension of an abelian variety by an affine group. After a review of the
Tannakian philosophy, the author provides short accounts of Lie algebras and finite
group schemes. The later chapters treat reductive algebraic groups over arbitrary fields,
including the Borel-Chevalley structure theory. Solvable algebraic groups are studied
in detail. Prerequisites have been kept to a minimum so that the book is accessible to
non-specialists in algebraic geometry.
J. S. Milne is professor emeritus at the University of Michigan, Ann Arbor. His
previous books include Étale Cohomology and Arithmetic Duality Theorems.
|
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| indexdate | 2024-07-10T07:55:40Z |
| institution | BVB |
| isbn | 9781107167483 |
| language | English |
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| physical | xvi, 644 Seiten |
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| publisher | Cambridge University Press |
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| series | Cambridge studies in advanced mathematics |
| series2 | Cambridge studies in advanced mathematics |
| spelling | Milne, J. S. 1942- Verfasser (DE-588)1077746881 aut Algebraic groups the theory of group schemes of finite type over a field J. S. Milne, University of Michigan, Ann Arbor Cambridge, United Kingdom Cambridge University Press 2017 xvi, 644 Seiten txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 170 Algebraische Gruppe (DE-588)4001164-1 gnd rswk-swf Algebraische Gruppe (DE-588)4001164-1 s DE-604 Erscheint auch als Online-Ausgabe 978-1-316-71173-6 Cambridge studies in advanced mathematics 170 (DE-604)BV000003678 170 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029950166&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029950166&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Milne, J. S. 1942- Algebraic groups the theory of group schemes of finite type over a field Cambridge studies in advanced mathematics Algebraische Gruppe (DE-588)4001164-1 gnd |
| subject_GND | (DE-588)4001164-1 |
| title | Algebraic groups the theory of group schemes of finite type over a field |
| title_auth | Algebraic groups the theory of group schemes of finite type over a field |
| title_exact_search | Algebraic groups the theory of group schemes of finite type over a field |
| title_full | Algebraic groups the theory of group schemes of finite type over a field J. S. Milne, University of Michigan, Ann Arbor |
| title_fullStr | Algebraic groups the theory of group schemes of finite type over a field J. S. Milne, University of Michigan, Ann Arbor |
| title_full_unstemmed | Algebraic groups the theory of group schemes of finite type over a field J. S. Milne, University of Michigan, Ann Arbor |
| title_short | Algebraic groups |
| title_sort | algebraic groups the theory of group schemes of finite type over a field |
| title_sub | the theory of group schemes of finite type over a field |
| topic | Algebraische Gruppe (DE-588)4001164-1 gnd |
| topic_facet | Algebraische Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029950166&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029950166&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000003678 |
| work_keys_str_mv | AT milnejs algebraicgroupsthetheoryofgroupschemesoffinitetypeoverafield |