Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field

660 Pages·2017·3.692 MB·English

by  J. S. Milne

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 170
EditorialBoard
B. BOLLOBA´S, W. FULTON, F. KIRWAN,
P. SARNAK, B. SIMON, B. TOTARO
ALGEBRAICGROUPS
Algebraic groups play much the same role for algebraists as Lie groups play for
analysts. This book is the firstcomprehensive introduction tothe theory ofalgebraic
group schemes over fields that includes the structure theory of semisimple algebraic
groupsandiswritteninthelanguageofmodernalgebraicgeometry.
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
groupasanextensionofanabelianvarietybyanaffinegroup.Afterareviewofthe
Tannakian philosophy, the author provides short accounts of Lie algebras and finite
groupschemes.Thelaterchapterstreatreductivealgebraicgroupsoverarbitraryfields,
includingtheBorel–Chevalleystructuretheory.Solvablealgebraicgroupsarestudied
indetail.Prerequisiteshavebeenkepttoaminimumsothatthebookisaccessibleto
non-specialistsinalgebraicgeometry.
J. S. Milne is professor emeritus at the University of Michigan, Ann Arbor. His
previousbooksincludeE´taleCohomologyandArithmeticDualityTheorems.

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131 D.A.CravenThetheoryoffusionsystems
132 J.Va¨a¨na¨nenModelsandgames
133 G.Malle&D.TestermanLinearalgebraicgroupsandfinitegroupsofLietype
134 P.LiGeometricanalysis
135 F.MaggiSetsoffiniteperimeterandgeometricvariationalproblems
136 M.Brodmann&R.Y.SharpLocalcohomology(2ndEdition)
137 C.Muscalu&W.SchlagClassicalandmultilinearharmonicanalysis,I
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139 B.HelfferSpectraltheoryanditsapplications
140 R.Pemantle&M.C.WilsonAnalyticcombinatoricsinseveralvariables
141 B.Branner&N.FagellaQuasiconformalsurgeryinholomorphicdynamics
142 R.M.DudleyUniformcentrallimittheorems(2ndEdition)
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144 I.Arzhantsev,U.Derenthal,J.Hausen&A.LafaceCoxrings
145 M.VianaLecturesonLyapunovexponents
146 J.-H.Evertse&K.Gyo¨ryUnitequationsinDiophantinenumbertheory
147 A.PrasadRepresentationtheory
148 S.R.Garcia,J.Mashreghi&W.T.RossIntroductiontomodelspacesandtheiroperators
149 C.Godsil&K.MeagherErdo¨s–Ko–Radotheorems:Algebraicapproaches
150 P.MattilaFourieranalysisandHausdorffdimension
151 M.Viana&K.OliveiraFoundationsofergodictheory
152 V.I.Paulsen&M.RaghupathiAnintroductiontothetheoryofreproducingkernelHilbertspaces
153 R.Beals&R.WongSpecialfunctionsandorthogonalpolynomials
154 V.JurdjevicOptimalcontrolandgeometry:Integrablesystems
155 G.PisierMartingalesinBanachspaces
156 C.T.C.WallDifferentialtopology
157 J.C.Robinson,J.L.Rodrigo&W.SadowskiThethree-dimensionalNavier–Stokesequations
158 D.HuybrechtsLecturesonK3surfaces
159 H.Matsumoto&S.TaniguchiStochasticanalysis
160 A.Borodin&G.OlshanskiRepresentationsoftheinfinitesymmetricgroup
161 P.WebbFinitegrouprepresentationsforthepuremathematician
162 C.J.Bishop&Y.PeresFractalsinprobabilityandanalysis
163 A.BovierGaussianprocessesontrees
164 P.SchneiderGaloisrepresentationsand(φ,(cid:3))-modules
165 P.Gille&T.SzamuelyCentralsimplealgebrasandGaloiscohomology(2ndEdition)
166 D.Li&H.QueffelecIntroductiontoBanachspaces,I
167 D.Li&H.QueffelecIntroductiontoBanachspaces,II
168 J.Carlson,S.Mu¨ller-Stach&C.PetersPeriodmappingsandperioddomains(2ndEdition)
169 J.M.LandsbergGeometryandcomplexitytheory
170 J.S.MilneAlgebraicgroups
The picture illustrates Grothendieck’s vision of a pinned reductive
group: the body is a maximal torus T, the wings are the opposite
BorelsubgroupsB,andthepinsrigidifythesituation.(“Demazurenous
indique que, derrie`re cette terminologie [e´pinglage], il y a l’image du
papillon(queluiafournieGrothendieck):lecorpsestuntoremaximal
T, les ailes sont deux sous-groupes de Borel oppose´es par rapport a` T,
onde´ploielepapillonene´talantlesailes,puisonfixedese´le´mentsdans
les groupes additifs (des e´pingles) pour rigidifier la situation.” SGA 3,
XXIII,p.177.)
The background image is a Blue Morpho butterfly. Credit: LPETTET/DigitalVision
Vectors/GettyImages.
Algebraic Groups
The Theory of Group Schemes of Finite
Type over a Field
J. S. MILNE
UniversityofMichigan,AnnArbor
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Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Conventionsandnotation . . . . . . . . . . . . . . . . . . . . . . . . 3
1 DefinitionsandBasicProperties 6
a Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
b Basicpropertiesofalgebraicgroups . . . . . . . . . . . . . . . 12
c Algebraicsubgroups . . . . . . . . . . . . . . . . . . . . . . . 18
d Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
e Kernelsandexactsequences . . . . . . . . . . . . . . . . . . . 23
f Groupactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
g Thehomomorphismtheoremforsmoothgroups . . . . . . . . . 28
h Closedsubfunctors: definitionsandstatements. . . . . . . . . . 29
i Transporters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
j Normalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
k Centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
l Closedsubfunctors: proofs . . . . . . . . . . . . . . . . . . . . 35
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2 ExamplesandBasicConstructions 39
a Affinealgebraicgroups . . . . . . . . . . . . . . . . . . . . . . 39
b E´talegroupschemes . . . . . . . . . . . . . . . . . . . . . . . 44
c Anti-affinealgebraicgroups . . . . . . . . . . . . . . . . . . . 45
d Homomorphismsofalgebraicgroups. . . . . . . . . . . . . . . 46
e Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
f Semidirectproducts . . . . . . . . . . . . . . . . . . . . . . . . 50
g Thegroupofconnectedcomponents . . . . . . . . . . . . . . . 51
h Thealgebraicsubgroupgeneratedbyamap . . . . . . . . . . . 53
i Restrictionofscalars . . . . . . . . . . . . . . . . . . . . . . . 57
j Torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 AffineAlgebraicGroupsandHopfAlgebras 64
a Thecomultiplicationmap . . . . . . . . . . . . . . . . . . . . . 64
viii Contents
b Hopfalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
c Hopfalgebrasandalgebraicgroups . . . . . . . . . . . . . . . 66
d Hopfsubalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 67
e HopfsubalgebrasofO.G/versussubgroupsofG . . . . . . . . 68
f SubgroupsofG.k/versusalgebraicsubgroupsofG . . . . . . . 68
g Affinealgebraicgroupsincharacteristiczeroaresmooth . . . . 70
h Smoothnessincharacteristicp¤0 . . . . . . . . . . . . . . . . 72
i FaithfulflatnessforHopfalgebras . . . . . . . . . . . . . . . . 73
j Thehomomorphismtheoremforaffinealgebraicgroups . . . . 74
k Formsofalgebraicgroups . . . . . . . . . . . . . . . . . . . . 76
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4 LinearRepresentationsofAlgebraicGroups 83
a Representationsandcomodules. . . . . . . . . . . . . . . . . . 83
b Stabilizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
c Representationsareunionsoffinite-dimensionalrepresentations 86
d Affinealgebraicgroupsarelinear. . . . . . . . . . . . . . . . . 86
e Constructingallfinite-dimensionalrepresentations . . . . . . . 88
f Semisimplerepresentations . . . . . . . . . . . . . . . . . . . . 90
g Charactersandeigenspaces . . . . . . . . . . . . . . . . . . . . 92
h Chevalley’stheorem. . . . . . . . . . . . . . . . . . . . . . . . 94
i Thesubspacefixedbyagroup . . . . . . . . . . . . . . . . . . 96
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5 GroupTheory;theIsomorphismTheorems 98
a Theisomorphismtheoremsforabstractgroups. . . . . . . . . . 98
b Quotientmaps . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
c Existenceofquotients . . . . . . . . . . . . . . . . . . . . . . . 102
d Monomorphismsofalgebraicgroups . . . . . . . . . . . . . . . 106
e Thehomomorphismtheorem . . . . . . . . . . . . . . . . . . . 108
f Theisomorphismtheorem . . . . . . . . . . . . . . . . . . . . 111
g Thecorrespondencetheorem . . . . . . . . . . . . . . . . . . . 112
h Theconnected-e´taleexactsequence . . . . . . . . . . . . . . . 114
i Thecategoryofcommutativealgebraicgroups . . . . . . . . . . 115
j Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
k Theisomorphismtheoremsforfunctorstogroups . . . . . . . . 118
l Theisomorphismtheoremsforsheavesofgroups . . . . . . . . 118
m Theisomorphismtheoremsforalgebraicgroups . . . . . . . . . 119
n Somecategorytheory . . . . . . . . . . . . . . . . . . . . . . . 121
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6 SubnormalSeries;SolvableandNilpotentAlgebraicGroups 124
a Subnormalseries . . . . . . . . . . . . . . . . . . . . . . . . . 124
b Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
c Compositionseriesforalgebraicgroups . . . . . . . . . . . . . 127
Contents ix
d Thederivedgroupsandcommutatorgroups . . . . . . . . . . . 129
e Solvablealgebraicgroups . . . . . . . . . . . . . . . . . . . . . 131
f Nilpotentalgebraicgroups . . . . . . . . . . . . . . . . . . . . 133
g Existenceofalargestalgebraicsubgroupwithagivenproperty . 134
h Semisimpleandreductivegroups . . . . . . . . . . . . . . . . . 135
i Astandardexample . . . . . . . . . . . . . . . . . . . . . . . . 136
7 AlgebraicGroupsActingonSchemes 138
a Groupactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
b Thefixedsubscheme . . . . . . . . . . . . . . . . . . . . . . . 138
c Orbitsandisotropygroups . . . . . . . . . . . . . . . . . . . . 139
d Thefunctordefinedbyprojectivespace . . . . . . . . . . . . . 141
e Quotientsofaffinealgebraicgroups . . . . . . . . . . . . . . . 141
f Linearactionsonschemes . . . . . . . . . . . . . . . . . . . . 145
g Flagvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8 TheStructureofGeneralAlgebraicGroups 148
a Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
b Normalaffinealgebraicsubgroups . . . . . . . . . . . . . . . . 149
c Pseudo-abelianvarieties . . . . . . . . . . . . . . . . . . . . . 149
d Localactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
e Anti-affinealgebraicgroupsandabelianvarieties . . . . . . . . 151
f Rosenlicht’sdecompositiontheorem. . . . . . . . . . . . . . . . 151
g Rosenlicht’sdichotomy . . . . . . . . . . . . . . . . . . . . . . 153
h TheBarsotti–Chevalleytheorem . . . . . . . . . . . . . . . . . 154
i Anti-affinegroups . . . . . . . . . . . . . . . . . . . . . . . . . 156
j Extensionsofabelianvarietiesbyaffinealgebraicgroups: asurvey159
k Homogeneousspacesarequasi-projective . . . . . . . . . . . . 160
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
9 TannakaDuality;JordanDecompositions 163
a Recoveringagroupfromitsrepresentations . . . . . . . . . . . 163
b Jordandecompositions . . . . . . . . . . . . . . . . . . . . . . 166
c Characterizingcategoriesofrepresentations . . . . . . . . . . . 171
d Categoriesofcomodulesoveracoalgebra . . . . . . . . . . . . 174
e ProofofTheorem9.24 . . . . . . . . . . . . . . . . . . . . . . 178
f Tannakiancategories . . . . . . . . . . . . . . . . . . . . . . . 183
g PropertiesofG versusthoseofRep.G/ . . . . . . . . . . . . . 184
10 TheLieAlgebraofanAlgebraicGroup 186
a Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
b TheLiealgebraofanalgebraicgroup . . . . . . . . . . . . . . 188
c BasicpropertiesoftheLiealgebra . . . . . . . . . . . . . . . . 190
d Theadjointrepresentation;definitionofthebracket . . . . . . . 191
x Contents
e DescriptionoftheLiealgebraintermsofderivations . . . . . . 194
f Stabilizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
g Centres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
h Centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
i AnexampleofChevalley . . . . . . . . . . . . . . . . . . . . . 198
j Theuniversalenvelopingalgebra . . . . . . . . . . . . . . . . . 199
k Theuniversalenvelopingp-algebra . . . . . . . . . . . . . . . 204
l Thealgebraofdistributions(hyperalgebra)ofanalgebraicgroup 207
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
11 FiniteGroupSchemes 209
a Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
b Locallyfreefinitegroupschemesoverabasering . . . . . . . . 211
c Cartierduality . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
d Finitegroupschemesoforderp . . . . . . . . . . . . . . . . . 215
e DerivationsofHopfalgebras . . . . . . . . . . . . . . . . . . . 216
f Structureoftheunderlyingschemeofafinitegroupscheme . . 218
g Finitegroupschemesofordernarekilledbyn . . . . . . . . . 220
h Finitegroupschemesofheightatmostone . . . . . . . . . . . 222
i TheVerschiebungmorphism . . . . . . . . . . . . . . . . . . . 224
j TheWittschemesW . . . . . . . . . . . . . . . . . . . . . . . 226
n
k Commutativegroupschemesoveraperfectfield . . . . . . . . . 227
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
12 GroupsofMultiplicativeType;LinearlyReductiveGroups 230
a Thecharactersofanalgebraicgroup . . . . . . . . . . . . . . . 230
b ThealgebraicgroupD.M/ . . . . . . . . . . . . . . . . . . . . 230
c Diagonalizablegroups . . . . . . . . . . . . . . . . . . . . . . 233
d Diagonalizablerepresentations . . . . . . . . . . . . . . . . . . 234
e Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
f Groupsofmultiplicativetype . . . . . . . . . . . . . . . . . . . 236
g Classificationofgroupsofmultiplicativetype . . . . . . . . . . 239
h Representationsofagroupofmultiplicativetype . . . . . . . . 241
i Densityandrigidity . . . . . . . . . . . . . . . . . . . . . . . . 242
j Centraltoriasalmost-factors . . . . . . . . . . . . . . . . . . . 245
k Mapstotori . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
l Linearlyreductivegroups . . . . . . . . . . . . . . . . . . . . . 248
m Unirationality . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
13 ToriActingonSchemes 254
a Thesmoothnessofthefixedsubscheme . . . . . . . . . . . . . 254
b Limitsinschemes . . . . . . . . . . . . . . . . . . . . . . . . . 258
c Theconcentratorschemeintheaffinecase . . . . . . . . . . . . 260
d Limitsinalgebraicgroups . . . . . . . . . . . . . . . . . . . . 263