Basic Theory of Affine Group Schemes - James Milne

275 Pages·2012·1.9 MB·English

Basic Theory
of
Affine Group Schemes
J.S. Milne
Version1.00
March11,2012
This is a modern exposition of the basic theory of affine group schemes. Although the
emphasis is on affine group schemes of finite type over a field, we also discuss more gen-
eral objects: affine group schemes not of finite type; base rings not fields; affine monoids
not groups; group schemes not affine, affine supergroup schemes (very briefly); quantum
groups(evenmorebriefly). “Basic”meansthatwedonotinvestigatethedetailedstructure
ofreductivegroupsusingrootdataexceptinthefinalsurveychapter(whichisnotyetwrit-
ten). Prerequisiteshavebeenheldtoaminimum: allthereaderreallyneedsisaknowledge
ofsomebasiccommutativealgebraandalittleofthelanguageofalgebraicgeometry.
BibTeXinformation:
@misc{milneAGS,
author={Milne, James S.},
title={Basic Theory of Affine Group Schemes},
year={2012},
note={Available at www.jmilne.org/math/}
}
v1.00 March11,2012,275pages.
Pleasesendcommentsandcorrectionstomeattheaddressonmywebsite
http://www.jmilne.org/math/.
The photo is of the famous laughing Buddha on The Peak That Flew Here, Hangzhou,
Zhejiang,China.
Copyright(cid:13)c 2012J.S.Milne.
Singlepapercopiesfornoncommercialpersonalusemaybemadewithoutexplicitpermis-
sionfromthecopyrightholder.
Table of Contents
TableofContents 3
I Definitionofanaffinegroup 17
1 Motivatingdiscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Somecategorytheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Affinegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Affinemonoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5 Affinesupergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
II AffineGroupsandHopfAlgebras 29
1 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Thedualityofalgebrasandcoalgebras . . . . . . . . . . . . . . . . . . . . 31
4 Bi-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5 AffinegroupsandHopfalgebras . . . . . . . . . . . . . . . . . . . . . . . 34
6 Abstractrestatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7 Commutativeaffinegroups . . . . . . . . . . . . . . . . . . . . . . . . . . 36
8 Quantumgroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
9 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
III AffineGroupsandGroupSchemes 41
1 Thespectrumofaring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Affinegroupsasaffinegroupschemes . . . . . . . . . . . . . . . . . . . . 44
4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
IV Examples 47
1 Examplesofaffinegroups . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2 Examplesofhomomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 54
3 Appendix: Arepresentabilitycriterion . . . . . . . . . . . . . . . . . . . . 54
V SomeBasicConstructions 57
1 Productsofaffinegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2 Fibredproductsofaffinegroups . . . . . . . . . . . . . . . . . . . . . . . 57
3 Limitsofaffinegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3
4 Extensionofthebasering(extensionofscalars) . . . . . . . . . . . . . . . 59
5 Restrictionofthebasering(Weilrestrictionofscalars) . . . . . . . . . . . 60
6 Transporters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7 Galoisdescentofaffinegroups . . . . . . . . . . . . . . . . . . . . . . . . 67
8 TheGreenbergfunctor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
VI Affinegroupsoverfields 71
1 Affinek-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2 Schemesalgebraicoverafield . . . . . . . . . . . . . . . . . . . . . . . . 71
3 Algebraicgroupsasgroupsinthecategoryofaffinealgebraicschemes . . . 73
4 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6 Reducedalgebraicgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7 Smoothalgebraicschemes . . . . . . . . . . . . . . . . . . . . . . . . . . 77
8 Smoothalgebraicgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
9 Algebraicgroupsincharacteristiczeroaresmooth(Cartier’stheorem) . . . 79
10 Smoothnessincharacteristicp¤0 . . . . . . . . . . . . . . . . . . . . . . 81
11 Appendix: ThefaithfulflatnessofHopfalgebras . . . . . . . . . . . . . . 82
VII GroupTheory: SubgroupsandQuotientGroups. 87
1 Acriteriontobeanisomorphism . . . . . . . . . . . . . . . . . . . . . . . 87
2 Injectivehomomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3 Affinesubgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4 Kernelsofhomomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5 Densesubgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6 Normalizers;centralizers;centres. . . . . . . . . . . . . . . . . . . . . . . 97
7 Quotientgroups;surjectivehomomorphisms . . . . . . . . . . . . . . . . . 100
8 Existenceofquotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
9 Semidirectproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
10 Smoothalgebraicgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
11 Algebraicgroupsassheaves . . . . . . . . . . . . . . . . . . . . . . . . . 106
12 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
VIII RepresentationsofAffineGroups 111
1 Finitegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
2 Definitionofarepresentation . . . . . . . . . . . . . . . . . . . . . . . . . 112
3 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4 Comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5 Thecategoryofcomodules . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6 Representationsandcomodules . . . . . . . . . . . . . . . . . . . . . . . . 118
7 ThecategoryofrepresentationsofG . . . . . . . . . . . . . . . . . . . . . 123
8 Affinegroupsareinverselimitsofalgebraicgroups . . . . . . . . . . . . . 123
9 Algebraicgroupsadmitfinite-dimensionalfaithfulrepresentations . . . . . 125
10 Theregularrepresentationcontainsall . . . . . . . . . . . . . . . . . . . . 126
11 EveryfaithfulrepresentationgeneratesRep.G/ . . . . . . . . . . . . . . . 127
12 Stabilizersofsubspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4
13 Chevalley’stheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
14 Sub-coalgebrasandsubcategories . . . . . . . . . . . . . . . . . . . . . . 133
15 Quotientgroupsandsubcategories . . . . . . . . . . . . . . . . . . . . . . 134
16 Charactersandeigenspaces . . . . . . . . . . . . . . . . . . . . . . . . . . 134
17 Everynormalaffinesubgroupisakernel . . . . . . . . . . . . . . . . . . . 136
18 VariantoftheproofofthekeyLemma17.4 . . . . . . . . . . . . . . . . . 138
19 ApplicationsofCorollary17.6 . . . . . . . . . . . . . . . . . . . . . . . . 138
IX GroupTheory: theIsomorphismTheorems 141
1 Reviewofabstractgrouptheory . . . . . . . . . . . . . . . . . . . . . . . 141
2 Theexistenceofquotients . . . . . . . . . . . . . . . . . . . . . . . . . . 142
3 Thehomomorphismtheorem . . . . . . . . . . . . . . . . . . . . . . . . . 143
4 Theisomorphismtheorem . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5 Thecorrespondencetheorem . . . . . . . . . . . . . . . . . . . . . . . . . 145
6 TheSchreierrefinementtheorem . . . . . . . . . . . . . . . . . . . . . . . 146
7 Thecategoryofcommutativealgebraicgroups . . . . . . . . . . . . . . . . 148
8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
X CategoriesofRepresentations(TannakaDuality) 151
1 Recoveringagroupfromitsrepresentations . . . . . . . . . . . . . . . . . 151
2 ApplicationtoJordandecompositions . . . . . . . . . . . . . . . . . . . . 154
3 Characterizationsofcategoriesofrepresentations . . . . . . . . . . . . . . 159
4 Homomorphismsandfunctors . . . . . . . . . . . . . . . . . . . . . . . . 166
XI TheLieAlgebraofanAffineGroup 169
1 DefinitionofaLiealgebra . . . . . . . . . . . . . . . . . . . . . . . . . . 169
2 Theisomorphismtheorems . . . . . . . . . . . . . . . . . . . . . . . . . . 170
3 TheLiealgebraofanaffinegroup . . . . . . . . . . . . . . . . . . . . . . 171
4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5 DescriptionofLie.G/intermsofderivations . . . . . . . . . . . . . . . . 176
6 Extensionofthebasefield . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7 TheadjointmapAdWG !Aut.g/ . . . . . . . . . . . . . . . . . . . . . . 178
8 Firstdefinitionofthebracket . . . . . . . . . . . . . . . . . . . . . . . . . 179
9 Seconddefinitionofthebracket . . . . . . . . . . . . . . . . . . . . . . . 179
10 ThefunctorLiepreservesfibredproducts . . . . . . . . . . . . . . . . . . 180
11 CommutativeLiealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 182
12 Normalsubgroupsandideals . . . . . . . . . . . . . . . . . . . . . . . . . 182
13 AlgebraicLiealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
14 Theexponentialnotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
15 Arbitrarybaserings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
16 MoreontherelationbetweenalgebraicgroupsandtheirLiealgebras . . . . 185
XII FiniteAffineGroups 191
1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
2 E´taleaffinegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
3 Finiteflataffinep-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
4 Cartierduality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5
XIII TheConnectedComponentsofanAlgebraicGroup 203
1 Idempotentsandconnectedcomponents . . . . . . . . . . . . . . . . . . . 203
2 E´talesubalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
3 Algebraicgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
4 Affinegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6 Whereweare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
XIV GroupsofMultiplicativeType;Tori 215
1 Group-likeelements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
2 Thecharactersofanaffinegroup . . . . . . . . . . . . . . . . . . . . . . . 216
3 TheaffinegroupD.M/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
4 Diagonalizablegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
5 Groupsofmultiplicativetype . . . . . . . . . . . . . . . . . . . . . . . . . 223
6 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7 Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
8 Groupschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
XV UnipotentAffineGroups 231
1 Preliminariesfromlinearalgebra . . . . . . . . . . . . . . . . . . . . . . . 231
2 Unipotentaffinegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
3 Unipotentaffinegroupsincharacteristiczero . . . . . . . . . . . . . . . . 237
4 Groupschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
XVI SolvableAffineGroups 241
1 Trigonalizableaffinegroups . . . . . . . . . . . . . . . . . . . . . . . . . 241
2 Commutativealgebraicgroups . . . . . . . . . . . . . . . . . . . . . . . . 243
3 Thederivedgroupofanalgebraicgroup . . . . . . . . . . . . . . . . . . . 246
4 Solvablealgebraicgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
5 Structureofsolvablegroups . . . . . . . . . . . . . . . . . . . . . . . . . 251
6 Splitsolvablegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
7 Toriinsolvablegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
XVII TheStructureofAlgebraicGroups 255
1 Radicalsandunipotentradicals . . . . . . . . . . . . . . . . . . . . . . . . 255
2 Definitionofsemisimpleandreductivegroups . . . . . . . . . . . . . . . . 256
3 Thecanonicalfiltrationonanalgebraicgroup . . . . . . . . . . . . . . . . 258
4 Thestructureofsemisimplegroups . . . . . . . . . . . . . . . . . . . . . . 259
5 Thestructureofreductivegroups . . . . . . . . . . . . . . . . . . . . . . . 261
6 Pseudoreductivegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
7 PropertiesofG versusthoseofRep .G/: asummary . . . . . . . . . . . . 265
k
XVIIIBeyondthebasics 267
Bibliography 269
Index 273
6
Preface
For one who attempts to unravel the story, the
problems are as perplexing as a mass of hemp
withathousandlooseends.
DreamoftheRedChamber,TsaoHsueh-Chin.
Algebraicgroupsaregroupsdefinedbypolynomials. Thosethatweshallbeconcerned
with in this book can all be realized as groups of matrices. For example, the group of
matricesofdeterminant1isanalgebraicgroup, asistheorthogonalgroupofasymmetric
bilinear form. The classification of algebraic groups and the elucidation of their structure
were among the great achievements of twentieth century mathematics (Borel, Chevalley,
Titsandothers,buildingontheworkofthepioneersonLiegroups). Algebraicgroupsare
used in most branches of mathematics, and since the famous work of Hermann Weyl in
the 1920s they have also played a vital role in quantum mechanics and other branches of
physics(usuallyasLiegroups).
The goal of the present work is to provide a modern exposition of the basic theory of
algebraic groups. It has been clear for fifty years, that in the definition of an algebraic
group, the coordinate ring should be allowed to have nilpotent elements,1 but the standard
expositions2 donotallowthis.3 Whatwecallanaffinealgebraicgroupisusuallycalledan
affinegroupschemeoffinitetype. Inrecentyears,thetannakianduality4betweenalgebraic
groupsandtheircategoriesofrepresentationshascometoplayaroleinthetheoryofalge-
braic groups similar to that of Pontryagin duality in the theory of locally compact abelian
groups. Weincorporatethispointofview.
Let k be a field. Our approach to affine group schemes is eclectic.5 There are three
mainwaysviewingaffinegroupschemesoverk:
˘ asrepresentablefunctorsfromthecategoryofk-algebrastogroups;
˘ ascommutativeHopfalgebrasoverk;
˘ asgroupsinthecategoryofschemesoverk.
Allthreepointsofviewareimportant: thefirstisthemostelementaryandnatural;thesec-
ondleadstonaturalgeneralizations,forexample,affinegroupschemesinatensorcategory
and quantum groups; and the third allows one to apply algebraic geometry and to realize
1See,forexample,Cartier1962. WithoutnilpotentsthecentreofSLp incharacteristicp isvisibleonly
through its Lie algebra. Moreover, the standard isomorphism theorems fail (IX, 4.6), and so the intuition
provided by group theory is unavailable. While it is true that in characteristic zero all algebraic groups are
reduced,thisisatheoremthatcanonlybestatedwhennilpotentsareallowed.
2TheonlyexceptionsIknowofareDemazureandGabriel1970,Waterhouse1979,andSGA3.
3Worse, much of the expository literature is based, in spirit if not in fact, on the algebraic geometry of
Weil’sFoundations(Weil1962).Thusanalgebraicgroupoverkisdefinedtobeanalgebraicgroupoversome
largealgebraicallyclosedfieldtogetherwithak-structure. Thisleadstoaterminologyinconflictwiththatof
modernalgebraicgeometry,inwhich,forexample,thekernelofahomomorphismofalgebraicgroupsovera
fieldkneednotbeanalgebraicgroupoverk. Moreover,itpreventsthetheoryofsplitreductivegroupsbeing
developedintrinsicallyoverthebasefield.
When Borel first introduced algebraic geometry into the study of algebraic groups in the 1950s, Weil’s
foundationsweretheyonlyonesavailabletohim.WhenhewrotehisinfluentialbookBorel1969,hepersisted
inusingWeil’sapproachtoalgebraicgeometry,andsubsequentauthorshavefollowedhim.
4Strictly,thisshouldbecalledthe“dualityofTannaka,Krein,Milman,Hochschild,Grothendieck,Saave-
draRivano,Deligne,etal.,”but“tannakianduality”isshorter. InhisRe´coltesetSemailles,1985-86,18.3.2,
Grothendieckarguesthat“Galois-Poincare´”wouldbemoreappropriatethan“Tannaka”.
5Eclectic: Designating, of, or belonging to a class of ancient philosophers who selected from various
schoolsofthoughtsuchdoctrinesaspleasedthem.(OED).
7
affinegroupschemesasexamplesofgroupsinthecategoryofallschemes. Weemphasize
thefirstpointofview,butmakeuseofallthree. Wealsouseafourth: affinegroupschemes
aretheTannakadualsofcertaintensorcategories.
TERMINOLOGY
For readers familiar with the old terminology, as used for example in Borel 1969, 1991,
we point out some differences with our terminology, which is based on that of modern
(post-1960)algebraicgeometry.
˘ Weallowourringstohavenilpotents,i.e.,wedon’trequirethatouralgebraicgroups
bereduced.
˘ ForanalgebraicgroupG overk andanextensionfieldK, G.K/denotesthepoints
of G with coordinates in K and G denotes the algebraic group over K obtained
K
fromG byextensionofthebasefield.
˘ We do not identify an algebraic group G with its k-points G.k/, even when the
ground field k is algebraically closed. Thus, a subgroup of an algebraic group G is
analgebraicsubgroup,notanabstractsubgroupofG.k/.
˘ An algebraic group G over a field k is intrinsically an object over k, and not an
object over some algebraically closed field together with a k-structure. Thus, for
example,ahomomorphismofalgebraicgroupsoverk istrulyahomomorphismover
k,andnotoversomelargealgebraicallyclosedfield. Inparticular,thekernelofsuch
a homomorphism is an algebraic subgroup over k. Also, we say that an algebraic
group over k is simple, split, etc. when it simple, split, etc. as an algebraic group
over k, not over some large algebraically closed field. When we want to say that G
is simple over k and remains simple over all fields containing k, we say that G is
geometrically(orabsolutely)simple.
Beyonditsgreatersimplicityanditsconsistencywiththeterminologyofmodernalgebraic
geometry, there is another reason for replacing the old terminology with the new: for the
studyofgroupschemesoverbasesotherthanfieldsthereisnooldterminology.
Notations;terminology
We use the standard (Bourbaki) notations: N D f0;1;2;:::g; Z D ring of integers; Q D
fieldofrationalnumbers;RDfieldofrealnumbers;CDfieldofcomplexnumbers;F D
p
Z=pZD field with p elements, p a prime number. For integers m and n, mjn means that
mdividesn,i.e.,n2mZ. Throughoutthenotes,p isaprimenumber,i.e.,pD2;3;5;:::.
Throughoutk isthegroundring(alwayscommutative,andoftenafield),andRalways
denotes a commutative k-algebra. Unadorned tensor products are over k. Notations from
commutativealgebraareasinmyprimerCA(seebelow). Whenk isafield,ksep denotesa
separablealgebraicclosureofk andkal analgebraicclosureofk. ThedualHom .V;k/
k-lin
ofak-moduleV isdenotedbyV_. ThetransposeofamatrixM isdenotedbyMt.
Weusetheterms“morphismoffunctors”and“naturaltransformationoffunctors”inter-
0
changeably. ForfunctorsF andF fromthesamecategory,wesaythat“ahomomorphism
F.X/ ! F0.X/ is natural in X” when we have a family of such maps, indexed by the
objects X of the category, forming a natural transformation F !F0. For a natural trans-
formation˛WF !F0, weoftenwrite˛ forthemorphism˛.X/WF.X/!F0.X/. When
X
its action on morphisms is obvious, we usually describe a functor F by giving its action
8
X (cid:32) F.X/ on objects. Categories are required to be locally small (i.e., the morphisms
between any two objects form a set), except for the category A_ of functors A!Set. A
diagram A!B ⇒C is said to be exact if the first arrow is the equalizer of the pair of
arrows;inparticular,thismeansthatA!B isamonomorphism(cf. EGAI,Chap.0,1.4).
Hereisalistofcategories:
Category Objects Page
Alg commutativek-algebras
k
A_ functorsA!Set
Comod.C/ finite-dimensionalcomodulesoverC p.118
Grp (abstract)groups
Rep.G/ finite-dimensionalrepresentationsofG p.112
Rep.g/ finite-dimensionalrepresentationsofg
Set sets
Vec finite-dimensionalvectorspacesoverk
k
Throughout the work, we often abbreviate names. In the following table, we list the
shortenednameandthepageonwhichwebeginusingit.
Shortenedname Fullname Page
algebraicgroup affinealgebraicgroup p.28
algebraicmonoid affinealgebraicmonoid p.28
bialgebra commutativebi-algebra p.37
Hopfalgebra commutativeHopfalgebra p.37
groupscheme affinegroupscheme p.75
algebraicgroupscheme affinealgebraicgroupscheme p.75
groupvariety affinegroupvariety p.75
subgroup affinesubgroup p.109
representation linearrepresentation p.113
When working with schemes of finite type over a field, we typically ignore the nonclosed
points. Inotherwords,weworkwithmaxspecsratherthanprimespecs,and“point”means
“closedpoint”.
Weusethefollowingconventions:
X (cid:26)Y X isasubsetofY (notnecessarilyproper);
X DdefY X isdefinedtobeY,orequalsY bydefinition;
X (cid:25)Y X isisomorphictoY;
X 'Y X andY arecanonicallyisomorphic(orthereisagivenoruniqueisomorphism);
Passages designed to prevent the reader from falling into a possibly fatal error are sig-
nalledbyputtingthesymbolAinthemargin.
ASIDES may be skipped; NOTES should be skipped (they are mainly reminders to the
author). Thereissomerepetitionwhichwillberemovedinlaterversions.
9
Prerequisites
Althoughthetheoryofalgebraicgroupsispartofalgebraicgeometry,mostpeoplewhouse
it are not algebraic geometers, and so I have made a major effort to keep the prerequisites
to a minimum. The reader needs to know the algebra usually taught in first-year graduate
courses(andinsomeadvancedundergraduatecourses),plusthebasiccommutativealgebra
tobefoundinmyprimerCA.Familiaritywiththeterminologyofalgebraicgeometry,either
varietiesorschemes,willbehelpful.
References
Inadditiontothereferenceslistedattheend(andinfootnotes),Ishallrefertothefollowing
ofmynotes(availableonmywebsite):
CA APrimerofCommutativeAlgebra(v2.22,2011).
GT GroupTheory(v3.11,2011).
FT FieldsandGaloisTheory(v4.22,2011).
AG AlgebraicGeometry(v5.22,2012).
CFT ClassFieldTheory(v4.01,2011).
ThelinkstoCA,GT,FT,andAGinthepdffilewillworkifthefilesareplacedinthesame
directory.
Also,Iusethefollowingabbreviations:
BourbakiA Bourbaki,Alge`bre.
BourbakiAC Bourbaki,Alge`breCommutative(I–IV1985;V–VI1975;VIII–IX1983;X
1998).
BourbakiLIE Bourbaki,GroupesetAlge`bresdeLie(I1972;II–III1972;IV–VI1981).
DG DemazureandGabriel,GroupesAlge´briques,TomeI,1970.
EGA Ele´mentsdeGe´ome´trieAlge´brique,Grothendieck(avecDieudonne´).
SGA3 Sche´mas en Groupes (Se´minaire de Ge´ome´trie Alge´brique, 1962-64, Demazure,
Grothendieck,etal.);2011edition.
monnnnn http://mathoverflow.net/questions/nnnnn/
Sources
I list some of the works which I have found particularly useful in writing this book, and
whichmaybeusefulalsotothereader: DemazureandGabriel1970;Serre1993;Springer
1998;Waterhouse1979.
Acknowledgements
The writing of these notes began when I taught a course at CMS, Zhejiang University,
Hangzhou in Spring, 2005. I thank the Scientific Committee and Faculty of CMS for the
invitationtolectureatCMS,andthoseattendingthelectures,especiallyDingZhiguo,Han
Gang,LiuGongxiang,SunShenghao,XieZhizhang,YangTian,ZhouYangmei,andMunir
Ahmed,fortheirquestionsandcommentsduringthecourse.
I thank the following for providing comments and corrections for earlier versions of
thesenotes: BrianConrad,DarijGrinberg,LucioGuerberoff,FlorianHerzig,TimoKeller,
Chu-WeeLim,VictorPetrov,DavidVogan,JonathanWang,XiandongWang.
10