Algebraic Groups- An Introduction to the Theory of Algebraic Group Schemes Over Fields

Algebraic GroupsAn introduction to the theory of

algebraic group schemes over fields

J.S. Milne

DraftJanuary 29, 2015

An algebraic group is a matrix group defined by polynomial conditions. More abstractly,it is a group scheme of finite type over a field. These notes are a modern introduction tothe theory of algebraic groups assuming only the knowledge of algebraic geometry usuallyacquired in a first course.

Once the current revision is finished, the book will be complete mathematically, and itwill only remain to improve the exposition and add exercises for publication.

BibTeX information

@misc{milneiAG,

author={Milne, James S.},

title={Algebraic Groups (v1.20)},

year={2015},

note={Available at www.jmilne.org/math/},

pages={373}

}

v1.00 (July 31, 2014). First published on the web.

v1.20 (January 29, 2015). Revised Parts A,B.Available at www.jmilne.org/math/Please send comments and suggestions for improvements to me at .

The photo is of a grotto on The Peak That Flew Here, Hangzhou, Zhejiang, China.

Copyright c 2014, 2015 J.S. Milne.Single paper copies for noncommercial personal use may be made without explicit permissionfrom the copyright holder.

This book was written on a 2005 vintage Thinkpad T42p, the quality of whose keyboard and screen have not been surpassed.

Preface

This book represents my attempt to write a modern successor to the three standard works, alltitled Linear Algebraic Groups, by Borel, Humphreys, and Springer. More specifically,it is an introduction to the theory of algebraic group schemes over fields, based on modernalgebraic geometry, but with minimal prerequisites.

It has been clear for fifty years (at least, to people like Cartier) that such a work has beenneeded. When Borel, Chevalley, and others introduced algebraic geometry into the theory ofalgebraic groups, the foundations they used were those of Weil since these were the onlyones available to them, and most subsequent authors on algebraic groups have followedthem. Apart from a confusing conflict with the terminology of modern algebraic geometry,the main defect of this approach is that it doesnt allow the structure rings to have nilpotents.

Roughly speaking, the old approach chooses an algebraically closed field K containingthe base field k, and defines an algebraic group over k to be a subgroup of Kn described bypolynomials with coefficients in k. The modern approach defines an algebraic group over kto be a functor from k-algebras to groups, again described by polynomials with coefficientsin k. From a different perspective, the modern theory studies algebraic group schemes overa field, whereas the old theory considers only smooth algebraic group schemes.

The advantages of the modern approach are manifold. For example, the infinitesimaltheory is built into it from the start instead of entering only in an ad hoc fashion, through theLie algebra. The Noether isomorphisms theorems hold for algebraic group schemes, andso the intuition from abstract group theory applies. The kernel of a homomorphism over abase field k exists over k, and behaves as one would expect. The kernels of infinitesimalhomomorphisms become visible as algebraic group schemes.

The first systematic exposition of the theory of group schemes was in SGA 3. As wasnatural for its authors (Demazure, Grothendieck, . . . ), they worked over an arbitrary basescheme and they used the full theory of schemes (EGA and SGA).1 Most subsequent authorson group schemes have followed them. The only books I know of that give an elementarytreatment of group schemes are Waterhouse 1979 and Demazure and Gabriel 1970. Inwriting this book, I have relied heavily on both, but neither goes very far. For example,neither treats the structure theory of reductive groups, which is essential for the applications.

As noted, the modern theory is more general than the old theory. The extra generalitygives a more attractive and powerful theory, but it does not come for free: some proofs aremore difficult (because they prove stronger statements). In this work, I have avoided anyappeal to advanced scheme theory by passing to the algebraic closure where possible and byan occasional use of Hopf algebras. Unpleasantly technical arguments that I have not (sofar) been able to avoid have been placed in separate sections where they can be ignored byall but the most serious students. By considering only schemes algebraic over a field, weavoid many of the technicalities that plague the general theory. Also, the theory over a fieldhas many special features that do not generalize to arbitrary bases.

As much as possible, we work over an arbitrary base field, except that we have largelyignored the problems arising from inseparability in characteristic p, which would requireanother book.2

1They also assumed the main classification results of the old theory.2Happily, such a book exists: Conrad et al. 2010.

3

The experienced reader is cautioned that, throughout the text, algebraic group schemeis shortened to algebraic group, nonclosed points are ignored, and a group variety is asmooth algebraic group.

Equivalently, a group variety is group in the category of algebraic varieties (geometricallyreduced separated schemes of finite type over a field). However, it is important to note thatvarieties are always regarded as special algebraic schemes. For example, fibres of maps areto be taken in the sense of schemes, and the kernel of a homomorphism of group varietiesis an algebraic group which is not necessarily a group variety (it need not be smooth). Astatement here may be stronger than a statement in Borel 1991 or Springer 1998 even whenthe two are word for word the same.3

We use the terminology of modern (post 1960) algebraic geometry; for example, foralgebraic groups over a field k; a homomorphism is (automatically) defined over k, not oversome large algebraically closed field.

The following works have been especially useful to me.Demazure, Michel; Gabriel, Pierre. Groupes algebriques. Tome I: Geometrie algebrique,

generalites, groupes commutatifs. Masson & Cie, Editeur, Paris; North-Holland PublishingCo., Amsterdam, 1970. xxvi+700 pp.

Seminaire Heidelberg-Strasbourg 196566 (Groupes Algebriques), multigraphie parlInstitut de Mathematique de Strasbourg (Gabriel, Demazure, et al.). 407 pp.

The expository writings of Springer, especially: Springer, T. A., Linear algebraic groups.Second edition. Progress in Mathematics, 9. Birkhauser Boston, Inc., Boston, MA, 1998.xiv+334 pp.

Waterhouse, William C., Introduction to affine group schemes. Graduate Texts inMathematics, 66. Springer-Verlag, New York-Berlin, 1979. xi+164 pp.

Notes of Ngo and Perrin have also been useful.

3An example is Chevalleys theorem on representations; see 5.21.

4

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Contents 5Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1 Basic definitions and properties 15Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Properties of algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Algebraic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23The algebraic subgroup generated by a map (1) . . . . . . . . . . . . . . . . . . 24The algebraic subgroup generated by a map (2) . . . . . . . . . . . . . . . . . . 25Closed subfunctors: definitions and statements . . . . . . . . . . . . . . . . . . . 26Transporters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Normalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Closed subfunctors: proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Examples of algebraic groups and morphisms 35The comultiplication map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Examples of affine algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . 36Examples of homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Some basic constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Restriction of the base field (Weil restriction of scalars) . . . . . . . . . . . . . . 39Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Affine algebraic groups and Hopf algebras 45Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Hopf algebras and algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . 46Hopf subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Hopf subalgebras of O.G/ versus algebraic subgroups of G . . . . . . . . . . . . 48Subgroups of G.k/ versus algebraic subgroups of G . . . . . . . . . . . . . . . . 48Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Algebraic groups in characteristic zero are smooth 51

5

Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51The theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Smoothness in characteristic p 0 . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Linear representations of algebraic groups 55Representations and comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Stabilizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56All representations are unions of finite-dimensional representations . . . . . . . . 57Affine algebraic groups are linear . . . . . . . . . . . . . . . . . . . . . . . . . . 58Constructing all finite-dimensional representations . . . . . . . . . . . . . . . . . 58Semisimple representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Characters and eigenspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Chevalleys theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63The subspace fixed by a group . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 Group theory; the isomorphism theorems 67Normal algebraic subgroups are kernels . . . . . . . . . . . . . . . . . . . . . . 67The homomorphism theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Semidirect products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74The isomorphism theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75The correspondence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76The category of commutative algebraic groups . . . . . . . . . . . . . . . . . . . 77Sheaf theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Faithful flatness for Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . 81Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7 Subnormal series; solvable and nilpotent algebraic groups 85Subnormal series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Composition series for algebraic groups . . . . . . . . . . . . . . . . . . . . . . 87Solvable and nilpotent algebraic groups . . . . . . . . . . . . . . . . . . . . . . 89

8 Finite algebraic groups 95Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Etale algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Commutative finite algebraic groups over a perfect field . . . . . . . . . . . . . . 98Cartier duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

9 The connected components of an algebraic group 103Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Existence of a greatest connected normal subgroup variety with a given property . 108Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

10 Algebraic groups acting on schemes 111Group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111The fixed subvariety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

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The fixed subscheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Orbits and isotropy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113The functor defined by projective space . . . . . . . . . . . . . . . . . . . . . . 116Quotients: definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Quotients: construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Existence of G=H when G is not reduced . . . . . . . . . . . . . . . . . . . . . 120Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

11 The structure of general algebraic groups 123Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124Local actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Anti-affine algebraic groups and abelian varieties . . . . . . . . . . . . . . . . . 126Rosenlichts decomposition theorem. . . . . . . . . . . . . . . . . . . . . . . . . 127Rosenlichts dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128The Barsotti-Chevalley theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 129Anti-affine groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

12 Tannaka duality; Jordan decompositions 133Recovering a group from its representations . . . . . . . . . . . . . . . . . . . . 133Application to Jordan decompositions . . . . . . . . . . . . . . . . . . . . . . . 136Characterizations of categories of representations . . . . . . . . . . . . . . . . . 141

13 The Lie algebra of an algebraic group 145Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145The Lie algebra of an algebraic group . . . . . . . . . . . . . . . . . . . . . . . 146Basic properties of the Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . 148The adjoint representation; definition of the bracket . . . . . . . . . . . . . . . . 149Stabilizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Centres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Normalizers and centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153An example of Chevalley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154The universal enveloping algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 155The universal enveloping p-algebra . . . . . . . . . . . . . . . . . . . . . . . . 156Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

14 Tori; groups of multiplicative type 161The characters of an algebraic group . . . . . . . . . . . . . . . . . . . . . . . . 161The algebraic group D.M/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Diagonalizable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Diagonalizable representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166Groups of multiplicative type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166Representations of a group of multiplicative type . . . . . . . . . . . . . . . . . 168Criteria for an algebraic group to be of multiplicative type . . . . . . . . . . . . . 168Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

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Unirationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174Linearly reductive groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

15 Unipotent algebraic groups 177Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177Preliminaries from linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 177Unipotent algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Unipotent algebraic groups in characteristic zero . . . . . . . . . . . . . . . . . . 185Unipotent algebraic groups in nonzero characteristic . . . . . . . . . . . . . . . . 189Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

16 Cohomology and extensions 195Nonabelian cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Abelian cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196Applications to homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 202Applications to centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203Calculation of some extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

17 The structure of solvable algebraic groups 215Trigonalizable algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 215Commutative algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Structure of trigonalizable algebraic groups . . . . . . . . . . . . . . . . . . . . 220Solvable algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224Nilpotent algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226Split solvable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228Complements on unipotent algebraic groups . . . . . . . . . . . . . . . . . . . . 229Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

18 Borel subgroups 231Borel fixed point theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231Borel subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234The density theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239Centralizers of tori are connected . . . . . . . . . . . . . . . . . . . . . . . . . . 240The normalizer of a Borel subgroup . . . . . . . . . . . . . . . . . . . . . . . . 243Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

19 Algebraic groups of semisimple rank at most 1 247Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247Review of Borel subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248Actions of tori on algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . . 249Actions of tori on a projective space . . . . . . . . . . . . . . . . . . . . . . . . 251Homogeneous curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252The automorphism group of the projective line . . . . . . . . . . . . . . . . . . . 252A generalization of the Borel fixed point theorem . . . . . . . . . . . . . . . . . 254Limits in solvable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254Algebraic groups of semisimple rank one (proof of 19.3) . . . . . . . . . . . . . 255Proof of Theorem 19.5: the kernel of the homomorphism to PGL2 is the centre of G.257

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Proof of the classification theorem 19.6 . . . . . . . . . . . . . . . . . . . . . . 259General base fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

20 The variety of Borel subgroups 265The variety of Borel subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 265Chevalleys theorem: intersection of the Borel subgroups containing a maximal torus267Proof of Chevalleys theorem (Luna) . . . . . . . . . . . . . . . . . . . . . . . . 268Regular tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271Proof of Chevalleys theorem (SHS) . . . . . . . . . . . . . . . . . . . . . . . . 272The big cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

21 Semisimple groups, reductive groups, and central isogenies 279Definition of semisimple and reductive groups . . . . . . . . . . . . . . . . . . . 279The canonical filtration on an algebraic group . . . . . . . . . . . . . . . . . . . 282Central isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283Pseudoreductive groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289Reductive groups in characteristic zero . . . . . . . . . . . . . . . . . . . . . . . 290Properties of G versus those of Repk.G/: a summary . . . . . . . . . . . . . . . 293Levi factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294Appendix: Proof of Theorem 21.30 . . . . . . . . . . . . . . . . . . . . . . . . . 294

22 Reductive algebraic groups and their root data 297Maximal tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297The Weyl group of .G;T / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298Root data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299The roots of a split reductive group . . . . . . . . . . . . . . . . . . . . . . . . . 299Split reductive groups of rank 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 302The root datum of a split reductive group . . . . . . . . . . . . . . . . . . . . . . 304The centre of a reductive group . . . . . . . . . . . . . . . . . . . . . . . . . . . 306Semisimple and toral root data . . . . . . . . . . . . . . . . . . . . . . . . . . . 307Statement of the main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

23 Semisimple algebraic groups and their root systems 309Split semisimple algebraic groups and their root systems. . . . . . . . . . . . . . 309Automorphisms of a semisimple algebraic group . . . . . . . . . . . . . . . . . 309The decomposition of a semisimple algebraic group . . . . . . . . . . . . . . . . 310Complements on reductive groups . . . . . . . . . . . . . . . . . . . . . . . . . 312Simply connected semisimple algebraic groups . . . . . . . . . . . . . . . . . . 312Classification of split almost-simple algebraic groups: statements . . . . . . . . . 313The root data of the classical semisimple groups . . . . . . . . . . . . . . . . . . 314Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

24 Root data and their classification 319Equivalent definitions of a root datum . . . . . . . . . . . . . . . . . . . . . . . 319Deconstructing root data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321Semisimple root data and root systems . . . . . . . . . . . . . . . . . . . . . . . 321Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

9

25 Representations of split reductive groups 331Classification in terms of roots and weights . . . . . . . . . . . . . . . . . . . . 331Grothendieck groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331Semisimplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

26 The isogeny theorem 333Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

27 The existence theorem 337

28 Further Topics 339

A Review of algebraic geometry 341Affine algebraic schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341Algebraic schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343Subschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344Algebraic schemes as functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 345Fibred products of algebraic schemes . . . . . . . . . . . . . . . . . . . . . . . . 348Algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348The dimension of an algebraic scheme . . . . . . . . . . . . . . . . . . . . . . . 349Tangent spaces; smooth points; regular points . . . . . . . . . . . . . . . . . . . 350Galois descent for closed subschemes . . . . . . . . . . . . . . . . . . . . . . . 352On the density of points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353Dominant maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353Separated maps; affine maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354Finite schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354Finite algebraic varieties (etale schemes) . . . . . . . . . . . . . . . . . . . . . . 355The algebraic variety of connected components of an algebraic scheme . . . . . . 355Flat maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356Finite maps and quasi-finite maps . . . . . . . . . . . . . . . . . . . . . . . . . 356The fibres of regular maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357Etale maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358Smooth maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358Complete algebraic schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359Proper maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359Algebraic schemes as flat sheaves . . . . . . . . . . . . . . . . . . . . . . . . . 360Restriction of the base field (Weil restriction of scalars) . . . . . . . . . . . . . . 361

B Dictionary 363Demazure and Gabriel 1970 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363Borel 1969/1991; Springer 1981/1998 . . . . . . . . . . . . . . . . . . . . . . . 364Waterhouse 1979 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

C Solutions to the exercises 365

Bibliography 367

Index 371

10

Notations and conventions

Throughout, k is a field and R is a k-algebra.4 All algebras over a field or ring are requiredto be commutative and finitely generated unless it is specified otherwise. Unadorned tensorproducts are over k. An extension of k is a field containing k. When V is a vector spaceover k, we often write VR or V.R/ for RV . The symbol kal denotes an algebraic closureof k, and ksep denotes the separable closure of k in kal.

An algebraic scheme over k (or algebraic k-scheme) is a scheme of finite type over k(EGA I, 6.5.1). An algebraic variety is a geometrically-reduced separated algebraic scheme.A point of an algebraic scheme or variety means closed point.5 For an algebraic scheme.X;OX / over k, we often let X denote the scheme and jX j the underlying topological spaceof closed points. When the base field k is understood, we write algebraic scheme foralgebraic scheme over k.

Let R be a finitely generated k-algebra. We let AlgR denote the category of finitelygenerated R-algebras. We fix a family of symbols .Ti /i2N indexed by N, and let Alg0Rdenote the category of R-algebras of the form RT0; : : : ;Tn=a for some n 2N and ideal a inRT0; : : : ;Tn. We call the objects of Alg0R small R-algebras. We fix a bijection N$ NN.When R is a small k-algebra, this allows us to realize a small R-algebra as a small k-algebra,and so define a forgetful functor Alg0R! Alg

0k

.A functor is said to be an equivalence of categories if it is fully faithful and essentially

surjective. For example, the inclusion Alg0k,! Algk is an equivalence because every finitely

generated k-algebra is isomorphic to a small k-algebra. The axiom of global choice impliesthat there exists a quasi-inverse to this inclusion functor specifically, one has to choose afinite ordered set of generators for every finitely generated k-algebra. Once a quasi-inversehas been chosen, every functor on Alg0

khas a well-defined extension to Algk (we dont need

this).6

An element g of a partially ordered set P is a greatest element if, for every element a inP , a g. An element m in P is maximal if, for a in P , m a implies aDm. If a partiallyordered set has a greatest element, it must be the unique maximal element, but otherwisethere can be more than one maximal element (or none). Least and minimal elements aredefined similarly.

A diagram A! B C is said to be exact if the first arrow is the equalizer of the pairof arrows.

After p.132, all algebraic groups are affine. (The reader may wish to assume throughoutthat all algebraic groups are affine, and skip Chapter 11.)

Foundations

We use the von NeumannBernaysGodel (NBG) set theory with the axiom of choice, whichis a conservative extension of ZermeloFraenkel set theory with the axiom of choice (ZFC).

4Except when it is root system.5Let X be an algebraic scheme over a field, and let X0 be the set of closed points in X with the induced

topology. Then the map U 7! U \X0 is a bijection from the set of open subsets of X onto the set of opensubsets of X0. Thus, for example, X is connected if and only if X0 is connected.

6When working with the category of functors on k-algebras, it is important that they be functors on smallk-algebras (otherwise the Hom sets need not be sets). Our point of view is that an algebraic scheme over afield k is determined by the functor it defines on small k-algebras, and defines a functor on all k-algebras (cf.A.27). An alternative approach is to use universes, but this requires assuming additional set-theoretic axioms.The reader is invited to ignore such questions.

11

This means that a sentence that doesnt quantify over proper classes is a theorem of NBG ifand only if it is a theorem of ZFC. The advantage of NBG is that it allows us to speak ofclasses.

We allow the objects of a category to form a class, but require the morphisms from oneobject to a second to form a set. When the objects form a set, we say that the category small.For example, Alg0

kis small.

Prerequisites

A first course in algebraic geometry. Since these vary greatly, we review the definitions andstatements that we need from algebraic geometry in Appendix A.

References

In addition to the references listed at the end (and in footnotes), I shall refer to the followingof my notes (available on my website):

AG Algebraic Geometry (v6.00, 2014).

CA A Primer of Commutative Algebra (v4.01, 2014).

LAG Lie Algebras, Algebraic Groups, and Lie Groups (v2.00, 2013).I also refer to:

DG Demazure, Michel; Gabriel, Pierre. Groupes algebriques. Tome I: Geometrie algebrique,generalites, groupes commutatifs. Masson & Cie, Editeur, Paris; North-Holland Pub-lishing Co., Amsterdam, 1970. xxvi+700 pp.

SHS Seminaire Heidelberg-Strasbourg 196566 (Groupes Algebriques), multigraphie parlInstitut de Mathematique de Strasbourg (Gabriel, Demazure, et al.). 407 pp.

SGA 3 Schemas en Groupes, Seminaire de Geometrie Algebriques du Bois Marie 196264,dirige par M. Demazure et A. Grothendieck. Revised edition (P. Gille and P. Poloeditors), Documents Mathematiques, SMF, 2011.

EGA Elements de Geometrie Algebrique, A. Grothendieck; J. A. Dieudonne; I, Le langagedes schemas (Springer Verlag 1971); II, III, IV Inst. Hautes Etudes Sci. Publ. Math. 8,11, 17, 20, 24, 28, 32 , 19611967.

A reference monnnn is to question nnnn on mathoverflow.net.

Introduction

The work can be divided roughly into six parts.

A. BASIC THEORY (CHAPTERS 111; p.15p.132)

The first eleven chapters cover the general theory of algebraic groups, emphasizing affinealgebraic groups. After defining algebraic groups and giving some examples, we show thatmost of the basic theory of abstract groups (subgroups, normal subgroups, normalizers,centralizers, Noether isomorphism theorems, subnormal series, etc.) carries over with littlechange to algebraic group schemes. We relate affine algebraic groups to Hopf algebras, and

12

Introduction 13

we prove that all affine algebraic groups in characteristic zero are smooth. We study thelinear representations of algebraic groups and the actions of algebraic groups on algebraicschemes. We show that every algebraic group is an extension of a finite etale algebraic groupby a connected algebraic group, and that every connected group variety over a perfect fieldis an extension of an abelian variety by an affine group variety (Barsotti-Chevalley theorem).

B. SOLVABLE ALGEBRAIC GROUPS (CHAPTERS 12-17; p.133p.230)

The next six chapters study solvable algebraic groups. Among these are the diagonalizablegroups and the unipotent groups.

An algebraic group G is diagonalizable if every linear representation r WG! GLV ofG is a direct sum of one-dimensional representations. In other words if, relative to somebasis for V , r.G/ lies in the algebraic subgroup Dn of diagonal matrices in GLn. Analgebraic group that becomes diagonalizable over an extension of the base field is said to beof multiplicative type.

An algebraic group G is unipotent if every nonzero representation V of G containsa nonzero fixed vector. This implies that, relative to some basis for V , r.G/ lies in thealgebraic subgroup Un of strictly upper triangular matrices in GLn.

Every smooth connected solvable algebraic group over a perfect field is an extension ofa group of multiplicative type by a unipotent group.

C. THE STRUCTURE OF AFFINE ALGEBRAIC GROUPS (CHAPTERS 1823;p.231p.318)

The next six chapters are the heart of the book.

D. THE CLASSIFICATION OF SPLIT REDUCTIVE GROUPS AND THEIRREPRESENTATIONS (CHAPTERS 24-28; p.319p.338)

The next five chapters prove the fundamental classification theorems in terms of roots andweights.

E. SURVEY OF FURTHER TOPICS (CHAPTER 28)

The final chapter surveys the following topics: the Galois cohomology of algebraic groups;classification of the forms of an algebraic group; description of the classical algebraicgroups in terms of algebras with involution; relative root systems and the anistropic kernel;classification of (nonsplit) reductive groups (Satake-Tits).

APPENDICES

In an appendix, we review the algebraic geometry needed.

CHAPTER 1Basic definitions and properties

Recall that k is a field, and that an algebraic k-scheme is a scheme of finite type over k.

Definition

An algebraic group over k is a group object in the category of algebraic schemes over k.

DEFINITION 1.1 Let G be an algebraic scheme over k and let mWGG!G be a regularmap. The pair .G;m/ is an algebraic group1 over k if there exist regular maps

eW !G; invWG!G (1)

such that the following diagrams commute:

GGG GG

GG G

mid

idm

m

m

G GG G

G

eid

'm

ide

'

(2)

G GG G

G

.inv;id/

m

.id;inv/

e e

(3)

Here is the one-point variety Spm.k/. When G is a variety, we call .G;m/ a group variety.A homomorphism 'W.G;m/! .G0;m0/ of algebraic groups is a regular map 'WG! G0such that ' mDm0 .''/.

Similarly, an algebraic monoid over k is an algebraic scheme M over k together withregular maps mWM M !M and eW !M such that the diagrams (2) commute.

An algebraic group G is trivial if eW ! G is an isomorphism, and a homomorphism'W.G;m/! .G0;m0/ is trivial if it factors through e0W !G0.

For example,

SLndefD SpmkX11;X12; : : : ;Xnn=.det.Xij /1/

1As we note elsewhere (p.3, p.4, 1.40, 2.13, 6.41, p.363) in most of the current literature, an algebraic groupover a field k is defined to be a group variety over some algebraically closed field K containing k together witha k-structure. In particular, nilpotents are not allowed.

15

16 1. Basic definitions and properties

becomes a group variety with the usual matrix multiplication,

.aij /; .bij / 7! .cij /; cij DPl ailblj .

DEFINITION 1.2 An algebraic subgroup of an algebraic group .G;mG/ is an algebraicgroup .H;mH / such that H is an algebraic subscheme of G and the inclusion map is ahomomorphism of algebraic groups. An algebraic subgroup that is a variety is called asubgroup variety.

NOTES

1.3 For an algebraic scheme X over k, we write jX j for the underlying topological spaceof X , and .x/ for the residue field at a point x of jX j (it is a finite extension of k). Weidentify X.k/ with the set of points x of jX j such that .x/D k. Let .G;m/ be an algebraicgroup over k. The map m.k/WG.k/G.k/!G.k/ makes G.k/ into a group with neutralelement e./ and inverse map inv.k).

When k is algebraically closed, G.k/ D jGj, and so mWG G ! G makes jGj intoa group. The maps x 7! x1 and x 7! ax (a 2 G.k/) are automorphisms of jGj as atopological space.

In general, when k is not algebraically closed, m does not make jGj into a group, andeven when k is algebraically closed, it does not make jGj into a topological group.

1.4 Let G be an algebraic group over a field k, and let k0 be a field containing k.

(a) We say that G.k0/ is dense in G if the only closed algebraic subscheme Z of G suchthat Z.k0/DG.k0/ is G itself.

(b) If G.k0/ is dense in G, then a homomorphism G!H of algebraic groups is deter-mined by its action on G.k0/ (algebraic groups are separated as schemes 1.8).

(c) If G.k0/ is dense in G, then G is reduced. Conversely, if G is geometrically reduced,then G.k0/ is dense in G if and only if it is dense in the topological space jGk0 j.

(d) If G is a group variety, then G.ksep/ is dense in G.

(e) G.k/ is dense in G if and only if G is reduced and G.k/ is dense jGj.

See A.55A.59.

1.5 Let .G;m/ be an algebraic group over k. For each a 2G.k/, there is a translation map

laWG ' fagGm!G; x 7! ax.

For a;b 2G.k/,la lb D lab

and le D id. Therefore la la1 D idD la1 la , and so la is an isomorphism sending e toa. Hence G is homogeneous2 when k is algebraically closed (but not in general otherwise;see 1.12).

2An algebraic scheme X over k is said to be homogeneous if the group of automorphisms of X actstransitively on jX j.

Properties of algebraic groups 17

1.6 Let R be a (finitely generated) k-algebra. An algebraic R-scheme is an algebraicscheme X over k together with a morphism X ! SpmR. A morphism of algebraic R-schemes is a morphism of algebraic k-schemes compatible with the maps to SpmR (seeA.31). To give the structure of an R-scheme on an algebraic k-scheme X is the same asgiving an R-algebra structure on OX compatible with its k-algebra structure. Let G be analgebraic scheme over R and let mWGG!G be an R-morphism. The pair .G;m/ is analgebraic group over R is there exist R-morphisms eWSpm.R/!G and invWG!G suchthat the diagrams (2) and (3) commute. For example, an algebraic group .G;m/ over k givesrise to an algebraic group .GR;mR/ over R by extension of scalars.

Properties of algebraic groups

PROPOSITION 1.7 The maps e and inv in (1.1) are uniquely determined by .G;m/. If'W.G;mG/! .H;mH / is a homomorphism of algebraic groups, then ' eG D eH and' invG D invH '.

PROOF. It suffices to prove the second statement. For a k-algebra R, the map '.R/ is ahomomorphism of abstract groups .G.R/;mG.R//! .H.R/;mH .R//, and so it maps theneutral element of G.R/ to that of H.R/ and the inversion map on G.R/ to that on H.R/.The Yoneda lemma (A.27) now shows that the same is true for '. 2

PROPOSITION 1.8 Algebraic groups are separated (as algebraic schemes).

PROOF. Let .G;m/ be an algebraic group. The diagonal inGG is the inverse image of theclosed point e 2G.k/ under the map m .id inv/WGG!G sending .g1;g2/ to g1g12 ,and so it is closed. 2

Therefore group variety = geometrically reduced algebraic group.Recall that an algebraic scheme over a field is a finite disjoint union of its (closed-open)

connected components (A.14). For an algebraic group G, we let G denote the connectedcomponent of G containing e, and we call it the identity (or neutral) component of G.

PROPOSITION 1.9 Let G be an algebraic group. The identity component G of G is analgebraic subgroup of G. Its formation commutes with extension of the base field: for everyfield k0 containing k,

Gk0' .Gk0/

:

In particular, G is connected if and only if Gk0 is connected, and G is geometricallyconnected; a connected algebraic group is geometrically connected.

For the proof, we shall need the following elementary lemma. Recall (A.79) that, amongthe regular maps from an algebraic scheme X to a zero-dimensional algebraic variety,there is one X ! 0.X/ that is universal. The fibres of jX j ! j0.X/j are the connectedcomponents of jX j.

LEMMA 1.10 Let X be a connected algebraic scheme over k such that X.k/ ;. Then Xis geometrically connected; moreover, for any algebraic scheme Y over k,

0.X Y /' 0.Y /:

In particular, X Y is connected if Y is connected.


PROOF. Because 0.X/ is a zero-dimensional algebraic variety, it equals Spm.A/ for someetale k-algebra A (A.77). Because X is connected, A is a separable field K extension of k,and because X.k/ is nonempty, K D k. Now

0.Xkal/A.79D 0.X/kal D Spm.k

al/,

which shows that Xkal is connected, and

0.X Y /A.79' 0.X/0.Y /' 0.Y /;

as required. 2

PROOF (OF 1.9) The identity componentG ofG has a k-point, namely, e, and soGG

is a connected component of GG (1.10). As m maps .e;e/ to e, it maps GG into G.Similarly, inv maps G into G. It follows that G is an algebraic subgroup of G. For anyextension k0 of k,

.G! 0.G//k0 'Gk0 ! 0.Gk0/

(see A.79). As G is the fibre over e, this implies that .G/k0 ' .Gk0/. In particular,.G/kal ' .Gkal/

, and so G is geometrically connected. 2

COROLLARY 1.11 A connected algebraic group is irreducible.

PROOF. It suffices to show thatG is geometrically irreducible. Thus, we may suppose that kis algebraically closed, and hence that G is homogeneous (1.5). By definition, no irreduciblecomponent is contained in the union of the remainder. Therefore, there exists a point thatlies on exactly one irreducible component. By homogeneity, all points have this property,and so the irreducible components are disjoint. As jGj is connected, there must be only one,and so G is irreducible. 2

1.12 We saw in (1.9) that the identity component of an algebraic group is geometricallyconnected. The other connected components need not by geometrically connected. Consider,for example, 3 D Spm.QT =T 31/. This becomes an algebraic group over Q with theobvious multiplication map (2.7). Note that

Spm.QT =.T 31/D Spm.Q/tSpm.QT =.T 2CT C1//

(disjoint union of one-point sets). The identity component, Spm.Q/, of 3 is geometricallyconnected, but the remaining connected component has two geometric components:

Spm.QT =.T 2CT C1//Spm.Q/ Spm.Qal/D Spm..QalT =.T 2CT C1///D Spm.Qal/tSpm.Qal/;

This reflects the fact that the algebraic group 3 over Q is not homogeneous.

PROPOSITION 1.13 Let G be an algebraic group over k.

(a) If G is reduced and k is perfect, then G is geometrically reduced (hence a groupvariety).

(b) If G is geometrically reduced, then it is smooth (and conversely).

Properties of algebraic groups 19

PROOF. (a) This is true for any algebraic scheme (A.37).(b) It suffices to show that Gkal is smooth, but some point of Gkal is smooth (A.50), and

so every point is smooth because Gkal is homogeneous (1.5). 2

Therefore group variety = smooth algebraic group. In characteristic zero, all al-gebraic groups are smooth (see 4.4 below for a proof in the affine case and 11.36 for thegeneral case).

1.14 There do exist reduced algebraic groups that are not geometrically reduced (see 2.13below).

DEFINITION 1.15 An algebraic group .G;m/ is commutative if m t Dm, where t is thetransposition map .x;y/ 7! .y;x/WGG!GG.

PROPOSITION 1.16 An algebraic group G is commutative if and only if G.R/ is commuta-tive for all k-algebras R. A group variety G is commutative if G.ksep/ is commutative.

PROOF. According to the Yoneda lemma (A.27), m t Dm if and only if m.R/ t .R/Dm.R/ for all k-algebras R, i.e., if and only if G.R/ is commutative for all R. The proves thefirst statement. Let G be a group variety. If G.ksep/ is commutative, then m t and m agreeon .GG/.ksep/, which is dense in GG (1.4). 2

PROPOSITION 1.17 The following conditions on an algebraic group G are equivalent:

(a) G is smooth;

(b) G is smooth;

(c) the local ring OG;e is regular;(d) the tangent space Te.G/ to G at e has dimension dimG;

(e) G is geometrically reduced;

(f) for all k-algebras R and all ideals I in R such that I 2 D 0, the map G.R/!G.R=I /is surjective.

PROOF. (a)H) (b)H) (c)H) (d): These implications are obvious (see A.46, A.49).(d)H) (a). The condition implies that the point e is smooth on G (A.49), and hence on

Gkal . By homogeneity (1.5), all points on Gkal are smooth, which means that G is is smooth.(a) (e). This was proved in (1.13).(a) (f). This is a standard criterion for an algebraic scheme to be smooth (A.51).2

COROLLARY 1.18 For an algebraic group G,

dimTe.G/ dimG;

with equality if and only if G is smooth.

PROOF. In general, for a point e on an algebraic k-scheme G with .e/D k, dimTe.G/dimG with equality if and only if OG;e is regular (A.46). But we know (1.17), that OG;e isregular if and only if G is smooth. 2


Algebraic subgroups

EXAMPLE 1.19 Let G be a group variety over an algebraically closed field k, and let S bea subgroup of G.k/. The Zariski-closure NS of S is also a subgroup of G.k/.3 The (unique)algebraic subvariety H of G such that jH j D NS is preserved by the regular maps m, e, inv(because H.k/ is dense in H ), and so H a subgroup variety of G.

Recall that for an algebraic scheme X , Xred denotes the (unique) reduced subscheme ofX with the same underlying topological space (A.24).

PROPOSITION 1.20 Let G be an algebraic group over k. If Gred is geometrically reduced,then it is an algebraic subgroup of G.

PROOF. If Gred is geometrically reduced, then GredGred is reduced (A.37), and so therestriction of m to GredGred factors through Gred ,!G:

GredGredmred!Gred ,!G.

Similarly, e and inv induce maps !Gred and Gred!Gred, and these make the diagrams(2, 3), p.15, commute for .Gred;mred/. 2

COROLLARY 1.21 LetG be an algebraic group over k. If k is perfect, thenGred is a smoothalgebraic subgroup of G.

PROOF. When k is perfect, Gred is geometrically reduced (1.13), hence an algebraic sub-group (1.20), and hence smooth (1.17). 2

PROPOSITION 1.22 Every algebraic subgroup of an algebraic group is closed (in the Zariskitopology).

PROOF. Let H be an algebraic subgroup of G. We may suppose that k is algebraicallyclosed, and then that H and G are group varieties (because passing to the reduced algebraicsubgroup doesnt change the underlying topological space). By definition, jH j is locallyclosed, i.e., open in its closure S . Now S is a subgroup of jGj (see 1.19), and it is a finitedisjoint union of the cosets of jH j in S . As each coset is open, each coset is also closed. Inparticular, H is closed in S , and so equals it. 2

COROLLARY 1.23 The algebraic subgroups of an algebraic group satisfy the descendingchain condition.

PROOF. For any algebraic scheme X , the topological space jX j is noetherian (A.14), and soits closed subsets satisfy the descending chain condition. 2

COROLLARY 1.24 Every algebraic subgroup of an affine algebraic group is affine.

PROOF. Closed subschemes of affine algebraic schemes are affine (A.19). 2

COROLLARY 1.25 Let H and H 0 be subgroup varieties of an algebraic group G. IfH.ksep/DH 0.ksep/, then H DH 0.

3Let S be a subgroup of G.k/. The map f WG.k/G.k/! G, f .x;y/D xy1, is continuous, and sof 1. NS/ is closed. We have S S f 1. NS/, and so NS NS S S f 1. NS/. In other words, f . NS NS/ NSas required.

Algebraic subgroups 21

PROOF. The condition implies that H.ksep/D .H \H 0/.ksep/. But H \H 0 is closed inH (1.22). As H is a variety, H.ksep/ is dense in H (A.59), and so H \H 0 DH . Similarly,H \H 0 DH 0. 2

ASIDE 1.26 Let k be an infinite perfect field. Then G.k/ is dense in G if G is a connected groupvariety over k (see 3.22 below). Therefore, if H and H 0 are connected subgroup varieties of analgebraic group G over k such that H.k/DH 0.k/, then H DH 0.

DEFINITION 1.27 Let G be an algebraic group.(a) An algebraic subgroupH ofG is normal ifH.R/ is normal inG.R/ for all k-algebras

R.

(b) An algebraic subgroup H of G is characteristic if .HR/DHR for all k-algebrasR and all automorphisms of GR.

It is only necessary to require that the conditions hold for small k-algebras, in which casethey hold for all k-algebras. In (b) GR and HR can be interpreted as functors from thecategory of (small) finitely generated R-algebras to the category of groups, or as algebraicR-schemes (i.e., as algebraic k-schemes equipped with a morphism to Spm.R/ (A.31)).Because of the Yoneda lemma (loc. cit.), the two interpretations give the same condition.

PROPOSITION 1.28 The identity component G of a group variety G is a characteristicsubgroup (in particular, a normal subgroup).

PROOF. As G is the unique connected open subgroup of G containing e, every automor-phism of G fixing e maps G into itself. Let k0 be a field containing k. As .G/k0 D .Gk0/,every automorphism of Gk0 fixing e maps .G/k0 into itself.

Let R be a k-algebra and let be an automorphism of GR. We regard GR and GR asalgebraic R-schemes. It suffices to show that .GR/ G

R, and, because G

R is an open

subscheme of GR, for this it suffices to show that .jGRj/ jGRj. Let x 2 jG

Rj, and let s

be the image of x in Spm.R/. Then x lies in the fibre G.s/ of GR over s:

GR G.s/

Spm.R/ Spm..s//:

In fact, x 2 jGR\G.s/j D jG.s/j. From the first paragraph of the proof, .s/.x/ 2 jG.s/j,

and so .x/ 2 jGRj, as required. 2

1.29 Let H be an algebraic subgroup of G. If .HR/ HR for all k-algebras R andendomorphisms of GR, then H is characteristic. To see this, let be an automorphism ofGR. Then 1.HR/HR, and so HR .HR/HR.

ASIDE 1.30 Let N be an algebraic subgroup of G. It is possible that .N /D N for all automor-phisms of G without N being characteristic, or even normal. For example, when k is perfect, Gredis an algebraic subgroup of G, and .Gred/D Gred for all automorphisms of G (obviously), butGred is not necessarily normal in G (see 1.35 below). As another example, a commutative algebraicgroup G over a perfect field contains a greatest (unique maximal) unipotent subgroup U . ClearlyU D U for all automorphisms of G, but U need not be characteristic (see 17.22 below).

ASIDE 1.31 The definition of characteristic subgroup agrees with DG II, 1, 3.9, p.166. The proofthat G is characteristic is from DG II, 5, 1.1, p.334.


NOTES

1.32 (Descent). LetG be an algebraic scheme over a field k, and let k0 be a field containingk. Let G0 DGk0 , and let H 0 be an algebraic subgroup of Gk0 .

(a) There exists at most one algebraic subgroup H of G such that Hk0 D H 0 (as analgebraic subgroup of Gk0). When such an H exists, we say that H 0 is defined over k(as an algebraic subgroup of G0).

(b) Let k0 be a Galois extension of k (possibly infinite), and let DGal.k0=k/. Then H 0

is defined over k if and only if it is stable under the action of on G0, i.e., the sheafof ideals defining it is stable under the action of on OG0 .

(c) Let k0 D ksep. A subgroup variety H 0 is stable under the action of on G0 (hencedefined over k) if and only if H 0.k0/ is stable under the action of on G.k0/.

Apply (A.53, A.54).

1.33 A closed algebraic subscheme H of an algebraic group G is an algebraic subgroup ofG if and only if H.R/ is a subgroup of G.R/ for all k-algebras R. This follows from theYoneda lemma (A.27).

1.34 (a) Let k be nonperfect of characteristic p, let a 2 krkp , and let G be the algebraicsubgroup of Ga defined by the equation Xp

2

D aXp. Then Gred is not an algebraic groupfor any regular map mWGredGred!Gred (see Exercise 9-7; SGA 3, VIA, 1.3.2a).

(b) Let k be nonperfect of characteristic p 3, let a 2 krkp , and let G be the algebraicsubgroup of GaGaGa defined by the equations

XpaY p D 0D Y p tZp:

Then G is a connected algebraic group such that Gred is not an algebraic group for any mapm (see SGA 3, VIA, 1.3.2b; mo38891).

1.35 When k is perfect, Gred is an algebraic subgroup of G, but it need not be normal. Forexample, over a field k of characteristic 3, let G D 3o .Z=2Z/k for the (unique) nontrivialaction of .Z=2Z/k on 3 (see Chapter 6 for semi-direct products); then Gred D .Z=2Z/k ,which is not normal in G.4 For an example with G connected, consider poGm for theobvious nontrivial action of Gm on p (SGA 3, VIA, 0.2, p.296.)

1.36 The formation of Gred doesnt commute with change of the base field; in particular,we may have .Gred/kal .Gkal/red. For example, G may be reduced without Gkal beingreduced; see (2.13) below.

1.37 Let G be an algebraic group over a field k of characteristic p 0, and let

k0 D kp1 defD fx 2 kal j 9m 1 such that xp

m

2 kg:

Then k0 is the smallest perfect subfield of kal containing k, and .Gk0/red is a smooth algebraicsubgroup ofGk0 (1.21). The algebraic variety .Gk0/red and its multiplication map are definedover a finite subextension of k0. This shows that there exists a finite purely inseparableextension K of k in kal such that ..GK/red/kal D .Gkal/red.

ASIDE 1.38 One may ask whether there is a scheme parametrizing the algebraic subgroups of analgebraic group. The answer is usually no. See mo188712.

4Let R be a k-algebra with no nontrivial idempotents but containing a primitive cube root of 1. Let denote the nonneutral element of .Z=2Z/k.R/D Z=2Z. By definition D 1 D 2. Therefore D 2,and 1 D 2 D 4 D .Z=2Z/k .R/.

Kernels 23

Kernels

Let 'WG!H be a homomorphism of algebraic groups, and let

Ker.'/DGH

G H

e

'

Then Ker.'/ is a closed subscheme of G such that

Ker.'/.R/D Ker.'.R//

for all k-algebras R. Therefore Ker.'/ is an algebraic subgroup of G (see 1.33). It is calledthe kernel of '. When G and H are affine, so also is N D Ker.'/, and

O.N /DO.G/O.H/ k 'O.G/=IHO.G/

where IH D Ker.O.H/f 7!f .e/! k/ is the augmentation ideal of H .

PROPOSITION 1.39 A surjective homomorphism 'WG!H of group varieties is smooth ifand only if Ker.'/ is smooth.

PROOF. We may suppose that k is algebraically closed. Recall (A.100), that a dominant map'WY !X of smooth algebraic varieties is smooth if and only if the maps .d'/y WTy.Y /!T'.y/.X/ on the tangent spaces are surjective for all y 2 Y .

Let N D Ker.'/. The exact commutative diagram

0 N.k"/ G.k"/ H.k"/

0 N.k/ G.k/ H.k/:

gives an exact sequence of kernels

0! Te.N /! Te.G/! Te.H/:

The fibres of ' are the cosets of N in G, which all have the same dimension, and so

dimN D dimGdimH

(A.92). On the other hand (1.18),

dimG D dimTe.G/

dimH D dimTe.H/

dimTe.N / dimN , with equality if and only if N is smooth.

Thus, we see that dimTe.N /D dimN (and N is smooth) if and only if .d'/eWTe.G/!Te.H/ is surjective. It remains to note that, by homogeneity (1.5), if .d'/e is surjective,then .d'/g is surjective for all g 2G. 2


ASIDE 1.40 Let 'WG!H be a homomorphism of group varieties over k. Borel 1991 et al. definethe kernel of ' to be the subgroup variety Ker.'kal/red of Gkal , which need not be defined over k(see 2.13). Springer 1998, 12.1.3 writes:

Let WG! G0 be a homomorphism5 of group varieties over k. If k is perfect or thetangent map .d/e is surjective, then the kernel is defined over k.

In the first case, Ker./red is geometrically reduced (A.37), and so Ker.kal/red D .Ker./red/kal ; inthe second case, Ker./ is smooth, and so Ker.kal/red D Ker./kal .

ASIDE 1.41 In the language of EGA/SGA, our algebraic groups over k are algebraic group schemesover k, i.e., group schemes over k whose underlying scheme is of finite type over k (SGA 3, VIA,p.295). Some of the above results hold without finiteness conditions. For example, group schemesover a field are always separated (ibid. 0.3, p.296). For a quasicompact morphism uWG!H ofgroup schemes locally of finite type over k, the following conditions are equivalent:

(a) u is a closed immersion;

(b) u is a monomorphism (i.e., u.R/ is injective for all k-algebras R);

(c) Ker.u/ is trivial;

in particular, every subgroup scheme of H is closed (SGA 3, VIB , 1.4.2, p.341). However, let .Z/Qbe the constant group scheme over Q; a nonzero homomorphism of .Z/Q!Ga of group schemesover Q has trivial kernel but is not a closed immersion (ibid. 1.4.3, p.341). As another example, overan algebraically closed field k there is a zero-dimensional (nonaffine) reduced group scheme G withG.k/D k; the obvious homomorphism k!Ga of group schemes is a homomorphism, and it is bothmono and epi, but it is not an isomorphism.

The algebraic subgroup generated by a map (1)

PROPOSITION 1.42 Let f WX!G be a regular map from a geometrically reduced algebraicscheme X over k to an algebraic group G. Assume that inv.f .X// f .X/, and let f n

denote the map.x1; : : : ;xn/ 7! f .x1/ f .xn/WX

n!G:

The reduced algebraic subscheme of G with underlying set the closure ofSn Im.f

n/ is asmooth algebraic subgroup of G.

PROOF. Because X is geometrically reduced, so also is Xn (A.37). The map f nWXn!His schematically dominant for n large because it is dominant and H is reduced (A.63). Itfollows that H is geometrically reduced and that its formation commutes with extension ofthe base field (A.64, A.65). Therefore, in proving that H is an algebraic subgroup of G, wemay suppose that k is algebraically closed. Let Z be the closure of m.H H/ in G. Theintersection of m1.ZrH/ with H H is an open subset of H H , which is nonemptyif m.H H/ is not contained in H . In that case, there exist x1; : : : ;xn;y1; : : : ;yn 2 X.k/such that

.f .x1/ f .xn/;f .y1/ f .yn// 2m1.ZrH/

(because Im.f n/ Im.f n/ is constructible, and therefore contains an open subset of itsclosure; A.61). But this is absurd, because

m.f .x1/ f .xn/;f .y1/ f .yn//D f .x1/ f .xn/f .y1/ f .yn/ 2H.k/:

5He means k-homomorphism.

The algebraic subgroup generated by a map (2) 25

The condition inv.f .X// f .X/ implies that inv maps H into H , and so H is analgebraic subgroup of G. It is smooth because it is geometrically reduced. 2

PROPOSITION 1.43 Let .fi WXi !G/i2I be a finite family of regular maps from geometri-cally reduced algebraic schemes Xi over k to an algebraic group G. There exists a smallestalgebraic group H of G such that all fi factor through H . Moreover, H is smooth.

PROOF. Let X DFi2I Xi tXi , and let f WX!G be the map whose restriction to Xi tXi

is fi on the first component and invfi on the second component. Then inv.f .X// f .X/,and the algebraic subgroup H attached to f in (1.42) has the required properties. 2

We call H the algebraic subgroup of G generated by the fi (or Xi ).

PROPOSITION 1.44 Let f WX !G be a regular map from a geometrically reduced schemeX over k to an algebraic group G. If X is geometrically connected and f .X/ contains e,then the algebraic subgroup of G generated by f is connected.

PROOF. Let f 0 be the mapX 0 defDX tX!G acting as f on the first component and invfon the second. The hypotheses imply that

SIm.f 0n/ is connected, and so its closure H is

connected. 2

The algebraic subgroup generated by a map (2)

Let f WX!G be a regular map from an affine algebraic scheme X (not necessarily reduced)to an affine algebraic group G. Assume that the image of f contains e, say f .o/D e. Let Inbe the kernel of the homomorphism O.G/!O.Xn/ of k-algebras defined by the regularmap

.x1; : : : ;xn/ 7! f .x1/ f .xn/WXn!G:

From the regular maps

X !X2! !Xn! !G;

.x/ 7! .x;o/ 7!

we get inclusionsI1 I2 In ;

and we let I DTIn.

PROPOSITION 1.45 Assume that inv.f .X.R// f .X.R// for all R. Then the subschemeH of G defined by I is an algebraic subgroup of G. It is the smallest algebraic subgroupH of G such that H.R/ contains f .X.R// for all k-algebras R. In other words, it is thesmallest algebraic subgroup such that f WX !G factors through H .

PROOF. From the diagram of algebraic schemes

Xn Xn X2n

G G G,mult


we get a diagram of k-algebras

O.Xn/ O.Xn/ O.X2n/

O.G/ O.G/ O.G/.

The image of O.G/ in O.Xn/ is O.G/=In and its image in O.X2n/ is O.G/=I2n, and sothe diagram shows that

WO.G/!O.G/=InO.G/=Infactors through O.G/!O.G/=I2n. It follows that

WO.G/!O.G/=I O.G/=I

factors through O.G/! O.G/=I , and defines a multiplication map mH WH H ! H .The triple .H;mH ; e/ is the smallest algebraic submonoid of G such that H.R/ containsf .X.R// for all k-algebras R.

The hypothesis inv.f .X.R// f .X.R// implies that inv.H/ has the same property,and so equals H . Therefore .H;mH / is an algebraic subgroup of G. It clearly has therequired properties. 2

We write hX;f i for the algebraic subgroupH in the proposition, and call it the algebraicsubgroup generated by f (or X/.

PROPOSITION 1.46 Let K be a field containing k. Then hX;f iK D hXK ;fKi.

PROOF. The definition of I commutes with extension of the base field. 2

PROPOSITION 1.47 If X is geometrically connected (resp. geometrically reduced), thenhX;f i is connected (resp. geometrically reduced).

PROOF. We may suppose that k is algebraically closed. An affine scheme Y is connected ifand only if O.Y / has no nontrivial idempotent (CA 14.2). Assume that X is connected. IfO.G/=I had a nontrivial idempotent, then so would O.G/=In for some n, but (by definition)the homomorphism of k-algebras O.G/=In!O.Xn/ is injective. As X is connected andk is algebraically closed, Xn is connected, and so this is a contradiction. The proof of theremaining statement is similar. 2

Closed subfunctors: definitions and statements

By a functor in this section, we mean a functor Alg0k! Set.

1.48 Let A be a k-algebra, and let hA denote the functor R Hom.A;R/. Let a be anideal in A. The set of zeros of a in hA.R/ is

Z.R/D f'WA!R j '.a/D 0 for all ' 2 ag:

A homomorphism of k-algebras R! R0 defines a map Z.R/! Z.R0/, and these mapsmake R Z.R/ into a subfunctor of hA, called the functor of zeros of a. For example, ifAD kT1; : : : ;Tn, then hA DAn, and the set of zeros of an ideal is the set of zeros in Rn ofa some finite family of polynomials in kT1; : : : ;Tn

Transporters 27

1.49 Let Z be a subfunctor of a functor X . From a map of functors f WhA!X , we obtaina subfunctor h1.Z/ defDZX hA of hA, namely,

R fa 2 hA.R/ j f .R/.a/ 2Z.R/g:

We say that Z is a closed subfunctor of X if, for every map f WhA! X , the subfunctorf 1.Z/ of hA is the functor of zeros of some ideal a in A.

Later in this chapter, we shall prove the following statements.

1.50 Let X be an algebraic scheme over k. The closed subfunctors of hX are exactly thoseof the form hZ with Z a closed subscheme of X (1.69). Recall that hX denotes the functorR X.R/.

1.51 Let Z be a closed subfunctor of a functor X . For every map Y ! X of functors,ZX Y is a closed subfunctor of Y (1.70).

Let R be a small k-algebra. For a functor X , we let XR denote the functor of smallR-algebras defined by composing X with the forgetful functor Alg0R! Alg

0k

. For functorsY and X , we let Hom.Y;X/ denote the functor

R Hom.YR;XR/:

If Z is a subfunctor of X , then Hom.Y;Z/ is a subfunctor of Hom.Y;X/.

1.52 Let Z be a subfunctor of a functor X , and let Y be an algebraic scheme. If Z isclosed in X , then Hom.Y;Z/ is closed in Hom.Y;X/ (1.74).

Transporters

A left action of an algebraic group G on an algebraic scheme is a regular map GX !Xsuch that, for all k-algebras, the map G.R/X.R/! X.R/ is a left action of the groupG.R/ on the set X.R/. Given such an action and algebraic subschemes Y and Z of X , thetransporter TG.Y;Z/ of Y into Z is the functor

R fg 2G.R/ j gYR ZRg:

Here YR and ZR can be interpreted as algebraic R-schemes (A.31) or as functors onthe category of small R-algebras. Because of the Yoneda lemma (A.31), the differentinterpretations give the same condition. Explicitly,

gYR ZR gY.R0/Z.R0/ for all (small) R-algebras R0:

PROPOSITION 1.53 If Z is closed in X , then TG.Y;Z/ is represented by a closed sub-scheme of G.

PROOF. Consider the diagram:

TG.Y;Z/' Hom.Y;Z/Hom.Y;X/G G

Hom.Y;Z/ Hom.Y;X/

b

c


The map b is defined by the action of G on X , and c is defined by the inclusion of Z intoX . According to (1.52), Hom.Y;Z/ is a closed subfunctor of Hom.Y;X/, and so TG.Y;Z/is a closed subfunctor of X (1.51). Therefore it is represented by a closed subscheme of G(1.50). 2

Normalizers

Let G be an algebraic group over k.

PROPOSITION 1.54 Let H be an algebraic subgroup of G. There is a unique algebraicsubgroup NG.H/ of G such that

NG.H/.R/Dg 2G.R/ j gHRg

1DHR

for all k-algebras R.

In other words, NG.H/ represents the functor

R N.R/defD fg 2G.R/ j gH.R0/g1 DH.R0/ for all R-algebras R0g.

PROOF. The uniqueness follows from the Yoneda lemma (A.27). Clearly N.R/ is a sub-group of G.R/, and so it remains to show that N is represented by a closed subscheme of G(1.33). But

g H.R0/ g1 DH.R0/ g H.R0/ g1 H.R0/ and g1 H.R0/ g H.R0/;

and so, when we let G act on itself by conjugation,

N D TG.H;H/\TG.H;H/1.

It follows from (1.53) that N is represented by a closed subscheme of G. 2

The algebraic subgroup NG.H/ is called the normalizer of H in G. Directly from itsdefinition, one sees that the formation of NG.H/ commutes with extension of the base field.Clearly H is normal in G if and only of NG.H/DG.

PROPOSITION 1.55 Let H be a subgroup variety of G, and let k0 be a field containing k. IfH.k0/ is dense in H , then NG.H/.k/ consists of the elements of G.k/ normalizing H.k0/in G.k0/.

PROOF. Let g 2 G.k/ normalize H.k0/, and let gH denote the image of H under theisomorphism x 7! gxg1WG!G. Then gH \H is an algebraic subgroup of H such that

.gH \H/.k0/D gH.k0/\H.k0/DH.k0/:

As H.k0/ is dense in H , this implies that gH \H DH , and so gH DH . In particular,gH.R/g1DH.R/ for all k-algebrasR, and so g 2NG.H/.k/. The converse is obvious.2

COROLLARY 1.56 Let H be a subgroup variety of a group variety G. If H.ksep/ is normalin G.ksep/, then H is normal in G.

Centralizers 29

PROOF. BecauseH is a variety,H.ksep/ is dense inH , and so (1.55) shows thatNG.H/.ksep/DG.ksep/. Because G is a variety, this implies that NG.H/DG. 2

COROLLARY 1.57 Let H be a normal algebraic subgroup of a group variety G. If Hred is asubgroup variety of G, then it is normal in G.

PROOF. As H is normal, H.ksep/ is normal in G.ksep/, but H.ksep/DHred.ksep/ and sowe can apply (1.56). 2

The example in (1.35) shows that it is necessary to take G to be a group variety in (1.56)and (1.57).

DEFINITION 1.58 An algebraic subgroup H of an algebraic group G is weakly character-istic if, for all fields k0 containing k, Hk0 is stable under all automorphisms of Gk0 .

COROLLARY 1.59 Let N be a normal subgroup variety of a group variety G, and let H bea subgroup variety of N . If H is weakly characteristic in N , then it is normal in G.

PROOF. By hypothesis, Hksep is stable under all automorphisms of Nksep , in particular, bythose induced induced by an inner automorphism of Gksep . Therefore H.ksep/ is normal inG.ksep/, and so we can apply (1.56). 2

Centralizers

Let G be an algebraic group over k.

PROPOSITION 1.60 Let H be an algebraic subgroup of G. There is a unique algebraicsubgroup CG.H/ of G such that, for all k-algebras R,

CG.H/.R/D fg 2G.R/ j g centralizes H.R0/ in G.R0/ for all R-algebras R0g:

PROOF. Let G act on GG by

g.g1;g2/D .g1;gg2g1/; g;g1;g2 2G.R/:

Recall (1.8) that H is closed in H H , and hence in GG. Now

C D TG.H;H /;

which is represented by a closed subscheme G (by 1.53). 2

The algebraic subgroup CG.H/ is called the centralizer of H in G. Directly from itsdefinition, one sees that the formation of CG.H/ commutes with extension of the base field.The centre ZG of G is defined to be CG.G/.

PROPOSITION 1.61 Let H be a subgroup variety of G, and let k0 be a field containing k. IfH.k0/ is dense in H , then CG.H/.k/ consists of the elements of G.k0/ centralizing H.k0/in G.k0/.

PROOF. Let n be an element of G.k/ centralizing H.k0/. Then n 2NG.H/.k/ (1.55), andthe homomorphism x 7! nxn1WH !H coincides with the identity map on an algebraicsubgroupH 0 ofH such thatH 0.k0/DH.k0/. This implies thatH 0DH , and so n centralizesH . 2


COROLLARY 1.62 Let H be a subgroup variety of a group variety G. If H.ksep/ is con-tained in the centre of G.ksep/, then H is contained in the centre of G.

PROOF. We have to show that CG.H/DG. For this, we may replace k with ksep (1.32a),and so assume that k is separably closed. Because H is a variety, H.k/ is dense in H , andso (1.61) shows that CG.H/.k/D G.k/. Because G is a group variety, this implies thatCG.H/DG (1.4d). 2

NOTES

1.63 The centre Z.G/ of a smooth algebraic group need not be smooth (for an example,see 13.25 below). Similarly, CG.H/ and NG.H/ need not be smooth, even when H and Gare. (For some situations where they are smooth, see 16.23 and 16.25 below.)

1.64 Assume that k is perfect, and let H be a subgroup variety of a group variety G. Then

CG.H/red.kal/D CG.H/.k

al/(1.61)D CG.kal/.H.k

al//;

and so CG.H/red is the unique subgroup variety C of G such that C.kal/ is the centralizerof H.kal/ in G.kal/. Similarly, NG.H/red is the unique subgroup variety N of G such thatN.kal/ is the normalizer of H.kal/ in G.kal/.

1.65 When k has characteristic zero, all algebraic groups over k are smooth (4.4, 11.36below). It follows from (1.64) that, over a field of characteristic zero, CG.H/ is the uniquealgebraic subgroup C of G such that C.kal/ is the centralizer of H.kal/ in G.kal/, andNG.H/ is the unique algebraic subgroup N of G such that N.kal/ is the normalizer ofH.kal/ in G.kal/.

1.66 LetH be a subgroup variety of a group varietyG. In Borel 1991, p.52, the normalizerN of H in G is defined to be the subgroup variety NGkal .Hkal/red of Gkal , which need notbe defined over k. The centralizer is similarly defined to be a subgroup variety of Gkal .

Closed subfunctors: proofs

In this section, functor means functor Alg0k! Set unless indicated otherwise.

CLOSED SUBFUNCTORS

LEMMA 1.67 Let Z be a subfunctor of a functor X . Then Z is closed in X if and only if itsatisfies the following condition: for every k-algebra A and map of functors f WhA! Y , thesubfunctor f 1.Z/ of hA is represented by a quotient of A.

PROOF. This is a restatement of the definition. 2

According to the Yoneda lemma, a map of functors f WhA!X corresponds to an element 2X.A/. Explicitly, f .R/WhA.R/!X.R/ is the map sending ' 2 hA.R/D Hom.A;R/to X.'/.)2X.R/, and so

f 1.Z/.R/D f'WA!R jX.'/./ 2Z.R/g.

Closed subfunctors: proofs 31

Therefore, Z is closed in X if and only if, for every A and 2X.A/, the functor

R f'WA!R jX.'/./ 2Z.R/g

is represented by a quotient of A; in down-to-earth terms, this means that there exists anideal a A such that

X.'/./ 2Z.R/ '.a/D 0:

EXAMPLE 1.68 LetB be a k-algebra, and letZ be a subfunctor ofX D hB . For the identitymap f WhB !X , f 1.Z/DZ. It follows that, if Z is closed in hB , then it is representedby a quotient of B . Conversely, suppose that Z is represented by a quotient B=b of B , sothat

Z.R/D f'WB!R j '.b/D 0g:

Let 2X.A/D Hom.B;A/, and let f be the corresponding map f WhA!X . Then

f 1.Z/.R/D f'WA!R j ' 2Z.R/g

D f'WA!R j '..b//D 0g,

and so f 1.Z/ is represented by the quotient A=.b/ of A.We conclude that the closed subfunctors of hB are exactly those defined by closed

subschemes of Spm.B/.

EXAMPLE 1.69 Consider the functor hX WR X.R/ defined by an algebraic scheme Xover k. If Z is a closed subscheme of X , then certainly hZ is a closed subfunctor of hX .Conversely, let Z be a closed subfunctor of X . For each open affine subscheme U of X ,there is a unique ideal I.U / in O.U / such thatZ\hU D hO.U /=I.U / (apply 1.68). Becauseof the uniqueness, the sheaves on U and U 0 defined by I.U / and I.U 0/ coincide on U \U 0.Therefore, there exists a (unique) coherent sheaf I on X such that .U;I/D I.U / for allopen affines U in X . Now Z D hZ0 where Z0 is the closed subscheme of X defined by I(A.19).

We conclude that the closed subfunctors of hX are exactly those defined by closedsubschemes of X .

PROPOSITION 1.70 Let Z be a closed subfunctor of a functor X . For every map Y !Xof functors, ZX Y is a closed subfunctor of Y .

PROOF. Let f WhA! Y be a map of functors. Then

f 1.ZX Y /defD .ZX Y /Y h

ADZX h

A,

which is the functor of zeros of some a A because Z is closed in X . 2

RESTRICTION OF SCALARS

LEMMA 1.71 Let A and B be k-algebras, and let b be an ideal in BA. Among the idealsa in A such that Ba b, there exists a smallest one.


PROOF. Choose a basis .ei /i2I for B as k-vector space. Each element b of BA can beexpressed uniquely as a finite sum

b DX

ei ai ; ai 2 A;

and we let a0 denote the ideal in A generated by the coordinates ai of the elements b 2 b.Clearly Ba0 b. Let a be a second ideal such that Ba b. Then the coordinates ofall elements of b lie in a, and so a a0. 2

Let B be a small k-algebra, and let X be a functor X WAlg0k! Set. We define X to be

the functorR X.BR/WAlg0k! Set:

PROPOSITION 1.72 Let B be a small k-algebra, and let Z be a subfunctor of a functor X .If Z is closed in X , then Z is closed in X.

PROOF. Let A be a k-algebra, and 2X.A/. To prove that Z is closed in X we haveto show that there exists an ideal a A such that, for all homomorphisms 'WA!R,

X.'/./ 2Z.R/ '.a/D 0;

i.e.,X.B'/./ 2Z.BR/ '.a/D 0:

We can regard as an element of X.BA/. Because Z is closed in X , there exists anideal b in BA such that, for all homomorphisms '0WBA!R0,

X.'0/./ 2Z.R0/ '0.b/D 0:

In particular (taking '0 D B'/, we have

X.B'/./ 2Z.BR/ .B'/.b/D 0: (4)

According to (1.71), there exists an ideal a in A such that an ideal a0 of A contains a ifand only if b Ba0. On taking a0 D Ker', we find that

a Ker.'/ b BKer.'/D Ker.B'/: (5)

Now'.a/D 0

(5) .B'/.b/D 0

(4) X.B'/./ 2Z.BR/;

as required. 2

APPLICATION TO Hom

LEMMA 1.73 An intersection of closed subfunctors of a functor is closed.

PROOF. Let Zi , i 2 I , be closed subfunctors of X , and let f WhA!X be map of functors.For each i 2 I , there is an ideal ai of A such that f 1.Zi / hA.R/ is the functor of zerosof ai . Now f 1.

Ti2I Zi /D

Ti2I f

1.Zi / is the functor of zeros of aDPi2I ai . 2

THEOREM 1.74 Let Z be a subfunctor of a functor X , and let Y be an algebraic scheme. IfZ is closed in X , then Hom.Y;Z/ is closed in Hom.Y;X/.

Closed subfunctors: proofs 33

PROOF. Suppose first that Y D hB for some k-algebra B (which we may assume to besmall). Then, for every k-algebra R,

Hom.Y;X/.R/DX.BR/;

and so Hom.Y;X/DX. In this case, the theorem is proved in (1.72).Let Y D

Si Yi be a finite covering of Y by open affines, and consider the diagram

Hom.Y;X/i! Hom.Yi ;X/

[ [

Hom.Y;Z/ ! Hom.Yi ;Z/

in which i is the restriction map. We know that Hom.Yi ;Z/ is closed in Hom.Yi ;X/, hence1i .Hom.Yi ;Z// is closed in Hom.Y;X/ (1.70), and so (see 1.73) it remains to show that

Hom.Y;Z/D\

i1i .Hom.Yi ;Z// .

Let Hi D 1i .Hom.Yi ;Z//. Certainly, Hom.Y;Z/TiHi , and for the reverse inclu-

sion it suffices to show that the map of functors\iHi

Y !X

defined by the evaluation map

WHom.Y;X/Y !X

factors through Z. For each i , we know that Hi Yi !X factors through Z. By definition,Z will become a closed subscheme of an (affine) scheme X after we have pulled backby a map of functors hA! X . Then 1.Z/ is a closed subscheme of Hom.Y;X/Ycontaining

TiHi

Yi for all i , and hence containing

TiHi

Y . Since this holds for

all maps hA!X , it follows that 1.Z/T

iHiY . 2

ASIDE 1.75 In this section, we used that k is a field only to deduce in the proof of (1.71) that B isfree as a k-module. Thus, the same arguments suffice to prove the following more general statement:let k be a commutative ring, let X be a functor of k-algebras, and let Z be a closed subfunctor of X ;let Y be a locally free scheme over k, i.e., such that Y admits a covering by open affines Yi for whichO.Yi / is a free k-module; then Hom.Y;Z/ is a closed subfunctor of Hom.Y;X/. See DG I, 2, 7.5,p. 64; also Jantzen 1987, 1.15.

CHAPTER 2Examples of algebraic groups and

morphisms

Let G be an affine algebraic group. We call O.G/ the coordinate ring of G. When G isembedded as a closed subvariety of some affine space An, O.G/ is the ring of functions onG generated by the coordinate functions on An, whence the name.

The comultiplication map

Let A be a k-algebra, and let WA! AA be a homomorphism. For a k-algebra R andhomomorphisms f1;f2WA!R, we set

f1 f2 D .f1;f2/ (6)

where .f1;f2/WAA! R denotes the homomorphism .a1;a2/ 7! f1.a1/f2.a2/. Thisdefines a binary operation on hA.R/D Hom.A;R/. On the other hand, because

Spm.AA/' Spm.A/Spm.A/

(A.33), we can regard Spm./ as a map Spm.A/Spm.A/! Spm.A/.

PROPOSITION 2.1 The pair .Spm.A/;Spm.// is an algebraic group if and only if (6)makes Hom.A;R/ into a group for all k-algebras R.

PROOF. Let .G;m/D .SpmA;Spm/. From .A;/ we get a functor hAWR Hom.A;R/from k-algebras to the category of sets equipped with a binary operation (i.e., to magmas).

If hA takes values in the subcategory of groups, then there are natural transformationseW ! hA, and invWhA! hA making the diagrams (2, 3), p.15, commute. According tothe Yoneda lemma (A.26), these natural transformations arise from regular maps eW !G,invWG!G making the same diagrams commute, and so .G;m/ is an algebraic group.

Conversely, the existence of the regular maps e and inv implies that .hA;h/ takesvalues in the subcategory of groups. 2

For an affine algebraic group .G;m/, the homomorphism of k-algebras

WO.G/!O.G/O.G/

corresponding to mWGG!G is called the comultiplication map.

35

36 2. Examples of algebraic groups and morphisms

REMARKS

2.2 Let G be an affine algebraic group, and let O.G/ be its coordinate ring. Then

G.R/' Homk-algebra.O.G/;R/;

and so an f 2O.G/ defines an evaluation map

fRWG.R/!R; g 7! g.f /I (7)

i.e.,fR.g/D g.f /; f 2O.G/; g 2G.R/.

In this way, we get an isomorphism

O.G/' Nat.G;A1/ (8)

where A1 is the functor sending a k-algebra R to its underlying set. Similarly,

O.GG/' Nat.GG;A1/

With this interpretation

.f /R.g1;g2/D fR.g1 g2/; f 2O.G/; g1;g2 2G.R/: (9)

2.3 Every affine algebraic group over k defines a functor from k-algebras to groupswhose underlying functor to sets is representable, and every such functor arises from anessentially unique algebraic group. Indeed, let F be such a functor. The pair .A;a/,a 2 F.A/, representing the underlying functor is unique up to a unique isomorphism, andthe multiplication on the sets F.R/ arises from a comultiplication map on A, which satisfiesthe equivalent conditions of (2.1) (by the Yoneda lemma).

This gives a very convenient way of defining affine algebraic groups.

Examples of affine algebraic groups

As we saw in (2.3), to give an affine algebraic group over k amounts to giving a functor fromk-algebras to groups such that the underlying functor to sets is representable.

2.4 The additive group Ga is the functor R .R;C/. It is represented by O.Ga/D kT ,and the universal element in Ga.kT / is T : for r 2Ga.R/, there is a unique homomorphismkT !R such that the map Ga.kT /!Ga.R/ sends T to r . The comultiplication mapis the k-algebra homomorphism WkT ! kT kT such that

.T /D T 1C1T:

2.5 The multiplicative group Gm is the functorR .R; /. It is represented by O.Gm/DkT;T 1 k.T /, and the comultiplication map is the k-algebra homomorphism with.T /D T T .

2.6 The trivial algebraic group is the functor R feg. It is represented by O./D k,and the comultiplication map is the unique k-algebra homomorphism k! kk. Moregenerally, every finite group F can be regarded as a constant algebraic group Fk withcoordinate ring a product of copies of k indexed by the elements of F . We sometimes writee for the trivial algebraic group.

Examples of affine algebraic groups 37

2.7 For an integer n 1, n is the functor R fr 2 R j rn D 1g. It is represented byO.n/D kT =.T n1/, and the comultiplication map is induced by that of Gm.

2.8 When k has characteristic p 0, pm is the functor R fr 2R j rpm

D 0g. To showthat this set is a subgroup of .R;C/, use that .xCy/p D xpCyp in characteristic p. Thefunctor is represented by O.pm/D kT =.T p

m

/, and the comultiplication map is inducedby that of Ga.

2.9 For a finite-dimensional vector space V , Va is the functor R R V .1 It is analgebraic group, isomorphic to GdimVa . Such an algebraic group is called a vector group.

2.10 For integers m;n 1, Mm;n is the functor R Mm;n.R/ (additive group of mnmatrices with entries in R). It is represented by kT11;T12; : : : ;Tmn. For a vector space Vover k, we define EndV to be the functor

R End.VR/ (R-linear endomorphisms).

When V has finite dimension n, the choice of a basis for V determines an isomorphismEndV Mn;n, and so EndV is an algebraic group in this case.

2.11 The general linear group GLn is the functor R GLn.R/ (multiplicative group ofinvertible nn matrices with entries in R). It is represented by

O.GLn/DkT11;T12; : : : ;Tnn;T

.det.Tij /T 1/D kT11;T12; : : : ;Tnn;1=det;

and the universal element in GLn.kT11; : : :/ is the matrix .Tij /1i;jn: for any .aij /1i;jn 2GLn.R/, there is a unique homomorphism kT11; : : :!R such that the map GLn.kT11; : : :/!GLn.R/ sends .Tij / to .aij /. The comultiplication map is the k-algebra homomorphism

WkT11; : : :! kT11; : : :kT11; : : :

such thatTij D

X1ln

TilTlj : (10)

Symbolically,.Tij /i;j D .Til/i;l .Tlj /l;j :

More generally, for any vector space V over k, we define GLV to be the functor

R Aut.VR/ (R-linear automorphisms).

When V is has finite dimension n, the choice of a basis for V determines an isomorphismGLV GLn, and so GLV is an algebraic group in this case.

The following are algebraic subgroups of GLn:

TnWR f.aij / j aij D 0 for i > j g (upper triangular matrices)UnWR f.aij / j aij D 0 for i > j , aij D 1 for i D j gDnWR f.aij / j aij D 0 for i j g (diagonal matrices),

1Our notation Va is that of DG, II, 1, 2.1, p.147. Many other notations are used.

38 2. Examples of algebraic groups and morphisms

TnD

0BBBBBB@

: : :: : :

0

1CCCCCCA ; UnD0BBBBBB@1

1 : : :

: : :

0 1 1

1CCCCCCA ; DnD0BBBBBB@

0: : :

0

1CCCCCCA :

For example, Un is represented by the quotient of kT11;T12; : : : ;Tnn by the ideal generatedby the polynomials

Tij .i > j /; Ti i 1 (all i ):

Examples of homomorphisms

2.12 Let k be a field of characteristic p 0. For a k-algebra R, we let fR denote thehomomorphism a 7! apWR! R. When R D k, we omit the subscript on f . For a k-

algebra ki! R, we let fR denote the ring R regarded as a k-algebra by means of the

map kf! k

i! R. Let G be an algebraic group over k, and let G.p/ denote the functor

R G.fR/. WhenG is affine, this is represented by kf;kO.G/ (tensor product of O.G/with k relative to the map f Wk! k),

R

k kf;kO.G/

k O.G/;

f

ib

aa 2G.fR/

b 2 Homk-algebra.kf;kO.G/;R/

and so it is again an algebraic group. The k-algebra homomorphism fRWR! fR defines ahomomorphism G.R/! G.p/.R/, which is natural in R, and so arises from a homomor-phism F WG! G.p/ of algebraic groups, called the Frobenius map. When G is affine, itcorresponds to the homomorphism of Hopf algebras

ca 7! capWO.G.p//!O.G/:

The kernel of the Frobenius map is a characteristic subgroup of G (DG II, 7, 1.4, p.271).Similarly we define F nWG!G.p

n/ by replacing p with pn. Then F n is the composite

GF!G.p/

F!

F!G.p

n/:

If F n D 0, then the algebraic group G is said to have height n.

2.13 Let k have characteristic p 0, and let a 2 kr kp. Let G be the kernel of thehomomorphism

WGaGa!Ga; .x;y/ 7! ypaxp:

Then G is reduced but not geometrically reduced. Indeed O.G/D kX;Y =.Y p aXp/,which is an integral domain because Y paXp is irreducible in kX;Y , but it acquires a

Some basic constructions 39

nilpotent ya1p x when tensored with kal. (Over kal, G becomes the line Y D a

1pX with

multiplicity p.)Note that Ker./ is not a group variety even though is a homomorphism of group

varieties. In the old terminology, one defined the kernel of to be the subgroup varietyG0WY D a

1pX of .GaGa/kal , and noted that it is not defined over k (cf. Springer 1998,

12.1.6). In our terminology, G0 D Ker.kal/red.

2.14 An algebraic group is finite if it is affine and its coordinate ring is a finite k-algebra(see 8.2). A homomorphism WG!H of group varieties is an isogeny if its kernel is finiteand the image of jj contains jH j. An isogeny is separable (resp. central) if its kernel issmooth (resp. contained in the centre of G).

Some basic constructions

2.15 Let G1; : : : ;Gn be algebraic groups over k. Then G1 Gn is an algebraic group,called the product of the Gi . It represents the functor

R G1.R/ Gn.R/.

When the Gi are affine, G1 Gn is affine, and

O.G1 Gn/'O.G1/ O.Gn/.

2.16 Let G1!H G2 be homomorphisms of algebraic groups. Then G1H G2 is analgebraic group, called the fibred product of G1 and G2 over H . It represents the functor

R G1.R/H.R/G2.R/:

When G1, G2, and H are affine, G1H G2 is affine, and

O.G1H G2/'O.G1/O.H/O.G2/.

Directly from the definition, one sees that the formation of fibred products of algebraicgroups commutes with extension of the base field:

.G1H G2/k0 ' .G1/k0 Hk0 .G2/k0 :

Restriction of the base field (Weil restriction of scalars)

Throughout this section, k0 is a finite k-algebra and all algebraic groups are affine.

2.17 A functor F from k0-algebras to sets defines a functor

.F /k0=k WAlgk! Set; R F.k0R/:

If F WAlgk0 ! Set i

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