Algebraic Groups

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Algebraic Groups

The theory of group schemes of finite type over a field.

J.S. Milne

Version 2.00 December 20, 2015

This is a rough preliminary version of the book published by CUP in 2017, The final version is substantially

rewritten, and the numbering has changed.

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it is a group scheme of finite type over a field. These notes are a comprehensive modern introduction to the theory of algebraic groups assuming only the knowledge of algebraic geometry usually acquired in a first course.

This is still only a preliminary version, but is the last before the final version. It will be revised again before publication. In particular, repetitions, references to my website, and notational inconsistencies will be removed; exercises and examples will be added; errors will be corrected. The final version will be more tightly written, and will include 500 pages + front matter. I welcome suggestions for improvements to the final version, which can be

sent to me at jmilne at umich.edu.

BibTeX information

@misc{milneiAG,

author={Milne, James S.},

title={Algebraic Groups (v2.00)}, year={2015},

note={Available at www.jmilne.org/math/}, pages={528}

}

v1.00 (July 31, 2014). First published on the web, 331 pages.

v1.20 (January 29, 2015). Revised Parts A,B, 373 pages.

v2.00 (December 20, 2015). Significantly rewritten and completed.

Available at www.jmilne.org/math/

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 permission from the copyright holder.

This book was written on a 2007 vintage Thinkpad T60p, the quality of whose keyboard and screen have not been surpassed.

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Preface

For one who attempts to unravel the story, the problems are as perplexing as a mass of hemp with a thousand loose ends.

Dream of the Red Chamber, Tsao Hsueh-Chin.

This book represents my attempt to write a modern successor to the three standard works, all titled “Linear Algebraic Groups”, by Borel, Humphreys, and Springer. More specifically, it is an exposition of the theory of group schemes of finite type over a field, based on modern algebraic geometry, but with minimal prerequisites.

It has been clear for fifty years that such a work has been needed.¹ When Borel, Chevalley, and others introduced algebraic geometry into the theory of algebraic groups, the foundations they used were those of the period (e.g.,Weil 1946), and most subsequent writers on algebraic groups have followed them. Specifically, nilpotents are not allowed, and the terminology used conflicts with that of modern algebraic geometry. For example, algebraic groups are usually identified with their points in some large algebraically closed fieldK, and an algebraic group over a subfieldkofKis an algebraic group overKequipped with ak-structure. The kernel of ak-homomorphism of algebraick-groups is an object over K (notk)which need not be defined overk.

In the modern approach, nilpotents are allowed,²an algebraick-group is intrinsically defined overk, and the kernel of a homomorphism of algebraic groups overkis (of course) defined overk. Instead of the points in some “universal” field, it is more natural to consider the functor ofk-algebras defined by the algebraic group.

The advantages of the modern approach are manifold. For example, the infinitesimal theory is built into it from the start instead of entering only in an ad hoc fashion through the Lie algebra. The Noether isomorphisms theorems hold for algebraic group schemes, and so the intuition from abstract group theory applies. The kernels of infinitesimal homomorphisms become visible as algebraic group schemes.

The first systematic exposition of the theory of group schemes was in SGA 3. As was natural for its authors (Demazure, Grothendieck, . . . ), they worked over an arbitrary base scheme and they used the full theory of schemes (EGA and SGA).³Most subsequent authors on group schemes have followed them. The only books I know of that give an elementary treatment of group schemes are Waterhouse 1979 and Demazure and Gabriel 1970. In writing 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 a central part of the theory.

As noted, the modern theory is more general than the old theory. The extra generality gives a richer and more attractive theory, but it does not come for free: some proofs are more difficult (because they prove stronger statements). In this work, I have avoided any appeal to advanced scheme theory by passing to the algebraic closure where possible and by an occasional use of Hopf algebras. Unpleasantly technical arguments that I have not (so far) been able to avoid have been placed in separate sections where they can be ignored by all

1“Another remorse concerns the language adopted for the algebrogeometrical foundation of the theory ...

two such languages are briefly introduced ... the language of algebraic sets ... and the Grothendieck language of schemes. Later on, the preference is given to the language of algebraic sets ... If things were to be done again, I would probably rather choose the scheme viewpoint ... which is not only more general but also, in many respects, more satisfactory.” J. Tits, Lectures on Algebraic Groups, Fall 1966.

2To anyone who asked why we need to allow nilpotents, Grothendieck would say that they are already there in nature; neglecting them obscures our vision. And indeed they are there, for example, in the kernel of SL_p!PGL_p.

3They also assumed the main classification results of the old theory.

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many of the technicalities that plague the general theory. Also, the theory over a field has many special features that do not generalize to arbitrary bases.

The exposition incorporates simplifications to the general theory fromAllcock 2009,

¯Dokovi´c 1988,Iversen 1976,Luna 1999, andSteinberg 1998,1999and elsewhere.

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The experienced reader is cautioned that, throughout the text, “algebraic group scheme”

is shortened to “algebraic group”, nonclosed points are ignored, and a “group variety” is a smooth algebraic group.

Equivalently, a group variety is group in the category of algebraic varieties (geometrically reduced separated schemes of finite type over a field). However, it is important to note that varieties are always regarded as special algebraic schemes. For example, fibres of maps are to be taken in the sense of schemes, and the kernel of a homomorphism of group varieties is an algebraic group which is not necessarily a group variety (it need not be smooth). A statement here may be stronger than a statement inBorel 1991orSpringer 1998even when the two are word for word the same.⁴

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

To repeat: all constructions are to be understood as being in the sense of schemes.

In writing this book, I have depended heavily on the expository efforts of earlier authors.

The following works have been especially useful to me.

Demazure, Michel; Gabriel, Pierre. Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam, 1970. xxvi+700 pp.

Séminaire Heidelberg-Strasbourg 1965–66 (Groupes Algébriques), multigraphié par l’Institut de Mathématique 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. Birkh¨auser Boston, Inc., Boston, MA, 1998.

xiv+334 pp.

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

Notes of Ngo, Perrin, and Pink have also been useful.

Finally, I note that the new edition of SGA 3 is a magnificent resource.

4An example is Chevalley’s theorem on representations; see4.21.

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Throughout,kis a field andRis ak-algebra. All algebras over a field or ring are required to be commutative and finitely generated unless it is specified otherwise. Unadorned tensor products are overk. An extension ofkis a field containingk. WhenV is a vector space overk, we often writeVRorV .R/forR˝V. The symbolk^aldenotes an algebraic closure ofk, andk^sepdenotes the separable closure ofkink^al.

An algebraic scheme overk(or algebraick-scheme) is a scheme of finite type overk (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/overk, we often letX denote the scheme andjXjthe underlying topological space of closed points. When the base fieldk is understood, we write “algebraic scheme” for

“algebraic scheme overk”.

LetR be a finitely generatedk-algebra. We letAlg_R denote the category of finitely generatedR-algebras.

All categories are locally small (i.e., the objects may form a proper class, but the morphisms from one object to a second are required to form a set). When the objects form a set, the category is said to be small.

A functor is said to be an equivalence of categories if it is fully faithful and essentially surjective. A sufficiently strong version of the axiom of global choice then implies that there exists a quasi-inverse to the functor. We loosely refer to a natural transformation of functors as a map of functors.

An elementgof a partially ordered setP is agreatestelement if, for every elementain P,ag. An elementminP ismaximalif, forainP,maimpliesaDm. If a partially ordered set has a greatest element, it must be the unique maximal element, but otherwise there can be more than one maximal element (or none). Leastandminimalelements are defined similarly. When the partial order is inclusion, we often saysmallestfor least.

A diagramA!BC is said to beexactif the first arrow is the equalizer of the pair of arrows.

After p.161, all algebraic groups are affine. (The reader may wish to assume this throughout, and skip Chapters7and10.)

Foundations

We use the von Neumann–Bernays–G¨odel (NBG) set theory with the axiom of choice, which is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (ZFC).

This means that a sentence that doesn’t quantify over proper classes is a theorem of NBG if and only if it is a theorem of ZFC. The advantage of NBG is that it allows us to speak of classes.

It is not possible to define an “unlimited category theory” that includes the category of allsets, the category ofallgroups, etc., and also the categories of functors from one of these categories to another (Ernst 2015). Instead, one must consider only categories of functors from categories that are small in some sense. To this end, we fix a family of symbols.Ti/i2N

indexed byN, and letAlg⁰_k denote the category ofk-algebras of the formkŒT0; : : : ; Tn=a

5LetXbe an algebraic scheme over a field, and letX0be the set of closed points inXwith the induced topology. Then the mapU7!U\X0is a bijection from the set of open subsets ofX onto the set of open subsets ofX0. In particular,Xis connected if and only ifX0is connected. To recoverXfromX0, add a pointz for each proper irreducible closed subsetZofX0; the pointzlies in an open subsetUif and only ifU\Zis nonempty.

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for some n2Nand idealainkŒT0; : : : ; Tn. Thus the objects ofAlg⁰_k are indexed by the ideals in some subring kŒT0; : : : ; TnofkŒT0; : : :— in particular, they form a set, and so Alg⁰_kis small. We call the objects ofAlg⁰_k smallk-algebras. IfRis a smallk-algebra, then the categoryAlg⁰_Rof smallR-algebras has as objects pairs consisting of a smallk-algebraA and a homomorphismR!Aofk-algebras. Note that tensor products exist inAlg⁰_k — in fact, if we fix a bijectionN$NN, then˝becomes a well-defined bi-functor.

The inclusionAlg⁰_k,!^Algk is an equivalence of categories because every finitely generated k-algebra is isomorphic to a smallk-algebra. Choosing a quasi-inverse amounts to choosing an ordered set of generators for each finitely generated k-algebra. Once a quasi-inverse has been chosen, every functor onAlg⁰_khas a well-defined extension toAlg_k. Alternatively, readers willing to assume additional axioms in set theory, may use Mac Lane’s “one universe” solution to defining functor categories (Mac Lane 1969) or Grothendieck’s “multi universe” solution (DG, p.xxv), and take a smallk-algebra to be one that is small relative to the chosen universe.⁶

Prerequisites

A first course in algebraic geometry. Since these vary greatly, we review the definitions and statements that we need from algebraic geometry in AppendixA. In a few places, which can usually be skipped, we assume more algebraic geometry.

References

In addition to the references listed at the end (and in footnotes), I shall refer to the following of 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 algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. Masson & Cie, Éditeur, Paris; North-Holland Pub- lishing Co., Amsterdam, 1970. xxvi+700 pp.

SHS Séminaire Heidelberg-Strasbourg 1965–66 (Groupes Algébriques), multigraphié par l’Institut de Mathématique de Strasbourg (Gabriel, Demazure, et al.). 407 pp.

SGA 3 Schémas en Groupes, Séminaire de Géométrie Algèbriques du Bois Marie 1962–64, dirigé par M. Demazure et A. Grothendieck. Revised edition (P. Gille and P. Polo editors), Documents Mathématiques, SMF, 2011.

EGA Eléments de Géométrie Algébrique, A. Grothendieck; J. A. Dieudonné; I, Le langage des schémas (Springer Verlag 1971); II, III, IV Inst. Hautes Etudes Sci. Publ. Math. 8, 11, 17, 20, 24, 28, 32 , 1961–1967.

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

Introduction

The work can be divided roughly into six parts.

6Or they may simply ignore the problem, which is what most of the literature does.

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A. BASIC THEORY (CHAPTERS 1–10 )

The first ten chapters cover the general theory of algebraic groups (not necessarily affine).

After defining algebraic groups and giving some examples, we show that most of the basic theory of abstract groups (subgroups, normal subgroups, normalizers, centralizers, Noether isomorphism theorems, subnormal series, etc.) carries over with little change to algebraic group schemes. We relate affine algebraic groups to Hopf algebras, and we prove that all affine algebraic groups in characteristic zero are smooth. We study the linear representations of algebraic groups and the actions of algebraic groups on algebraic schemes. We show that every algebraic group is an extension of a finite ´etale algebraic group by a connected algebraic group, and that every connected group variety over a perfect field is an extension of an abelian variety by an affine group variety (Barsotti-Chevalley theorem).

B. PRELIMINARIES ON AFFINE ALGEBRAIC GROUPS (CHAPTERS11-13)

The next three chapters are preliminary to the more detailed study of affine algebraic groups in the later chapters. They cover Tannakian duality, in which the category of representations of an algebraic group plays the role of the topological dual of a locally compact abelian group; Jordan decompositions; the Lie algebra of an algebraic group; the structure of finite algebraic groups.

C. SOLVABLE ALGEBRAIC GROUPS (CHAPTERS14-17

The next four chapters study solvable algebraic groups. Among these are the diagonalizable groups and the unipotent groups.

An algebraic groupG is diagonalizable if every linear representationrWG!GLV of G is a direct sum of one-dimensional representations. In other words if, relative to some basis for V, r.G/ lies in the algebraic subgroupDn of diagonal matrices in GLn. An algebraic group that becomes diagonalizable over an extension of the base field is said to be of multiplicative type.

An algebraic groupG is unipotent if every nonzero representationV ofG contains a nonzero fixed vector. This implies that, relative to some basis forV, r.G/lies in the algebraic subgroupUnof strictly upper triangular matrices in GLn.

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

D. REDUCTIVE GROUPS (CHAPTERS 18-25) This is the heart of the book.

E. SURVEY CHAPTERS (CHAPTERS 26-27)

These describe the classification theorems of Satake-Selbach-Tits (the anistropic kernel etc.) and the Galois cohomology of algebraic groups (classification of the forms of an algebraic group; description of the classical algebraic groups in terms of algebras with involution;

etc.).

APPENDICES

In an appendix, we review the algebraic geometry needed.

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C

HAPTER

1 Basic definitions and properties

Recall thatkis a field, and that an algebraick-scheme is a scheme of finite type overk. We often omit thek. Morphisms ofk-schemes are required to bek-morphisms.

a. Definition

An algebraic group overkis a group object in the category of algebraic schemes overk. In detail, this means the following.

DEFINITION 1.1. LetGbe an algebraic scheme overkand letmWGG!G be a regular map. The pair.G; m/is analgebraic group overkif 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, and when G is an affine scheme, we call.G; m/ anaffine algebraic group.¹ A homomorphism'W.G; m/!.G⁰; m⁰/of algebraic groups is a regular map'WG !G⁰such that'ımDm⁰ı.''/.

Similarly, analgebraic monoidoverkis an algebraic schemeM overktogether with regular mapsmWMM !M andeW !M such that the diagrams (2) commute. An algebraic groupGistrivialifeW !Gis an isomorphism, and a homomorphism'W.G; m/!.G⁰; m⁰/ istrivialif it factors throughe⁰W !G⁰.

1As we note elsewhere (p.3, p.5,1.50,5.40, p.515) in most of the current literature, an algebraic group over a fieldkis defined to be a group variety over some algebraically closed fieldKcontainingktogether with a k-structure. In particular, nilpotents are not allowed.

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For example,

SLn

defDSpmkŒT11; T12; : : : ; Tnn=.det.Tij/ 1/

becomes a group variety with the usual matrix multiplication, .aij/; .bij/7!.cij/; cij DP

la_{i l}b_lj. For many more examples, see Chapter 2.

DEFINITION 1.2. Analgebraic subgroupof an algebraic group.G; m_G/overkis an algebraic group.H; mH/overksuch thatH is ak-subscheme ofG and the inclusion map is a homomorphism of algebraic groups. An algebraic subgroup that is a variety is called a subgroup variety.

HOMOGENEITY

1.3. For an algebraic schemeX overk, we writejXjfor the underlying topological space ofX, and.x/for the residue field at a pointx ofjXj(it is a finite extension ofk). We identifyX.k/with the set of pointsxofjXjsuch that.x/Dk. Let.G; m/be an algebraic group overk. The mapm.k/WG.k/G.k/!G.k/makesG.k/into a group with neutral elemente./and inverse map inv.k).

Whenk is algebraically closed,G.k/D jGj, and somWGG!G makesjGjinto a group. The maps x 7!x ¹ and x7!ax (a2G.k/) are automorphisms of jGj as a topological space.

In general, whenkis not algebraically closed,mdoes not makejGjinto a group, and even whenkis algebraically closed, it does not makejGjinto atopologicalgroup.²

1.4. Let.G; m/be an algebraic group overk. For eacha2G.k/, there is a translation map laWG' fag G ^m!G; x7!ax.

Fora; b2G.k/,

laıl_bDl_ab

andleDid. Thereforel_a ıl_a 1DidDl_a 1ıl_a , and solais an isomorphism sendingeto a. HenceGis homogeneous³whenkis algebraically closed (but not in general otherwise;

see1.7).

ALGEBRAIC GROUPS AS FUNCTORS

Since we allow nilpotents in the structure sheaf, the points of an algebraic group with coordinates in a field, even algebraically closed, do not convey much information about the group. Thus, it is natural to consider its points in ak-algebra. Once we do that, the points captureallinformation about the algebraic group.

2Assumekis algebraically closed. The mapjmjW jGGj ! jGjis continuous, andjGGj D jGj jGjas a set, butnotas a topological space. The multiplication mapjGj jGj ! jGj, i.e.,G.k/G.k/!G.k/, need not be continuous for the product topology.

3An algebraic schemeX overkis said to be homogeneous if the group of automorphisms ofX (as a k-scheme) acts transitively onjXj.

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a. Definition 19 1.5. An algebraic schemeX overkdefines a functor

XQW^Alg⁰k !^Set; R X.R/;

and the functorX XQ is fully faithful (Yoneda lemma,A.28); in particular,XQ determines X uniquely up to a unique isomorphism. We say that a functor fromk-algebras to sets is representableif it is of the formXQ.

Let.G; m/be an algebraic group overk. ThenR .G.R/; m.R//is a functor fromk- algebras to groups whose underlying functor to sets is representable, and every such functor arises from an essentially unique algebraic group. Thus, to give an algebraic group overk amounts to giving a functorAlg⁰_k!^Grpwhose underlying functor to sets is representable by an algebraic scheme. We sometimes writeGQ forGregarded as a functor.

We often describe a homomorphism of algebraic groups by describing its action on R-points. For example, when we say that invWG!Gis the mapx7!x ¹, we mean that, for allk-algebrasRand allx2G.R/, inv.x/Dx ¹.

From this perspective, SLnis the algebraic group overkwhoseR-points are thenn matrices with entries inRand determinant1.

1.6. An algebraic subschemeH of an algebraic groupGis an algebraic subgroupofGif and only ifH.R/is a subgroup ofG.R/for allk-algebrasR. In more detail, assume that H.R/is a subgroup ofG.R/for all (small)R; then the Yoneda lemma (A.28) shows that the maps

.h; h⁰/7!hh⁰WH.R/H.R/!H.R/

arise from a morphismmHWHH!H, and.H; mH/is an algebraic subgroup of.G; mG/.

1.7. Consider the functor ofk-algebras

3WR fa2Rja³D1g:

This is represented by Spm.kŒT =.T³ 1//, and so it is an algebraic group. We consider three cases.

(a) The fieldkis algebraically closed of characteristic¤3. Then

kŒT =.T³ 1/'kŒT =.T 1/kŒT =.T /kŒT =.T ²/

where1; ; ²are the cube roots of1ink. Thus,3is a disjoint union of three copies of Spm.k/indexed by the cube roots of1ink.

(b) The fieldkis of characteristic¤3but does not contain a primitive cube root of1.

Then

kŒT =.T³ 1/'kŒT =.T 1/kŒT =.T²CT C1/;

and so3 is a disjoint union of Spm.k/and Spm.kŒ/where is a primitive cube root of1. In particular,j3jis not homogeneous.

(c) The fieldkis of characteristic3. Then

T³ 1D.T 1/³;

and so 3 is not reduced. Although3.K/D1for all fieldsK containing k, the algebraic group3is not trivial.

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DENSITY OF POINTS

In general, thek-points of an algebraic group tell us little about the group, but sometimes they do. For example, a smooth algebraic groupGover a separably closed fieldkis commutative ifG.k/is commutative (1.21).

1.8. LetX be an algebraic scheme overk, and letSbe a subset ofX.k/ jXj. We say thatS isschematically denseinX if the family of homomorphisms

f 7!f .s/WO^X !.s/Dk; s2S;

is injective. LetSX.k/be schematically dense inX:

(a) ifZis a closed algebraic subscheme ofX such thatZ.k/containsS, thenZDX; (b) ifu; vWX!Y are regular maps fromX to a separated algebraic schemeY such that

u.s/Dv.s/for alls2S, thenuDv.

IfSX.k/is schematically dense inX, thenSis dense injXj, and the converse is true if X is reduced. A schematically dense subset remains schematically dense under extension of the base field. If an algebraic schemeX admits a schematically dense subsetSX.k/, then it is geometrically reduced. For a geometrically reduced schemeX, a subset ofX.k/is schematically dense inX if and only if it is dense injXj. See (A.62) et seq.

1.9. LetG be an algebraic group over a fieldk, and letk⁰ be a field containingk. We say that G.k⁰/ is dense in G if the only closed algebraic subscheme Z of G such that Z.k⁰/DG.k⁰/isG itself.

(a) IfG.k⁰/is dense inG, thenG is reduced. Conversely, ifG is geometrically reduced, thenG.k⁰/is dense inG if and only if it is dense in the topological spacejG_k0j. (A.59, A.60).

(b) IfG is smooth, thenG.k⁰/is dense inGwheneverk⁰contains the separable closure ofk(A.44).

(c) G.k/is dense inG if and only ifGis reduced andG.k/is densejGj. ALGEBRAIC GROUPS OVER RINGS

Although we are only interested in algebraic groups over fields, occasionally, we shall need to consider them over more general base rings.

1.10. LetRbe a (finitely generated) k-algebra. Formally, an algebraic scheme overR is a scheme X equipped with a morphism X !SpmRof finite type. Less formally, we can think ofX as an algebraic scheme overksuch thatO^X is equipped with anR-algebra structure compatible with its k-algebra structure. For example, affine algebraic schemes overRare the spectra finitely generatedR-algebrasA. A morphism of algebraicR-schemes 'WX !Y is a morphism of schemes compatible with theR-algebra structures, i.e., such thatO^Y !'O^X is a homomorphism of sheaves ofR-algebras. LetG be an algebraic scheme overRand letmWGG!G be a morphism ofR-schemes. The pair.G; m/is an algebraic group overRif there existR-morphismseWSpm.R/!Gand invWG!G such that the diagrams (2) and (3) commute. For example, an algebraic group.G; m/overkgives rise to an algebraic group.GR; mR/overRby extension of scalars.

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b. Basic properties of algebraic groups 21

b. Basic properties of algebraic groups

PROPOSITION1.11. The mapse andinvin (1.1) are uniquely determined by.G; m/. If 'W.G; mG/!.H; mH/is a homomorphism of algebraic groups, then 'ıeG DeH and 'ıinvGDinvHı'.

PROOF. It suffices to prove the second statement. For ak-algebraR, the map'.R/is a homomorphism of abstract groups.G.R/; mG.R//!.H.R/; mH.R//, and so it maps the neutral element ofG.R/to that ofH.R/and the inversion map onG.R/to that onH.R/.

The Yoneda lemma (A.28) now shows that the same is true for'. ₂ PROPOSITION1.12. Algebraic groups are separated (as algebraic schemes).

PROOF. Let.G; m/be an algebraic group. The diagonal inGGis the inverse image of the closed pointe2G.k/under the mapmı.idinv/WGG!G sending.g1; g2/tog1g₂¹,

and so it is closed. ₂

Therefore “group variety” = “geometrically reduced algebraic group”.

COROLLARY1.13. LetGbe an algebraic group overk, and letk⁰be a field containingk.

IfG.k⁰/is dense inG, then a homomorphismG!H of algebraic groups is determined by its action onG.k⁰/.

PROOF. Let'; '⁰be homomorphismsG!H. BecauseH is separated, the subscheme ZofGon which they agree is closed (seeA.37). If'.x/D'⁰.x/for allx2G.k⁰/, then

Z.k⁰/DG.k⁰/, and soZDG. ₂

Recall that an algebraic scheme over a field is a finite disjoint union of its (closed-open) connected components (A.14). For an algebraic groupG, we letG^ıdenote the connected component ofG containinge, and we call it theidentity(orneutral)componentofG. PROPOSITION1.14. LetG be an algebraic group. The identity componentG^ıofG is an algebraic subgroup ofG. Its formation commutes with extension of the base field: for every fieldk⁰containingk,

G^ı

k⁰'.G_k0/^ı:

In particular,G is connected if and only if G_k0 is connected; the algebraic groupG^ıis geometrically connected; every connected algebraic group is geometrically connected.

For the proof, we shall need the following elementary lemma. Recall (A.84) that the set of connected components of an algebraic scheme X can be given the structure of a zero-dimensional algebraic variety0.X /. Moreover,X !0.X /is a regular map whose fibres are the connected components ofX.

LEMMA1.15. LetX be a connected algebraic scheme overksuch thatX.k/¤ ;. ThenX is geometrically connected; moreover, for any algebraic schemeY overk,

0.XY /'0.Y /:

In particular,XY is connected ifY is connected.

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PROOF. Because0.X /is a zero-dimensional algebraic variety, it equals Spm.A/for some

´etalek-algebraA(A.82). IfAhad more than one factor,O.X /would contain nontrivial idempotents, andX would not be connected. Therefore, Ais a field containingk, and, becauseX.k/is nonempty, it equalsk. Now

0.X_kal/^A.84D 0.X /_kal DSpm.k^al/, which shows thatX_kal is connected, and

0.XY /^A.84' 0.X /0.Y /'0.Y /;

as required. ₂

PROOF(OF1.14). The identity component G^ı of G has a k-point, namely, e, and so G^ıG^ıis a connected component ofGG(1.15). Asmmaps.e; e/toe, it mapsG^ıG^ı intoG^ı. Similarly, inv mapsG^ıintoG^ı. ThereforeG^ıis an algebraic subgroup ofG. For any extensionk⁰ofk,

.G!0.G//_k0'G_k0!0.G_k0/

(seeA.84). As G^ı is the fibre over e, this implies that .G^ı/_k0 '.G_k0/^ı. In particular, .G^ı/_kal '.G_kal/^ı, and soG^ıis geometrically connected. ₂ COROLLARY1.16. A connected algebraic group is irreducible.

PROOF. It suffices to show thatGisgeometricallyirreducible. Thus, we may suppose thatk is algebraically closed, and hence thatGis homogeneous (1.4). By definition, no irreducible component is contained in the union of the remainder. Therefore, there exists a point that lies on exactly one irreducible component. By homogeneity, all points have this property, and so the irreducible components are disjoint. AsjGjis connected, there must be only one,

and soGis irreducible. ₂

SUMMARY1.17. The following conditions on an algebraic groupGoverkare equivalent:

(a) G is irreducible;

(b) G is connected;

(c) G is geometrically connected.

WhenGis affine, the conditions are equivalent to:

(d) the quotient ofO.G/by its nilradical is an integral domain.

Algebraic groups are unusual. For example, the subscheme ofA²defined by the equation X Y D0is connected but not irreducible (and hence is not the underlying scheme of an algebraic group).

PROPOSITION1.18. LetGbe an algebraic group overk.

(a) If G is reduced andk is perfect, thenG is geometrically reduced (hence a group variety).

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

PROOF. (a) This is true for every algebraic scheme (A.39).

(b) It suffices to show thatG_kal is smooth, but some point ofG_kal is smooth (A.52), and so every point is smooth becauseG_kal is homogeneous (1.4). ₂

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b. Basic properties of algebraic groups 23 Therefore

“group variety” = “smooth algebraic group”.

In characteristic zero, all algebraic groups are smooth (see 3.38below for a proof in the affine case and10.36for the general case).

EXAMPLE1.19. Letkbe a nonperfect field of characteristicp, and leta2kXk^p. LetG be the algebraic subgroup ofA¹defined by the equation

Y^p aX^pD0:

The ringADkŒX; Y =.Y^p aX^p/is reduced becauseY^p aX^pis irreducible inkŒX; Y , butAacquires a nilpotenty a^p¹xwhen tensored withk^al, and soGis not geometrically reduced. (Over the algebraic closure ofk, it becomes the lineY Da^p¹X with multiplicity p.)

DEFINITION 1.20. An algebraic group.G; m/iscommutativeifmıtDm, wheret is the transposition map.x; y/7!.y; x/WGG!GG.

PROPOSITION1.21. An algebraic groupG is commutative if and only ifG.R/is commutative for allk-algebrasR. A group varietyG is commutative ifG.k^sep/is commutative.

PROOF. According to the Yoneda lemma (A.28),mıtDmif and only ifm.R/ıt .R/D m.R/for allk-algebrasR, i.e., if and only ifG.R/is commutative for allR. The proves the first statement. LetGbe a group variety. IfG.k^sep/is commutative, thenmıt andmagree on.GG/.k^sep/, which is dense inGG(1.9). ₂ PROPOSITION1.22. The following conditions on an algebraic groupG are equivalent:

(a) G is smooth;

(b) G^ıis smooth;

(c) the local ringO^G;e is regular;

(d) the tangent spaceTe.G/toG atehas dimensiondimG;

(e) G is geometrically reduced;

(f) for allk-algebrasRand all idealsI inRsuch thatI²D0, the mapG.R/!G.R=I / is surjective.

PROOF. (a)H)(b)H)(c)H)(d): These implications are obvious (seeA.48,A.51).

(d)H)(a). The condition implies that the pointeis smooth onG (A.51), and hence on G_kal. By homogeneity (1.4), all points onG_kal are smooth, which means thatGis is smooth.

(a)”(e). This was proved in (1.18).

(a)”(f). This is a standard criterion for an algebraic scheme to be smooth (A.53).₂ COROLLARY1.23. For an algebraic groupG,

dimTe.G/dimG;

with equality if and only ifGis smooth.

PROOF. In general, for a pointeon an algebraick-schemeG with.e/Dk, dimTe.G/

dimGwith equality if and only ifO^G;e is regular (A.48). But we know (1.22), thatO^G;e is

regular if and only ifGis smooth. ₂

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c. Algebraic subgroups

For a closed subset S of an algebraic schemeX, we let S_red denote the reduced closed subscheme ofX withjS_redj DS. A morphismWY !X factors throughS_redifjjfactors throughS andY is reduced. SeeA.25.

PROPOSITION1.24. Let .G; m/ be an algebraic group over k. If G_red is geometrically reduced, then it is an algebraic subgroup ofG.

PROOF. IfG_red is geometrically reduced, thenG_redG_red is reduced (A.39), and so the restriction ofmtoG_redG_red factors throughG_red,!G:

G_redG_red ^m!^red G_red,!G.

Similarly,eand inv induce maps !G_redandG_red!G_red, and these make the diagrams

(2,3), p.17, commute for.G_red; m_red/. ₂

COROLLARY1.25. LetG be an algebraic group overk. Ifk is perfect, thenG_red is a smooth algebraic subgroup ofG.

PROOF. Over a perfect field, reduced algebraic schemes are geometrically reduced (1.46), and soG_redis geometrically reduced, hence an algebraic subgroup ofG, and hence smooth

(1.22). ₂

LEMMA1.26. LetGbe an algebraic group overk. The Zariski closureSN of a(n abstract) subgroupS ofG.k/is a subgroup ofG.k/.

PROOF. Fora2G.k/, the mapx7!axWG.k/!G.k/is a homeomorphism because its inverse is of the same form. Fora2S, we haveaS S NS, and soaSN D.aS / NS.

Thus, for a2 NS, we haveSa NS, and soS aN D.Sa/ NS. HenceSNSN NS. The map x7!x ¹WG.k/!G.k/is a homeomorphism, and so.S /N ¹D.S ¹/ D NS. ₂ PROPOSITION1.27. Every algebraic subgroup of an algebraic group is closed (in the Zariski topology).

PROOF. LetH be an algebraic subgroup of an algebraic groupG. IfH_kal is closed inG_kal

thenH is closed in G (seeA.10) and so we may suppose thatk is algebraically closed.

We may also suppose thatH andG are reduced, because passing to the reduced algebraic subgroup doesn’t change the underlying topological space. By definition,jHjis locally closed, i.e., open in its closureS. NowSis a subgroup ofjGj(1.26), and it is a finite disjoint union of cosets ofjHj. As each coset is open, it is also closed. ThereforeH is closed inS,

and so equals it. ₂

COROLLARY1.28. The algebraic subgroups of an algebraic group satisfy the descending chain condition.

PROOF. In fact, the closed subschemes of an algebraic scheme satisfy the descending chain

condition (A.19). ₂

COROLLARY1.29. Every algebraic subgroup of an affine algebraic group is affine.

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

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c. Algebraic subgroups 25 COROLLARY1.30. LetH andH⁰be subgroup varieties of an algebraic groupG overk.

ThenH DH⁰ifH.k⁰/DH⁰.k⁰/for some fieldk⁰containing the separable closure ofk.

PROOF. The condition implies that

H.k⁰/D H\H⁰

.k⁰/DH⁰.k⁰/: (4) ButH\H⁰is closed inH (1.27). AsH is a variety,H.k⁰/is dense inH (A.61), and so (4) implies thatH\H⁰DH. Similarly,H\H⁰DH⁰. ₂ PROPOSITION1.31. LetG be an algebraic group overk, and letSbe a closed subgroup of G.k/. There is a unique reduced algebraic subgroupH ofGsuch thatSDH.k/(andH is geometrically reduced). The algebraic subgroupsH ofG that arise in this way are exactly those for whichH.k/is schematically dense inH (i.e., such thatH is reduced andH.k/is dense injHj).

PROOF. LetH denote the reduced closed subscheme ofG such thatjHjis the closure of S injGj. ThenSDG.k/\ jHj DH.k/. AsH is reduced andH.k/is dense injHj, it is geometrically reduced (1.8). ThereforeHH is reduced, and so the mapmGWHH !G factors throughH. Similarly, invG restricts to a regular mapH !H and !G factors throughH. ThusH is an algebraic subgroup ofG. AlsoH.k/is schematically dense in H because it is dense andH is reduced. Conversely, ifH is a reduced algebraic subgroup ofGsuch thatH.k/is dense injHj, then the above construction starting withSDH.k/

gives backH. ₂

COROLLARY1.32. LetG be an algebraic group overk, and letS be a closed subgroup ofG.k/. There is a unique subgroup varietyH ofG such thatSDH.k/. The subgroup varietiesH ofG that arise in this way are exactly those for whichH.k/is dense injHj.

PROOF. This is a restatement of the proposition. ₂

COROLLARY1.33. LetGbe an algebraic group over a separably closed fieldk. The map H 7!H.k/is a bijection from the set of subgroup varieties ofG onto the set of closed (abstract) subgroups ofG.k/.

PROOF. Askis separably closed,H.k/is dense injHjfor every group subvariety ofG.₂ DEFINITION 1.34. LetG be an algebraic group overk, and letSbe a subgroup ofG.k/.

The unique subgroup varietyH ofG such thatH.k/is the Zariski closure ofS is called the Zariski closureofSinG.

ASIDE1.35. Letkbe an infinite perfect field. ThenH.k/is dense injHjfor any connected group varietyHoverk(cf.3.26below). LetGbe an algebraic group overk; then the mapH7!H.k/is a bijection from the set of connected subgroup varieties ofGto the set of closed subgroups ofG.k/

whose closures injGjare connected.

PROPOSITION1.36. Let.Hj/_j₂_J be a family of algebraic subgroups ofG. ThenH ^defD T

j2JHj is an algebraic subgroup ofG. IfG is affine, thenH is affine, and its coordinate ring isO.G/=I whereI is the ideal inO.G/generated by the idealsI.Hj/of theHj.

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PROOF. Certainly,H is a closed subscheme (A.19). Moreover, for allk-algebrasR, H.R/D\

j2JHj.R/ (intersection insideG.R/),

which is a subgroup ofG.R/, and soH is an algebraic subgroup ofG(1.6). Assume thatG is affine. For anyk-algebraR,

Hj.R/D fg2G.R/jfR.g/D0for allf 2I.Hj/g: Therefore,

H.R/D fg2G.R/jfR.g/D0for allf 2[

I.Hj/g

DHom.O.G/=I; R/: ₂

In fact, because of (1.28), every infinite intersection is equal to a finite intersection.

EXAMPLE1.37. (a) LetGDGLp over a field of characteristicp. Then SL_p and the group H of scalar matrices inGare smooth subgroups ofG, but SLp\H Dp is not reduced.

(b) LetGDG²a. ThenH1DGa f0gandH1D f.x; x²Cax⁴/gare smooth algebraic subgroups ofG, but their intersection is not reduced.

NORMAL AND CHARACTERISTIC SUBGROUPS DEFINITION 1.38. LetG be an algebraic group.

(a) An algebraic subgroupH ofGisnormalifH.R/is normal inG.R/for allk-algebras R.

(b) An algebraic subgroupH ofG ischaracteristicif˛ .HR/DHRfor allk-algebras Rand all automorphisms˛ofGR.

The conditions hold for allk-algebrasRif they hold for all smallk-algebras. In (b)GRand HRcan be interpreted as functors from the category of (small) finitely generatedR-algebras to the category of groups, or as algebraicR-schemes (i.e., as algebraick-schemes equipped with a morphism to Spm.R/(1.10)). Because of the Yoneda lemma (loc. cit.), the two interpretations give the same condition.

PROPOSITION1.39. The identity componentG^ıof an algebraic groupGis a characteristic subgroup ofG (hence a normal subgroup).

PROOF. AsG^ıis the unique connected open subgroup ofG containinge, every automorphism ofGfixingemapsG^ıinto itself. Letk⁰be a field containingk. As.G^ı/_k0D.G_k0/^ı, every automorphism ofGk⁰ fixingemaps.G^ı/k⁰into itself.

LetRbe ak-algebra and let˛ be an automorphism ofGR. We regardG_R^ı andGR as algebraicR-schemes. It suffices to show that˛.G_R^ı/G_R^ı, and, becauseG_R^ı is an open subscheme ofG_R, for this it suffices to show that˛.jG_R^ıj/ jG_R^ıj. Letx2 jG_R^ıj, and lets be the image ofxin Spm.R/. Thenxlies in the fibreG_.s/ofGRovers:

G_R G_.s/

Spm.R/ Spm..s//:

In fact,x2 jG_R^ı\G_.s/j D jG_.s/^ı j. From the first paragraph of the proof,˛_.s/.x/2 jG_.s/^ı j,

and so˛.x/2 jG_R^ıj, as required. ₂

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d. Examples 27 REMARK1.40. LetHbe an algebraic subgroup ofG. If˛.H_R/H_Rfor allk-algebrasR and endomorphisms˛ofGR, thenHis characteristic. To see this, let˛be an automorphism ofGR. Then˛ ¹.HR/HR, and soHR˛.HR/HR.

NOTES. The definition of characteristic subgroup agrees with DG II,1, 3.9, p.166. The proof that G^ıis characteristic is from DG II,5, 1.1, p.234.

DESCENT OF SUBGROUPS

1.41. LetGbe an algebraic scheme over a fieldk, and letk⁰be a field containingk. Let G⁰DG_k0, and letH⁰be an algebraic subgroup ofG_k0.

(a) There exists at most one algebraic subgroupH of G such that Hk⁰ DH⁰ (as an algebraic subgroup ofG_k0). When such anH exists, we say thatH⁰isdefined overk (as an algebraic subgroup ofG⁰).

(b) Letk⁰be a Galois extension ofk(possibly infinite), and let DGal.k⁰=k/. ThenH⁰ is defined overkif and only if it is stable under the action of onG⁰, i.e., the sheaf of ideals defining it is stable under the action of onO^G⁰.

(c) Letk⁰Dk^sep. A subgroup varietyH⁰is stable under the action of onG⁰(hence defined overk) if and only ifH⁰.k⁰/is stable under the action of onG.k⁰/.

Apply (A.55,A.56).

d. Examples

We give some examples to illustrate what can go wrong.

1.42. Letkbe a nonperfect field of characteristicp > 2, and lett2kXk^p. LetG be the algebraic subgroup ofA²defined by

Y^p Y DtX^p.

This is a connected group variety overkthat becomes isomorphic toA¹overk^al, butG.k/

is finite (and so not dense inG). IfkDk0.t /, thenG.k/D feg(Rosenlicht 1957, p.46).

1.43. Letkbe nonperfect of characteristicp, and lett2kXk^p. LetG be the algebraic subgroup ofA¹defined by the equation

X^p² tX^pD0:

ThenG_redis not an algebraic group for any mapmWG_redG_red!G_red(Exercise2-5; SGA 3, VIA, 1.3.2a).

1.44. Letkbe nonperfect of characteristicp3, and lett2kXk^p. LetGbe the algebraic subgroup ofA⁴defined by the equations

U^p tV^pD0DX^p t Y^p:

ThenGis aconnectedalgebraic group of dimension2, butG_redis singular at the origin, and hence not an algebraic group for any mapm(SGA 3, VIA, 1.3.2b).

1.45. We saw in (1.25) thatG_redis an algebraic subgroup ofG whenkis perfect. However, it need not be normal even whenG is connected. For examples, see (2.23) below.

Algebraic Groups