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Algebraic Groups, Lie Groups,

and their

Arithmetic Subgroups

J.S. Milne

Version 3. April 1, 2011

This work is a modern exposition of the theory of algebraic groups (affine group schemes), Lie groups, and their arithmetic subgroups.

BibTeX information:

@misc{milneALA, author={Milne, James S.}, title={Algebraic Groups, Lie Groups, and their Arithmetic Subgroups}, year={2011}, note={Available at www.jmilne.org/math/} }

v1.00 April 29, 2009. First version on the web (first two chapters only). v1.01 May 10, 2009. Minor fixes. v1.02 June 1, 2009. More minor fixes. v2.00 April 27, 2010. Posted all 6 chapters (378 pages). v3.00 April 1, 2011. Revised and expanded (422 pages).

Please send comments and corrections to me at the address on my website http://www.jmilne.org/math/.

The photo is of the famous laughing Buddha on The Peak That Flew Here, Hangzhou, Zhejiang, China.

Copyright c^ 2005, 2006, 2009, 2010, 2011 J.S. Milne.

Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder.

• Table of Contents
  • Preface
• I Basic Theory of Affine Groups
  • 1 Introductory overview
  • 2 Definitions
  • 3 Examples
  • 4 Some basic constructions
  • 5 Affine groups and Hopf algebras
  • 6 Affine groups and affine group schemes
  • 7 Group theory: subgroups and quotient groups.
  • 8 Representations of affine groups
  • 9 Group theory: the isomorphism theorems
  • 10 Recovering a group from its representations; Jordan decompositions
  • 11 Characterizations of categories of representations
  • 12 Finite flat affine groups
  • 13 The connected components of an algebraic group
  • 14 Groups of multiplicative type; tori
  • 15 Unipotent affine groups
  • 16 Solvable affine groups
  • 17 The structure of algebraic groups
  • 18 Example: the spin groups
  • 19 The classical semisimple groups
  • 20 The exceptional semisimple groups
  • 21 Tannakian categories
• II Lie Algebras and Algebraic Groups
  • 1 The Lie algebra of an algebraic group
  • 2 Lie algebras and algebraic groups
  • 3 Nilpotent and solvable Lie algebras
  • 4 Unipotent algebraic groups and nilpotent Lie algebras
  • 5 Semisimple Lie algebras and algebraic groups
  • 6 Semisimplicity of representations
    • acteristic Zero III The Structure of Semisimple Lie Algebras and Algebraic Groups in Char-
  • 1 Root systems and their classification
  • 2 Structure of semisimple Lie algebras and their representations
  • 3 Structure of semisimple algebraic groups and their representations
  • 4 Real Lie algebras and real algebraic groups
  • 5 Reductive groups
• IV Lie groups
  • 1 Lie groups
  • 2 Lie groups and algebraic groups
  • 3 Compact topological groups
• V The Structure of Reductive Groups: the split case
  • 1 Split reductive groups: the program
  • 2 The root datum of a split reductive group
  • 3 Borel fixed point theorem and applications
  • 4 Parabolic subgroups and roots
  • 5 Root data and their classification
  • 6 Construction of split reductive groups: the existence theorem
  • 7 Construction of isogenies of split reductive groups: the isogeny theorem
  • 8 Representations of split reductive groups
• VI The Structure of Reductive Groups: general case
  • 1 The cohomology of algebraic groups; applications
  • 2 Classical groups and algebras with involution
  • 3 Relative root systems and the anisotropic kernel.
• VII Arithmetic Subgroups
  • 1 Commensurable groups
  • 2 Definitions and examples
  • 3 Questions
  • 4 Independence of  and L.
  • 5 Behaviour with respect to homomorphisms
  • 6 Ad`elic description of congruence subgroups
  • 7 Applications to manifolds
  • 8 Torsion-free arithmetic groups
  • 9 A fundamental domain for SL
  • 10 Application to quadratric forms
  • 11 “Large” discrete subgroups
  • 12 Reduction theory
  • 13 Presentations
  • 14 The congruence subgroup problem
  • 15 The theorem of Margulis
  • 16 Shimura varieties
• Bibliography
• Index

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.

Algebraic groups are groups defined by polynomials. Those that we shall be concerned with in this book can all be realized as groups of matrices. For example, the group of matrices of determinant 1 is an algebraic group, as is the orthogonal group of a symmetric bilinear form. The classification of algebraic groups and the elucidation of their structure were among the great achievements of twentieth century mathematics (Borel, Chevalley, Tits and others, building on the work of the pioneers on Lie groups). Algebraic groups are used in most branches of mathematics, and since the famous work of Hermann Weyl in the 1920s they have also played a vital role in quantum mechanics and other branches of physics (usually as Lie groups). Arithmetic groups are groups of matrices with integer entries. They are a basic source of discrete groups acting on manifolds. The first goal of the present work is to provide a modern exposition of the theory of al- gebraic groups. It has been clear for fifty years, that in the definition of an algebraic group, the coordinate ring should be allowed to have nilpotent elements,^1 but the standard exposi- tions^2 do not allow this.^3 In recent years, the tannakian duality^4 between algebraic groups and their categories of representations has come to play a role in the theory of algebraic groups similar to that of Pontryagin duality in the theory of locally compact abelian groups. Chapter I develops the basic theory of algebraic groups, including tannakian duality. Lie algebras are a essential tool in studying both algebraic groups and Lie groups. In Chapter II develops the basic theory of Lie algebras and discusses the functor from alge- braic groups to Lie algebras. As Cartier (1956) noted, the relation between Lie algebras and algebraic groups in char- acteristic zero is best understood through their categories of representations. In Chapter III we review the classification of semisimple Lie algebras and their representations, and we

(^1) See, for example, Cartier 1962. Without nilpotents the centre of SLp in characteristic p is visible only through its Lie algebra. Moreover, the standard isomorphism theorems fail, and so the intuition provided by group theory is unavailable. Consider, for example, the subgroups H D SLp and N D Gm (diagonal) of GLp over a field of characteristic p. If nilpotents are not allowed, then N \ H D 1 , and the map H=H \ N! HN=N is the homomorphism SLp! PGLp , which is an inseparable isogeny of degree p; in particular, it is injective and surjective but not an isomorphism. While it is true that in characteristic zero all algebraic groups are reduced, this is a theorem that can only be stated when nilpotents are allowed. (^2) The only exceptions I know of are Demazure and Gabriel 1970, Waterhouse 1979, and SGA3. While the first two do not treat the classification of semisimple algebraic groups over fields, the third assumes it. (^3) Worse, much of the expository literature is based, in spirit if not in fact, on the algebraic geometry of Weil’s Foundations (Weil 1962). Thus an algebraic group over k is defined to be an algebraic group over some large algebraically closed field together with a k-structure. This leads to a terminology in conflict with that of modern algebraic geometry, in which, for example, the kernel of a homomorphism of algebraic groups over a field k need not be an algebraic group over k. Moreover, it prevents the theory of split reductive groups being developed intrinsically over the base field. When Borel first introduced algebraic geometry into the study of algebraic groups in the 1950s, Weil’s foundations were they only ones available to him. When he wrote his influential book Borel 1969b, he persisted in using Weil’s approach to algebraic geometry, and, with the exceptions noted in the preceding footnote, all subsequent authors have followed him. (^4) Strictly, this should be called the “duality of Tannaka, Krein, Milman, Hochschild, Grothendieck, Saave- dra Rivano, Deligne, et al.,” but “tannakian duality” is shorter. In his R´ecoltes et Semailles, 1985-86, 18.3.2, Grothendieck argues that “Galois-Poincar´e” would be more appropriate than “Tannaka”.

˘ We allow our rings to have nilpotents, i.e., we don’t require that our algebraic groups be reduced. ˘ We do not identify an algebraic group G with its points G.k/ with in k, even when the ground field k is algebraically closed. Thus, a subgroup of an algebraic group G is an algebraic subgroup, not an abstract subgroup of G.k/. ˘ An algebraic group G over a field k is intrinsically an object over k, and not an object over some algebraically closed field together with a k-structure. Thus, for example, a homomorphism of algebraic groups over k is truly a homomorphism over k, and not over some large algebraically closed field. In particular, the kernel of such a homomorphism is an algebraic subgroup over k. Also, we say that an algebraic group over k is simple, split, etc. when it simple, split, etc. as an algebraic group over k, not over some large algebraically closed field. When we want to say that G is simple over k and remains simple over all fields containing k, we say that G is geometrically (or absolutely) simple. ˘ For an algebraic group G over k and an extension field K, G.K/ denotes the points of G with coordinates in K and GK denotes the algebraic group over K obtained from G by extension of the base field.

Beyond its greater simplicity, there is another reason for replacing the old terminology with the new: for the study of group schemes over bases more general than fields there is no old terminology.

0a Notations; terminology

We use the standard (Bourbaki) notations: N D f0; 1; 2; : : :g; Z D ring of integers; Q D field of rational numbers; R D field of real numbers; C D field of complex numbers; Fp D Z=pZ D field with p elements, p a prime number. For integers m and n, mjn means that m divides n, i.e., n 2 mZ. Throughout the notes, p is a prime number, i.e., p D 2; 3; 5; : : :. Throughout k is the ground ring (always commutative, and usually a field), and R always denotes a commutative k-algebra. Unadorned tensor products are over k. Notations from commutative algebra are as in my primer CA (see below). When k is a field, ksep denotes a separable algebraic closure of k and kal^ an algebraic closure of k. The dual Homk-lin.V; k/ of a k-module V is denoted by V . The transpose of a matrix M is denoted by M t^. We use the terms “morphism of functors” and “natural transformation of functors” in- terchangeably. When F and F 0 are functors from a category, we say that “a homomorphism F .a/! F 0 .a/ is natural in a” when we have a family of such maps, indexed by the objects a of the category, forming a natural transformation F! F 0. For a natural transformation ˛W F! F 0 , we often write ˛R for the morphism ˛.R/W F .R/! F 0 .R/. When its action on morphisms is obvious, we usually describe a functor F by giving its action R F .R/ on objects. Categories are required to be locally small (i.e., the morphisms between any two objects form a set), except for the category A^ of functors A! Set. A diagram A! B ⇒ C is said to be exact if the first arrow is the equalizer of the pair of arrows; in particular, this means that A! B is a monomorphism (cf. EGA I, Chap. 0, 1.4). Here is a list of categories:

Category Objects Page Algk commutative k-algebras A_^ functors A! Set Comodk .C / finite-dimensional comodules over C p. 100 Grp (abstract) groups Repk .G/ finite-dimensional representations of G p. 95 Repk .g/ finite-dimensional representations of g Set sets Veck finite-dimensional vector spaces over k

In each case, the morphisms are the usual ones, and composition is the usual composition. Throughout the work, we often abbreviate names. In the following table, we list the shortened name and the page on which we begin using it.

AG Algebraic Geometry (v5.21, 2011). CFT Class Field Theory (v4.00, 2008).

The links to CA, GT, FT, and AG in the pdf file will work if the files are placed in the same directory. Also, I use the following abbreviations:

Bourbaki A Bourbaki, Algebre. Bourbaki AC Bourbaki, Algebre Commutative (I–IV 1985; V–VI 1975; VIII–IX 1983; X 1998). Bourbaki LIE Bourbaki, Groupes et Alg`ebres de Lie (I 1972; II–III 1972; IV–VI 1981). Bourbaki TG Bourbaki, Topologie G´en´erale. DG Demazure and Gabriel, Groupes Alg´ebriques, Tome I, 1970. EGA El´ements de G´eom´etrie Alg´ebrique, Grothendieck (avec Dieudonn´e). SGA S´eminaire de G´eom´etrie Alg´ebrique, Grothendieck et al. monnnnn http://mathoverflow.net/questions/nnnnn/ ê Subsection (so II, ê3c means Chapter II, Section 3, Subsection c).

0d Sources

I list some of the works that I have found particularly useful in writing this book, and which may be useful also to the reader.

Chapter I: Demazure and Gabriel 1970; Serre 1993; Springer 1998; Waterhouse 1979. Chapters II, III: Bourbaki LIE; Demazure and Gabriel 1970; Erdmann and Wildon 2006; Humphreys 1972; Serre 1965; Serre 1966. Chapter IV: Lee 2002. Chapter V: Conrad et al. 2010, Demazure and Gabriel 1970; SGA3; Springer 1979; Springer 1989; Springer 1998. Chapter VI: Kneser 1969. Chapter VII: Borel 1969a. History: Borel 2001; Hawkins 2000; Helgason 1990, 1994; chapter notes in Springer

0e Acknowledgements

The writing of these notes began when I taught a course at CMS, Zhejiang University, Hangzhou in Spring, 2005. I thank the Scientific Committee and Faculty of CMS for the invitation to lecture at CMS, and those attending the lectures, especially Ding Zhiguo, Han Gang, Liu Gongxiang, Sun Shenghao, Xie Zhizhang, Yang Tian, Zhou Yangmei, and Munir Ahmed, for their questions and comments during the course. I thank the following for providing comments and corrections for earlier versions of these notes: Darij Grinberg, Lucio Guerberoff, Florian Herzig, Chu-Wee Lim, Victor Petrov, David Vogan, Xiandong Wang.

DRAMATIS PERSONÆ

JACOBI (1804–1851). In his work on partial differential equations, he discovered the Jacobi identity. Jacobi’s work helped Lie to develop an analytic framework for his geometric ideas.

RIEMANN (1826–1866). Defined the spaces whose study led to the introduction of local Lie groups and Lie algebras.

LIE (1842–1899). Founded the subject that bears his name in order to study the solutions of differential equations.

KILLING (1847–1923). He introduced Lie algebras independently of Lie in order to un- derstand the different noneuclidean geometries (manifolds of constant curvature), and he classified the possible Lie algebras over the complex numbers in terms of root systems. In- troduced Cartan subalgebras, Cartan matrices, Weyl groups, and Coxeter transformations.

MAURER (1859–1927). His thesis was on linear substitutions (matrix groups). He charac- terized the Lie algebras of algebraic groups, and essentially proved that group varieties are rational (in characteristic zero).

ENGEL (1861–1941). In collaborating with Lie on the three-volume Theorie der Transfor- mationsgruppen and editing Lie’s collected works, he helped put Lie’s ideas into coherent form and make them more accessible.

E. CARTAN (1869–1951). Corrected and completed the work of Killing on the classifi- cation of semisimple Lie algebras over C, and extended it to give a classification of their representations. He also classified the semisimple Lie algebras over R, and he used this to classify symmetric spaces.

WEYL (1885–1955). Proved that the finite-dimensional representations of semisimple Lie algebras and Lie groups are semisimple (completely reducible).

NOETHER (1882–1935). HASSE (1898–1979). BRAUER (1901–1977). ALBERT (1905–1972).

They found a classification of semisimple algebras over number fields, which gives a classification of the classical algebraic groups over the same fields.

HOPF (1894–1971). Observed that a multiplication map on a manifold defines a comultipli- cation map on the cohomology ring, and exploited this to study the ring. This observation led to the notion of a Hopf algebra.

VON NEUMANN (1903–1957). Proved that every closed subgroup of a real Lie group is again a Lie group.

WEIL (1906–1998). Classified classical groups over arbitrary fields in terms of semisimple algebras with involution (thereby winning the all India cocycling championship for 1960).

CHEVALLEY (1909–1984). He proved the existence of the simple Lie algebras and of their representations without using the classification. One of the initiators of the systematic study of algebraic groups over arbitrary fields. Classified the split semisimple algebraic groups over any field, and in the process found new classes of finite simple groups.

KOLCHIN (1916–1991). Obtained the first significant results on matrix groups over arbi- trary fields as preparation for his work on differential algebraic groups.

IWASAWA (1917–1998). Found the Iwasawa decomposition, which is fundamental for the structure of real semisimple Lie groups.

HARISH-CHANDRA (1923–1983). Independently of Chevalley, he showed the existence of the simple Lie algebras and of their representations without using the classification. With

11

CHAPTER I

Basic Theory of Affine Groups

The emphasis in this chapter is on affine algebraic groups over a base field, but, when it requires no extra effort, we often study more general objects: affine groups (not of finite type); base rings rather than fields; affine algebraic monoids rather than groups; affine algebraic supergroups (very briefly); quantum groups (even more briefly). The base field (or ring) is always denoted k, and R is always a commutative k-algebra.

NOTES Most sections in this chapter are complete but need to be revised. The main exceptions are Sections 18 and 19, which need to be completed, and Section 20, which needs to be written.

1 Introductory overview.............................. 14 2 Definitions.................................... 18 3 Examples.................................... 29 4 Some basic constructions............................ 34 5 Affine groups and Hopf algebras........................ 41 6 Affine groups and affine group schemes.................... 53 7 Group theory: subgroups and quotient groups.................. 73 8 Representations of affine groups........................ 94 9 Group theory: the isomorphism theorems................... 121 10 Recovering a group from its representations; Jordan decompositions..... 128 11 Characterizations of categories of representations............... 137 12 Finite flat affine groups............................. 144 13 The connected components of an algebraic group............... 152 14 Groups of multiplicative type; tori....................... 163 15 Unipotent affine groups............................. 176 16 Solvable affine groups.............................. 183 17 The structure of algebraic groups........................ 194 18 Example: the spin groups............................ 203 19 The classical semisimple groups........................ 217 20 The exceptional semisimple groups....................... 232 21 Tannakian categories.............................. 233

14 I. Basic Theory of Affine Groups

1 Introductory overview

Loosely speaking, an algebraic group over a field k is a group defined by polynomials. Be- fore giving the precise definition in the next section, we look at some examples of algebraic groups. Consider the group SLn.k/ of n  n matrices of determinant 1 with entries in a field k. The determinant of a matrix .aij / is a polynomial in the entries aij of the matrix, namely,

det.aij / D

X

 2 Sn^ sign./^
^ a1.1/^


^ an.n/^ (Sn^ D^ symmetric group),

and so SLn.k/ is the subset of Mn.k/ D kn 2 defined by the polynomial condition det.aij / D

1. The entries of the product of two matrices are polynomials in the entries of the two matrices, namely,

.aij /.bij / D .cij / with cij D ai1b1j C


C ai nbnj ;

and Cramer’s rule realizes the entries of the inverse of a matrix with determinant 1 as poly- nomials in the entries of the matrix, and so SLn.k/ is an algebraic group (called the special linear group). The group GLn.k/ of n  n matrices with nonzero determinant is also an algebraic group (called the general linear group) because its elements can be identified with the n^2 C 1 -tuples ..aij / 1 i;j n; d / such that det.aij /
d D 1. More generally, for a finite-dimensional vector space V , we define GL.V / (resp. SL.V /) to be the group of au- tomorphisms of V (resp. automorphisms with determinant 1 ). These are again algebraic groups. To simplify the statements, for the remainder of this section, we assume that the base field k has characteristic zero.

1a The building blocks

We now list the five types of algebraic groups from which all others can be constructed by successive extensions: the finite algebraic groups, the abelian varieties, the semisimple algebraic groups, the tori, and the unipotent groups.

FINITE ALGEBRAIC GROUPS

Every finite group can be realized as an algebraic group, and even as an algebraic subgroup of GLn.k/. Let  be a permutation of f1; : : : ; ng and let I./ be the matrix obtained from the identity matrix by using  to permute the rows. For any n  n matrix A, the matrix I./A is obtained from A by using  to permute the rows. In particular, if  and ^0 are two permutations, then I./I.^0 / D I.^0 /. Thus, the matrices I./ realize Sn as a subgroup of GLn. Since every finite group is a subgroup of some Sn, this shows that every finite group can be realized as a subgroup of GLn, which is automatically defined by polynomial conditions. Therefore the theory of algebraic groups includes the theory of finite groups. The algebraic groups defined in this way by finite groups are called constant finite algebraic groups. More generally, to give an ´etale finite algebraic group over k is the same as giving a finite group together with a continuous action of Gal.kal=k/ — all finite algebraic groups in characteristic zero are of this type. An algebraic group is connected if its only finite quotient group is trivial.

16 I. Basic Theory of Affine Groups

GROUPS OF MULTIPLICATIVE TYPE; ALGEBRAIC TORI

An affine algebraic subgroup T of GL.V / is said to be of multiplicative type if, over kal, there exists a basis of V relative to which T is contained in the group Dn of all diagonal matrices (^0)

B BB BB @

C

CC

CC

A

In particular, the elements of an algebraic torus are semisimple endomorphisms of V. A connected algebraic group of multiplicative type is a torus.

UNIPOTENT GROUPS

An affine algebraic subgroup G of GL.V / is unipotent if there exists a basis of V relative to which G is contained in the group Un of all n  n matrices of the form 0 B BB BB @

C

CC

CC

A

In particular, the elements of a unipotent group are unipotent endomorphisms of V.

1b Extensions

We now look at some algebraic groups that are nontrivial extensions of groups of the above types.

SOLVABLE GROUPS

An affine algebraic group G is solvable if there exists a sequence of algebraic subgroups

G D G 0 


 Gi 


 Gn D 1

such that each GiC 1 is normal in Gi and Gi =GiC 1 is commutative. For example, the group Un is solvable, and the group Tn of upper triangular n  n matrices is solvable because it contains Un as a normal subgroup with quotient isomorphic to Dn. When k is algebraically closed, a connected subgroup G of GL.V / is solvable if and only if there exists a basis of V relative to which G is contained in Tn (Lie-Kolchin theorem 16.31).

REDUCTIVE GROUPS

A connected affine algebraic group is reductive if it has no connected normal unipotent subgroup other than 1. According to the table below, they are extensions of semisimple groups by tori. For example, GLn is reductive. It is an extension of the simple group PGLn by the torus Gm, 1! Gm! GLn! PGLn! 1:

1. Introductory overview 17

Here Gm D GL 1 and the map Gm! GLn sends it onto the subgroup of nonzero scalar matrices.

NONCONNECTED GROUPS

We give some examples of naturally occurring nonconnected algebraic groups.

The orthogonal group. For an integer n  1 , let On denote the group of n  n matrices A such that At^ A D I. Then det.A/^2 D det.At^ / det.A/ D 1 , and so det.A/ 2 f˙ 1 g. The matrix diag.1; 1; : : :/ lies in On and has determinant 1 , and so On is not connected: it contains

SOn^ def D Ker

On

det ! f˙ 1 g

as a normal algebraic subgroup of index 2 with quotient the

constant finite group f˙ 1 g.

The monomial matrices. Let M be the group of monomial matrices, i.e., those with ex- actly one nonzero element in each row and each column. This group contains both the algebraic subgroup Dn and the algebraic subgroup Sn of permutation matrices. Moreover, for any diagonal matrix diag.a 1 ; : : : ; an/;

I./
diag.a 1 ; : : : ; an/
I./^1 D diag.a.1/; : : : ; a.n//. (3)

As M D DnSn, this shows that Dn is normal in M. Clearly D \ Sn D 1 , and so M is the semi-direct product M D Dn o Sn

where W Sn! Aut.Dn/ sends  to the automorphism in (3).

1c Summary

Recall that we are assuming that the base field k has characteristic zero. Every algebraic group has a composition series whose quotients are respectively a finite group, an abelian variety, a semisimple group, a torus, and a unipotent group. More precisely:

(a) An algebraic group G contains a unique normal connected subgroup Gı^ such that G=Gı^ is a finite ´etale algebraic group (see 13.17). (b) A connected algebraic group G contains a largest^2 normal connected affine algebraic subgroup N ; the quotient G=N is an abelian variety (Barsotti, Chevalley, Rosen- licht).^3 (c) A connected affine algebraic group G contains a largest normal connected solvable algebraic subgroup N (see ê17a); the quotient G=N semisimple. (d) A connected solvable affine algebraic group G contains a largest connected normal unipotent subgroup N ; the quotient G=N is a torus (see 17.2; 16.33).

In the following tables, the group at left has a subnormal series whose quotients are the groups at right.

(^2) This means that it contains all other such algebraic subgroups; in particular, it is unique. (^3) The theorem is proved in Barsotti 1955b and in Rosenlicht 1956. Rosenlicht (ibid.) notes that it had been proved earlier with a different proof by Chevalley in 1953, who only published his proof in Chevalley 1960. A modern proof can be found in Conrad 2002.

2. Definitions 19

is a functor from the category of rings to groups. Essentially, this is our definition together with the requirement that the functor be “defined by polynomials”. Throughout this section, k is a commutative ring.

2a Motivating discussion

We first explain how a set of polynomials defines a functor. Let S be a subset of kŒX 1 ; : : : ; Xnç. For any k-algebra R, the zero-set of S in Rn^ is

S.R/ D f.a 1 ; : : : ; an/ 2 Rn^ j f .a 1 ; : : : ; an/ D 0 for all f 2 Sg:

A homomorphism of k-algebras R! R^0 defines a map S.R/! S.R^0 /, and these maps make R S.R/ into a functor from the category of k-algebras to the category of sets. This suggests defining an affine algebraic group to be a functor Algk! Grp that is isomorphic (as a functor to sets) to the functor defined by a set of polynomials in a finite number of symbols. For example, the functor R SLn.R/ satisfies this condition because it is isomorphic to the functor defined by the polynomial det.Xij / 1 where

det.Xij / D

X

 2 Sn sign./
X1.1/


Xn.n/ 2 kŒX 11 ; X 12 ; : : : ; Xnnç: (4)

The condition that G can be defined by polynomials is very strong: it excludes, for example, the functor with

G.R/ D

Z=2Z if R D k f 1 g otherwise. Now suppose that k is noetherian, and let S be a subset of kŒX 1 ; : : : ; Xnç. The ideal a generated by S consists of the finite sums X gi fi ; gi 2 kŒX 1 ; : : : ; Xnç; fi 2 S:

Clearly S and a have the same zero-sets for any k-algebra R. According to the Hilbert basis theorem (CA 3.6), every ideal in kŒX 1 ; : : : ; Xnç can be generated by a finite set of polynomials, and so an affine algebraic group is isomorphic (as a functor to sets) to the functor defined by a finite set of polynomials. We have just observed that an affine algebraic group G is isomorphic to the functor defined by an ideal a of polynomials in some polynomial ring kŒX 1 ; : : : ; Xnç. Let A D kŒX 1 ; : : : ; Xnç=a. For any k-algebra R, a homomorphism A! R is determined by the images ai of the Xi , and the n-tuples .a 1 ; : : : ; an/ that arise from a homomorphism are exactly those in the zero-set of a. Therefore the functor R a.R/ sending a k-algebra R to the zero-set of a in Rn^ is canonically isomorphic to the functor

R Homk-alg.A; R/:

Since the k-algebras that can be expressed in the form kŒX 1 ; : : : ; Xnç=a are exactly the finitely generated k-algebras, we conclude that the functors Algk! Set defined by a set of polynomials in a finite number of symbols are exactly the functors R Homk-alg.A; R/ defined by a finitely generated k-algebra A. Before continuing, it is convenient to review some category theory.

20 I. Basic Theory of Affine Groups

2b Some category theory

An object A of a category A defines a functor

hAW A! Set by

hA.R/ D Hom.A; R/; R 2 ob.A/; hA.f /.g/ D f ı g; f W R! R^0 ; g 2 hA.R/ D Hom.A; R/:

A morphism ˛W A^0! A of objects defines a map f 7! f ı ˛W hA.R/! hA 0 .R/ which is natural in R (i.e., it is a natural transformation of functors hA^! hA 0 ):

THE YONEDA LEMMA

Let F W A! Set be a functor from A to the category of sets, and let A be an object of A. A natural transformation T W hA^! F defines an element aT D TA.idA/ of F .A/.

2.1 (YONEDA LEMMA) The map T 7! aT is a bijection

Hom.hA; F / ' F .A/ (5)

with inverse a 7! Ta, where

.Ta/R.f / D F .f /.a/; f 2 hA.R/ D Hom.A; R/:

The bijection is natural in both A and F (i.e., it is an isomorphism of bifunctors).

PROOF. Let T be a natural transformation hA^! F. For any morphism f W A! R, the commutative diagram

hA.A/ hA.R/

F .A/ F .R/

hA.f /

TA TR F .f /

idA f

aT F .f /.aT / D TR.f /

shows that TR.f / D F .f /.aT /: (6)

Therefore T is determined by aT , and so the map T 7! aT is injective. On the other hand, for a 2 F .A/, .Ta/A.idA/ D F .idA/.a/ D a;

and so the map T 7! aT is surjective. The proof of the naturality of (5) is left as an (easy) exercise for the reader. (^2)

2.2 When we take F D hB^ in the lemma, we find that

Hom.hA; hB^ / ' Hom.B; A/:

In other words, the contravariant functor A hAW A! A_^ is fully faithful.