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UBC Theses and Dissertations

Algebraic monoids Renner, Lex Ellery 1982

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ALGEBRAIC MONOIDS by LEX ELLERY RENNER B.Sc., University of Saskatchewan, 1975 M.Sc., University of B r i t i s h Columbia, 1978 A THESIS SUBMITTED IN PARTIAL FULLFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS '-' We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1982 © Lex E l l e r y Renner, 1982 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a gree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Mathematics The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date A p r i l 26, 1982 DE-6 (3/81) i i Abstract Def in i t ion: Let k be a n - a l g e b r a i c a l l y closed f i e l d . An algebraic  monoid is a t r i p l e (E,m,l) such that E is an algebraic variety defined over k, m : ExE > E i s an associative morphism and 1 € E is a two—sided unit for m. The object of this thesis is to expose several fundamental topics in the theory of algebraic monoids. My results may be divided into three types; general theory of i r r e d u c i b l e a f f i n e monoids, structure and c l a s s i f i c a t i o n of semi—simple rank one reductive monoids, and theory of general monoid v a r i e t i e s (not necessarily a f f i n e ) . I General Theory of Affine Monoids ( 3.3.6 ) Existence of Affine Algebraic Monoids Let G be an irreducible a f f i n e algebraic group. Then the following are equivalent. (i) There exists an i r r e d u c i b l e algebraic monoid E such that G(E) = ( g £ E | g" 1 « E ] is isomorphic to G and E does not consist, e n t i r e l y of units. ( i i ) There exists an irreducible algebraic monoid E such that G = G(E) and 0 e E (0 * 1 ) . ( i i i ) X(G) = Hom(G,k*) is a n o n — t r i v i a l abelian group. (iv) rank R(G) > 0, where R(G) is the solvable r a d i c a l of G. ( 4.2.6 ) Nilpotent Algebraic Monoids Let E be a nilpotent irreducible algebraic monoid. Then the following are equivalent. (i) For a l l x e E there exists an idempotent e e E and an element x* e E, such that xx* = e = x*x, ex = xe and ex* = x*e. ( i i ) The morphism m : G(u)xE(s) > E, (u,s) —> us, is f i n i t e and dominant, where G(u) is the closed subgroup of unipotent elements and E(s) is the closed submonoid of semi—simple elements. In case E i s also a normal variety, m i s an isomorphism. ( 4.4.14 ) Reductive and Regular Monoids (a) Let E be a reductive algebraic monoid. Then E i s regular, i . e . For . a l l x e E there exists g e G(E) such that gx = e is an idempotent. (b) Let E be an irreducible algebraic monoid with 0. Then the following are equivalent. (i) E i s regular. ( i i ) E is reductive. ( i i i ) E has no n o n — t r i v i a l nilpotent ideals. ( 4.5.2 ) Connected Algebraic Monoids Let E be an algebraic monoid with 0. Then the following are equivalent. (i) E i s connected in the Zariski topology. ( i i ) There exist idempotents 1, e ( l ) , e(2),..., .e(m) = 0 such that e (i) > e(i+l) for a l l i = 0,...,m-1 and e(i+l) e e(i)Ee(i)° (the irreducible component of 1) for a l l i . If , in case ( i i ) we require that each idempotent be minimal, then the number m i s uniquely determined and each i v e ( i ) E e ( i ) is uniquely determined up to isomorphism. ( 5.2.1 ) Structure of Prime Ideals Let E be irreducible and a f f i n e . A prime ideal P of E, is a non-empty subset of of E such that EPE is a subset of P and E - P i s m u l t i p l i c a t i v e l y closed. (i) Suppose P i s a prime ideal of E. Then there exists a morphism x : E > k such that P = x ~ 1 ( 0 ) . ( i i ) Let T be a maximal torus of G(E), X i t s closure in E and W the Weyl group of T. Then there are canonical b i j e c t i o n s primes(E) <—> W-inv.primes(X) <—> W—inv.idempotents(X) II Reductive Monoids of Semi-simple Rank One (*) Let E be an i r r e d u c i b l e , reductive a f f i n e algebraic monoid with 0 such that dim ZG(E) = 1 and rkss G(E) = 1. The r e s t r i c t i o n on the center i s required to avoid the r e l a t i v e a r b i t r a r i n e s s of D—monoids. 1. Geometric Structure ( 7.2.3 ) The action GxGxE > E, (g,h,x) > gxh" 1 has three o r b i t s , G(E), ( E - G ) - { 0 } and { 0 }. ( 7.4.4 ) If E i s also normal then E i s a Cohen—Macaulay algebraic variety. 2. C l a s s i f i c a t i o n ( 7.5.17 ) C l a s s i f i c a t i o n I Let E be as in (*)• above. Then G(E) is isomorphic with one of G l ( k 2 ) , S l ( k 2 ) x k * or PGl(k 2)xk*. Let G be one of these groups and l e t Q+ denote the set of positive ra t i o n a l numbers. Then there i s a canonical b i j e c t i o n V 0/ <—> E(G) = { E | E as in (*), E normal, G(E) = G }. For G = Gl(k 2) the correspondence i s as follows. Given E, there i s a unique bicartesian diagram, E(nr 1 ) > End(k 2) a v v E > E(n) such that a l l morphisms are f i n i t e and dominant. If degree a = n is odd then degree(p) = m i s odd and (m,n) = 1. If degree(o) = 2n is even then degree(£) = 2m is even and (m,n) = 1 (and one of m and n is even). In any case, the map E(G) > Q+ given by E > deg(o)/degU) is well defined and one—to—one. Conversely, i f r t Q+ then r = m/n, where m,n > 0 and (m,n) = 1. It i s then possible to construct a bicartesian diagram as above such that deg(a) = n and deg(p) = m i f mn i s odd, or deg(a) = 2n and deg(^) = 2m i f mn i s even. Thus we obtain the inverse map Q+ > E(G), r — > E ( r ) . A l l normal monoids with group Sl(k 2 ) x k * are constructed from the monoids with group Gl ( k 2 ) using integral closure and the morphism m : S l ( k 2 ) x k * > G l ( k 2 ) , m(x,t) = xt. A l l normal monoids with group PGl(k 2)xk* are constructed v i from the monoids with group G l ( k 2 ) using f i n i t e group scheme quotients and the morphism c : Gl ( k 2 ) > PGl(k 2)xk*, c(x) = ([x],det(x)). ( 7.6.5 ) C l a s s i f i c a t i o n II Let E as in (*) above, be normal. Let T be a maximal torus and l e t * be the roots of the adjoint representation. Let Z denote the Zariski closure of T in E. From general p r i n c i p l e s (4.1.7 of the text) there exists F = { F(1), F(2) } contained in X(Z) such that <F(1),F(2)> > X(Z) is a f i n i t e , dominant morphism and each F ( i ) is non—zero modulo the square of the maximal ideal of functions that vanish at zero. F is ca l l e d the set of fundamental generators of X(Z) . (X(T),*(T),F(E)) i s the polyhedral root system of the pair (E, T) . E i s uniquely determined up to isomorphism by i t s polyhedral  root system. The following is a l i s t of a l l possible polyhedral root systems (X,4>,F) for the various groups G ((u, v) denotes the free abelian group on u and v, written a d d i t i v e l y ) . (i) G = Gl ( k 2 ) X = (u, v) a, B e Z # = { u-v, v-u } a >|B|> 0 F = { au+BV, OV + BU } {O.,B) - 1 v i i ( i i ) G = Sl(k 2 ) x k * X = (a, b) m,n e N * = { 2b, -2b } m,n > 0 F = { ma+nb, ma-nb } (m,n) = 1 ( i i i ) G = PGl(k 2)xk* X = (x, y) m, n e N * = { y, -y } m,n > 0 F = { mx+ny, mx-ny } (m,n) = 1 III General Monoid V a r i e t i e s Let E be an irreducible monoid variety (not necessarily af f ine). ( 8.1.4 ) If E is quasi—affine then E is a f f i n e . ( 8.2.3 ) If E is projective then E i s an abelian variety. v i i i TABLE OF CONTENTS INTRODUCTION . 1 I PRELIMINARIES 13 1.1 Algebraic Geometry And Commutative Algebra 13 1.2 Linear Algebraic Groups 14 1.3 F i n i t e D-group Actions 16 II AFFINE ALGEBRAIC MONOIDS 22 2.1 Preliminaries 22 2.2 Elements 24 2.3 Examples 27 III IRREDUCIBLE ALGEBRAIC MONOIDS 29 3.1 F i r s t P r i n c i p l e s 29 3.2 Integral Closure And Normalization 30 3.3 Existence Of Algebraic Monoids 32 3.4 Closure 35 IV TYPES OF MONOIDS 40 4.1 D-monoids 40 4.2 Nilpotent Monoids . 43 4.3 Solvable Monoids 45 4.4 Reductive And Regular Algebraic Monoids 48 4.5 Connected Monoids With Zero 57 V IDEALS 60 5.1 Preliminary Results 60 5.2 The Structure Of Prime Ideals 62 VI TWO-DIMENSIONAL REGULAR MONOIDS 65 6.1 Structural Properties 65 VII SEMI-SIMPLE RANK ONE, REDUCTIVE MONOIDS 69 ix 7.1 Rank Two, Semi-simple Rank One, Reductive Groups .... 69 7.2 Properties Of Semi—simple Rank One Monoids 71 7.3 Constructing Morphisms And Applications 74 7.4 Cohen-Macaulay Monoids 78 7.5 C l a s s i f i c a t i o n I 81 7.6 Polyhedral Root Systems And C l a s s i f i c a t i o n II ....... 96 VIII IRREDUCIBLE MONOID VARIETIES 101 8.1 Quasi-affine Monoids 101 8.2 Projective Monoids 103 IX APPLICATIONS TO RATIONAL HOMOTOPY THEORY 106 9.1 Algebraic Categories And Positive Weights Spaces ....106 9.2 Homotopy Types With Connected Endomorphism Monoid ...111 X Acknowledgements I would l i k e to thank my advisor, Dr. Roy Douglas for his encouragement and boundless enthusiasm, especially in the formative stages of this thesis when there was a scarcity of relevant l i t e r a t u r e . Dr. Klaus Hoechsmann i s to be thanked for suggesting several problems that I have considered in chapter 8. Also, I would l i k e to thank Dr. Larry Roberts for sharing with me, on several occasions, his superior knowledge of the available l i t e r a t u r e on algebraic geometry. F i n a l l y , I would l i k e to thank Dr. Mohan Putcha of -North Carolina State University. Over the past two years he has given me access to much of his unpublished work on algebraic monoids. 1 INTRODUCTION An algebraic monoid E i s an algebraic variety which also has the structure of a monoid m : ExE >E, in such a way that m is a morphism of v a r i e t i e s . Well known examples include the monoids associated with f i n i t e dimensional associative algebras. The most familiar non—linear example i s surely the cusp, { (x,y) e k 2 | x 2=y 3 }. Algebraic monoids also arise in many other contexts. For example, i f V i s a f i n i t e dimensional vector space over the f i e l d k and. f:V»V >V i s a linear map, then { t e End(k) | f(t(v)«t(w))=t(f(v»w)) for a l l v,w e k } is an algebraic submonoid of End(V)' ('•' denotes 'tensor product of vector spaces'). At t h i s time there i s no comprehensive theory of algebraic monoids nor is there an established paradigm as to what should be the aims of the theory. Toric monoids have been discussed in [ 1 2 ] and [14] mainly from the point of view of modern algebraic geometry. In [7] a f f i n e monoids are b r i e f l y encountered as part of the comprehensive introduction to algebraic group theory. Aside from the copious work of M. Putcha [18—26] very l i t t l e has been done from the point of view of modern semi—group theory. Truly there is no f u l l y developed ideal standard of structure in the theory of monoids. There are now available several books on algebraic group 2 theory which have properly d i s t i l l e d the necessary prerequisites so as to make the theory available to a wide audience. J. Humphreys' book [13] is a complete introduction to the linear theory over an a l g e b r a i c a l l y closed f i e l d and more recently W. Waterhouse [32] has written a coherent introductory text on the three basic approaches to this theory. It is always a source of c l a r i t y and depth to keep in mind the interplay (and equivalences) among, Hopf algebras, linear groups and group valued functors. For example, l e t G=k* be the m u l t i p l i c a t i v e group of units of k. As a linear group we have, G=G1(1). As a Hopf algebra we have, k[G]=k[T,T" 1] i(T)=T- 1 d(T)=T»T e(T)=1. As a group valued functor we have, G(R)=U(R), the units of R, where R i s any k—algebra. Generally speaking the f i r s t two viewpoints are equivalent by Hilbert's zeros theorem and the l a s t two are equivalent by Yoneda's lemma. This correspondence c a r r i e s over to the case of monoids as well. For example, let E=End(k). As a linear monoid E=End(k) . 3 As a bigebra k[E]=k[T] d(T)=T»T e ( T ) = 1 . As a monoid valued functor we have E(R)=R, with the obvious m u l t i p l i c a t i o n . P l a i n l y the only l o g i c a l difference from li n e a r groups is the absence of inverses. There is a motif in the theory of algebraic groups which is of fundamental importance for monoids as well. Stated as a problem for monoids th i s i s as follows. (I) Let E be an irreducible algebraic monoid and let G be i t s group of units. Assume G i s reductive. Let X be the closure in E of a maximal torus T of G and. l e t W be the Weyl group associated with T. To what extent can the structure of E be determined by X and int : w > Aut(X) ? In p a r t i c u l a r s , (i) Can E be determined by (W,X)? ( i i ) Are there axioms which characterize the class of D—monoids that are obtained from reductive monoids in this fashion? (One should keep in mind the overwhelming success of root systems in both modern and c l a s s i c a l Lie theory, and in algebraic group theory). The l i s t can be continued. Of course there are other important problems which are not completely encompassed by the above motif. The most important of these, in my opinion, i s the following. 4 (II) Find a complete l i s t of a l l algebraic monoids E such that E i s irreducible, reductive, normal, and rk(G)=rkss(G) + 1 . The condition, rk(G)=rkss(G) + 1, i s precisely what i s needed to avoid the r e l a t i v e a r b i t r a r i n e s s of central D—monoids. My thesis is guided almost e n t i r e l y by problems (I) and (II) above. The main results, which are f u l l y exposed in chapters 4 and 7, provide ample evidence that these problems are of fundamental importance in the theory of algebraic monoids. The most s i g n i f i c a n t result of chapter 4 asserts that a l l irreducible monoids are regular in the sense of von Neumann. As a dire c t consequence, problem (II) above i s solved completely in case rkss(G(E)) = 1. Chapter 1 contains prelimimary information from algebraic geometry and algebraic group theory. It i s offered p a r t l y to indicate the level of discourse and also to deduce some preliminary results concerning D—groups. Chapter 2 introduces the general theory and the basic notation. It i s proved that any n o n — t r i v i a l monoid possesses an abundance of idempotents. Thus, a m u l t i p l i c a t i v e Jordan decomposition i s possible for many non—units in the monoid E. In pa r t i c u l a r , the subset of semi—simple elements is well—defined and n o n — t r i v i a l . In chapter 3 irreducible monoids are discussed. The groups G which occur n o n - t r i v i a l l y as the group of units of an irreducible monoid are characterized (see Theorem 3 . 3 . 6 ) . The remainder of the chapter is more technical and mainly concerned with closure properties of irreducible monoids. Let E be 5 i r r e d u c i b l e , and T a maximal torus of G,the group of units of E. Let B be a Borel subgroup of G, B(G) = { B | B is a Borel subgroup of G }, and e € E an idempotent. The most important background result in t h i s regard (due to M. Putcha) asserts that eEe is in the closure of the c e n t r a l i z e r in G of e. From this i t follows that E(s) is equal to the union of the gXg~1 as g varies over G, where E(s) i s the set of semi-simple elements of E and X is the closure of T in E. Analagously, i t is proved that E is the union of the gZg~ 1 where Z i s the closure of B in E. Thus every element i s in the closure of a Borel subgroup. Furthermore, B(x) = { B e B(G) | x is contained in the closure of B } is closed in B(G). Thus Borel's fixed point theorem can be applied (to B(x)) to prove that i f G i s reductive then C(T)=X. Chapter 4 i s dedicated to the special properties of the five basic types of monoids. These are D—monoids, nilpotent monoids, solvable monoids regular monoids, and reductive monoids. Let E be an i r r e d u c i b l e D—monoid. Then k[E] = k[X(E)], the monoid algebra of X(E) = Hom(E,k) (this i s one standard d e f i n i t i o n of D—monoids). If E i s normal and 0 e E there exists { F ( i ) | i=1,...,n } contained in X(E) such that <F(1),...,F(n)> > X(E) i s f i n i t e and dominant, n is the number of minimal non—zero idempotents, and { F ( i ) } i s a l i n e a r l y independent subset modulo m2, where m i s the maximal . ideal of functions which vanish at 0 . Furthermore { F ( i ) } is the only such subset. { F ( i ) } i s c a l l e d the set of fundamental generators of X(E). In the discussion of semi—simple, rank one, monoids i t is seen that 6 the fundamental generators are precisely what is needed to synthesise with the root system, in order to c l a s s i f y these monoids in the s p i r i t of c l a s s i c a l Lie theory. Let E be an irreducible nilpotent monoid. It is interesting to know the conditions under which the well—known structure theorem for nilpotent groups can be generalized to monoids. If.G is an irreducible nilpotent algebraic group the theorem asserts that m : G(u)xG(s) > G is an isomorphism where G(u) is the subgroup of unipotent elements and G(s) is the subgroup of semi—simple elements. This theorem generalizes to the nilpotent monoid E i f and only i f E is a C l i f f o r d monoid (see Theorem 4.2.6) . Solvable irreducible monoids are important generally because of their r e l a t i v e s i m p l i c i t y combined with the fact that every ir r e d u c i b l e monoid is the union of i t s solvable irreducible submonoids. The main result of this section is a characterization of solvable monoids among irreducible monoids with 0. Let E be an irreducible monoid with 0. Then E is solvable i f and only i f i t s subset of nilpotent elements i s a two—sided i d e a l . This result i s o r i g i n a l l y due to M. Putcha. My proof i s s l i g h t l y d i f f e r e n t , making use of the universal D—monoid associated with a given solvable monoid. Reductive algebraic monoids are perhaps the most, important of a l l monoids. An irreducible monoid E is reductive i f G, i t s group of units is a reductive group. My preliminary discussion of reductive monoids i s mainly concerned with class functions, semi—simple elements and conjugacy classes. Let E be reductive and suppose x i s an element of E. Then, 7 (i) x is semi—simple i f and only i f the conjugacy class of x is closed in E. ( i i ) If T i s a maximal torus of G then the ce n t r a l i z e r of T in E is equal to the closure of T in E. ( i i i ) There is a one—to—one correspondence between semi-simple  conjugacy classes and orbits of the Weyl group action on the  closure of a maximal torus. (iv) c l [ E ] > k[E] » k[X] i d e n t i f i e s c l [ E ] = { f € k[E] | f(xy) = f(yx) for a l l x,y € E } with the ring of invariant functions of k[X] under the induced action of the Weyl group (X is the closure of a maximal torus). (v) If E is any irreducible monoid then there exists a morphism p : E > E' such that (o) p is dominant. (p) The kernel of p is the unipotent radi c a l of G. (p) If X and X' are the closures of respective maximal t o r i of E and E' such that p(X) = X' then p : X —-> X' is an isomorphism. Thus every irreducible monoid E maps to a reductive monoid E' so as to preserve as much of the o r i g i n a l structure as one could possibly hope for. This construction has two consequences recorded in the thes i s. ( 1 ) If E is a von Neumann regular monoid with 0 , then E is reductive. (2) If Q i s a prime ideal of E then Q i s the inverse image of some prime ideal of E' (p : E > E' as above). In my proof of the structure theorem for prime ideals 8 (Theorem 5.2.1), the most important step i s a synthesis of this second result and a result on class functions.' The f i n a l result of t h i s section i s the most s i g n i f i c a n t structure theorem of the thesis. Generalizing a theorem of M. Putcha we find that a l l reductive monoids are regular in the sense of von—Neumann (see Theorems 4.4.14 and 4.4.15). The structure theory of chapter 7 is a direct consequence. I have concluded chapter 4 with a short discusion of connected monoids with 0. A monoid is connected i f i t i s connected in the Zariski topology. Using the general theory of chapters 2 and 3, I have obtained the following result (Theorem 4.5.2). A monoid E with 0 is connected i f and only i f there exists a chain of idempotents 1 = e(0) > e(l) > ... > e(k) = 0 such that e(i+l) is an element of the irreducible component of e ( i ) E e ( i ) containing e ( i ) , for a l l i = 1,...,k—1. This i s reinterpreted in the context of rational homotopy theory in chapter 9. Chapter 5 i s exclusively devoted to finding the prime ideals of a given irreducible monoid E in terms of a maximal irreducible D—submonoid. Let E be an irreducible monoid and P be a prime ideal of E. Let T be a maximal torus and l e t X be i t s closure in E. Let P(T) be the intersection of P and X. P(T) is a prime ideal of X, invariant under the Weyl group W. Thus, we can construct a W—invariant character c of X such that c 1 ( 0 ) = P(T). Using the results on class functions there exists a character q on E such that P = q~ 1(0). Thus, the map P > P(T) induces a one—to—one correspondence between the prime ideals of ..E and the W—invariant prime ideals of X (see Theorem 9 5.2.1). Chapter 6 begins the descent towards the c l a s s i f i c a t i o n and structure theory of semi—simple rank one monoids. The structure theory requires a c l a s s i f i c a t i o n of two—dimensional non—commutative monoids without 0. There are two types. Let k* be a one—parameter muultiplicative subgroup of G. Then there exists either p : kxE > E extending k*xE > E, (t,x) > t x f 1 , or p : kxE > E extending k*xE > E, (t,x) > f ' x t , depending on whether gxh = gx or gxh = xh for a l l x in E and a l l g,h in G. Chapter 7 contains the main computations of my thesis. A l l reductive, normal semi—simple rank one monoids E, with one—dimensional center are c l a s s i f i e d in two ways. Using the results of chapter 4 and some representation theory, I construct an e s s e n t i a l l y unique bicartesian diagram, E ( 2 ) > E ( 1 ) v v E > E(3) such that a l l morphisms are f i n i t e and dominant, and have l i n e a r l y reductive kernels (possibly non—reduced). Since a l l the morphisms can be determined numerically, and E(1) i s the monoid of 2—by-2 matrices, the c l a s s i f i c a t i o n follows (see Theorem 7.6.17) . The second c l a s s i f i c a t i o n theorem is established in the proof of the f i r s t one. This i s accomplished by following the 10 relevant data (roots and fundamental generators) around the diagram from E(1) to E. It i s my be l i e f that a c l a s s i f i c a t i o n in the s p i r i t of c l a s s i c a l Lie theory should be possible for reductive algebraic monoids with one—dimensional center. The second c l a s s i f i c a t i o n theorem is offered in accordance with this b e l i e f . If E is reductive normal and has a 0 and a one—dimensional center then E is uniquely determined by i t s poyhedral root system (X(E),#(E),F(E)) in case the semi—simple rank i s one. A statement of t h i s theorem and a l i s t of a l l the polyhedral root systems is recorded in 7.7.5. One c o r o l l a r y of the bicartesian diagram above is the following. If E is as above then E is Cohen—Macaulay. This proof also requires the generalization of a theorem of P. Roberts [28] that I. have established in the l a t t e r part of chapter 1. Hochster has proven that i f E is an irreducible normal D—monoid then E is Cohen—Macaulay. The extent to which th i s result can be generalized to irreducible monoids i s not known. Chapter 8 i s a short discussion dedicated to general monoid v a r i e t i e s (not necessarily a f f i n e ) . . A well known structure theorem of C. Chevalley asserts that i f G is a smooth algebraic group then there exists a unique a f f i n e algebraic subgroup N of G such that G/N is an abelian variety. It is not known whether thi s result extends to algebraic monoid v a r i e t i e s . For example, i f G(E) is a f f i n e , is E affine? I have considered two special cases, irreducible quasi-affine monoids and irreducible projective monoids. If E is quasi—affine i t is possible to imbed E as an open sub-monoid of some irreducible af fine monoid E' , E > E' . Thus E'•—E is a 11 prime ideal of E'. It follows from the results of chapter 5 that E'—E i s a p r i n c i p l e d i v i s o r . Thus, E i s actually a f f i n e (see Theorem 8.1.4). Using the completeness property of projective v a r i e t i e s we see that a l l projective irreducible monoids are abelian v a r i e t i e s (see Theorem 8.2.3). In chapter 9, the f i n a l chapter, I have discussed a problem which has i t s origins in rat i o n a l homotopy theory. A rational homotopy type may be regarded, by Sullivan's theory [30], as a d i f f e r e n t i a l graded algebra M, defined over Q, which is minimal, and free as a graded algebra. The problem I have considered i s a special case of, "To what extent is the structure of M influenced by i t s algebraic monoid of endomorphisms?". I have discussed t h i s problem in a more general context so as to abstract from the p e c u l i a r i t i e s of rational homotopy theory. Let V(k) be the category of vector spaces over k and l e t S be an 'algebraic structure' on V(k). An algebraic structure a, i s a rule (or functor) which associates with the vector space V, a c o l l e c t i o n of linear transformations { a(s) | s t S } = a(S) in the union of the Hom(V(m),V(n)) (as m and n vary) s a t i s f y i n g various relations depending on S (V(m) denotes the tensor product of V with i t s e l f m times). The c o l l e c t i o n of pairs (V,a) are the objects of a category n(k,S). If V and W are objects of n(k,S) then Horn'(V,W) = { f e Hom(V,W) | a(s)of(m) = f(n)oa(s) for a l l s in S }. Assume n(k,S) has a zero object and each V in o(k,S) s a t i s f i e s suitable finiteness conditions. Then End'(V) i s an algebraic monoid and furthermore 0 c End'(V). D e f i n i t i o n : Let V be an object of n(k,S). Then V has positive  weights i f 0 i s in the Zariski closure of Aut(V) in End(V). 1 2 Equivalently, there exists a 1—p.s.g. t : k* > Aut(V) such that t extends to a morphism t : k -—> End(V) with t(0) = 0. The importance of this d e f i n i t i o n was f i r s t noticed by R. Body, R. Douglas and D. Sullivan in the context of rat i o n a l homotopy theory. If X is a f i n i t e simply—connected C.W. space then M(X), the minimal model of X, has positive weights i f and only i f , for every prime p, there exist maps f ( i ) : X > X such that the homotopy direct l i m i t of { f ( i ) : X > X | i e N } i s homotopy equivalent to the p — l o c a l i z a t i o n of X. R. Body and R. Douglas [2] have proven that i f X i s a rational homotopy type with p o s i t i v e weights then X s a t i s f i e s uniqueness of product decompositions in the sense of Krull—Schmidt. This result has since been generalized and dualized by R. Douglas and myself [9]. In chapter 9 I have sketched the main points of this arguement in the more general context the category n(k,S). In the l a s t section of chapter 9 I have discussed connected algebraic monoids in the context of rational homotopy. The characterization of connected monoids with 0 in chapter 4 f i t s in neatly with the main theorem of my master's thesis [27;Theorem 3.6.2]. Let M be a minimal algebra. Then End(M) is connected in the Zariski topology i f and only i f there i s a chain 1 = e(0) > e(l) > ... > e(k) = 0 of idempotents in End(M) such that e(i+l) is in the closure of Aut(e(i)(M)) for i 1 3 I PRELIMINARIES The theory of algebraic monoids requires much background information from algebraic geometry and algebraic group theory. In t h i s chapter I have assembled many of the prerequisite concepts and results that are needed in subsequent chapters. Occasionaly I have proven a result that is only t a c i t l y available in the l i t e r a t u r e but more often the results are stated with e x p l i c i t references and no proofs. 1.1 Algebraic Geometry And Commutative Algebra 1.1.1 Dimension Theorem [I3;p.30]: Suppose f : X > Y i s a dominant morphism of irreducible v a r i e t i e s , r = dimX — dimY. Suppose W is a closed and irreducible subset of Y and Z is a maximal irreducible component of f~1(W) which dominates W. Then dimZ > dimW + r. 1.1.2 Zariski's main theorem [I5;p.414]r Let f : X > Y be a dominant morphism of irreducible v a r i e t i e s . Suppose that every fi b r e of f i s f i n i t e . Then there exists a f a c t o r i z a t i o n of f, f = f ' o j , where j : X > Y' i s an open imbedding and f' : Y' > Y i s a f i n i t e morphism. Corollary: If f : X > Y, as in 1.1.2, i s b i r a t i o n a l and Y i s a normal variety then f i s an open imbedding. 1.1.3 Separability [I3;p.44]: Let f : X > Y be a dominant morphism of irreducible v a r i e t i e s . For x e X l e t T(x,X) denote the tangent space of x to X. If there is a smooth point x of X such that y = f(x) is a smooth point of Y, and df : T(x,X) > T(y,Y) i s surje c t i v e , then f is separable. 1.1.4 "Nakayama's lemma": Let A be a non—negatively graded ring such that A(0) i s a f i e l d . Let m be the unique graded maximal 1 4 ideal and suppose M i s a non—negatively graded A—module. Suppose further that { x(i) | i e I } are homogeneous elements of M which generate M/mM. Then { x(i) } generate M. Proof: Let x t M(i) be of minimal degree such that x i s not an element of <x(i)> (the submodule generated by { x(i) }). Now modulo m, x = E a ( i ) x ( i ) . So z = x - E a ( i ) x ( i ) c mM. Thus z = Zm(i)z(i), where m(i) e m and z ( i ) e M. But deg(z(i)) < deg(z) for a l l i . So z(j) e <x(i)> for a l l j . Thus z = x - Z a ( i ) x ( i ) € <x(i)> and so x = z + Za( i ) x ( i ) is also in <x(i)>. 1.1.5 Codimension 2 Lemma [I0;p.239]: Suppose X i s a normal a f f i n e variety and V i s a closed subset of codimension larger than or equal to two. Then any morphism from X—V to an affine variety extends uniquely to X. 1.2 Linear Algebraic Groups Throughout this section, G denotes a linear algebraic group. 1.2.1 Orbits [I7;p.66]: Let GxX > X be an action of the algebraic group G on the variety X. For x e X, l e t 0(x) = { gx I g 6 G } and G(x) = { g e G | gx = x }. Then (i) G(x) is a closed subgroup of G. ( i i ) For a l l x e X, 0(x) is open in i t s closure. ( i i i ) For a l l x e X, dimO(x) = dimG — dimG(x). (iv) For a l l n > 0, { x e X | dimO(x) > n } i s open in X. 1.2.2 Borel subgroups. Let G be an irreducible algebraic group. A subgroup B, of G is a Borel subgroup i f B is solvable and G/B is a complete variety. (i) Suppose f and g : G > H are morphisms of irreducible 1 5 algebraic groups such that f|B = g|B for some Borel subgroup B of G. Then f = g. Proof: f g ~ 1 i s a morphism of v a r i e t i e s from G to H which factors through G/B. Thus, fg~ 1(G) is complete, irreducible and a f f i n e . Hence, f = g. ( i i ) Normalizer theorem [13;p.143]: If B i s a Borel subgroup of G then N(B) = B (N(B) is the normalizer of B in G). ( i i i ) Borel fixed point theorem [13;p.13 4]: Suppose X i s a complete algebraic variety • and G is a solvable irreducible algebraic group. If GxX > X is a group action then F(X,G) (the fixed point set of this action) i s non-empty. (iv) Construction [13;p.145]: Let B(G) = { B | B is a Borel subgroup of G } and G/B' = { gB' | g e G } where B' is some fixed Borel subgroup. Then by the normalizer theorem s : B(G) > G/B', B > gB' (where B = gB'g~1) is well—defined and b i j e c t i v e . Further, GxB(G) > B(G), (h,B) -—> hBh"1 1 xs s GxG/B' > G/B', (h,gB') > hgB' commutes. Conclusion: The Borel fixed point theorem applies to the action GxB(G) > B(G) which is a p r i o r i only set—valued. 1.2.3 Closure: (i) [29;p.68]. Let GxX > X, be a group action, V a closed subset of X and B a Borel subgroup of G such that BV i s contained in V. Then GV is closed in X. ( i i ) [29;p.70]. Suppose GxX >X, X i s a f f i n e , and x € X s a t i s f i e s tx = x for a l l t e T a maximal torus of G. Then O(x), 1 6 the G—orbit of x, i s closed in X. 1.2.4 Reductive and geometrically reductive groups: G is reductive i f every unipotent normal subgroup of G i s t r i v i a l . G i s geometrically reductive i f for every morphism p : G > G1(V) and every non—zero element v of V l e f t invariant by G, there i s a homogeneous polynomial function f : V > k, invariant under G such that f(v) is non—zero. (i) [11]. If G i s reductive then G i s geometrically reductive. ( i i ) [I7;p.49]. Suppose G i s reductive and X is a f f i n e . If GxX > X then F(k[X],G) i s a f i n i t e l y generated k-algebra. Let p : X > Y be the morphism induced from F(k[X],G) > k[X] (so, k[Y] = F(k[X],G)). Then for a l l y in Y there is a unique closed G—orbit O(x) contained in p ~ 1 ( y ) . ( i i i ) If GxX > X then the union of the set of closed o r b i t s of maximal dimension i s open in X. 1.3 F i n i t e D-qroup Actions An a f f i n e group scheme is a generalized linear algebraic group. Technically, in the study of linear groups and monoids, one i s often led quite naturally to consider group schemes which are not necessarily reduced. For example, in c h a r a c t e r i s t i c p > 0 the category of affin e commutative algebraic groups is not an abelian category, but i f the commutative non—reduced group schemes are allowed as well, the resulting category is abelian [32;p.127,ex.12]. In chapter 7 I have been led to consider certain morphisms f : G > H such that kernel(f) is a (not necessarily reduced) f i n i t e D—group scheme. This w i l l lead to an important structure 1 7 theorem about semi—simple, rank one monoids. Def i n i t i o n : An affine algebraic group scheme G is a representable functor from the categoryof k—algebras to the category of groups. By Yoneda's lemma, a l l the group structures G(R) (as R varies over k—algebras), are the result of morphisms e : A > k (unit) d : A > A»A (multiplication (where '•' denotes 'tensor product over k')) i : A > A (inverse) where A is the representing object for G: G(R) = Horn(A,R). The group axioms imply that (d«l)od = (1»d)od ((noe)»l)od = 1 = O»(noe))od and ( i * 1 ) od = noe = (1»i)od where n : k > A is the unit of the k—algebra structure on A. A i s thus a Hopf algebra. If G i s an affine group scheme we write A = k[G] i f A is the Hopf algebra representing G. Def i n i t ion: Let G be an a f f i n e group scheme over k, an al g e b r a i c a l l y closed f i e l d . Then G i s a f i n i t e D-group i f (i) dim k[G] i s f i n i t e . ( i i ) X(G) = { a e k[G] | d(a) = a*a } i s a k—linear basis of A. ( Note: X(G) i s always a group.) X(G) i s the group of characters of G. Thus k[G] = k[X(G)], the group algebra of X(G) over k. 1.3.1 F i n i t e D-group actions: Let X be an af f i n e variety defined over k. Then there is a canonical b i j e c t i o n between actions of 18 the D—group G on X, GxX > X, and direct sum decompositions k[G] = Ik[G](c) such that a e X(G) and k[G](o).k[G](a) is contained in k[G](c+*). Proof; Let R = k[X]. Given R = ER(a) define f : R > R«k[G] as f(x) = x»a for x e R(c). Clearly this determines an action GxX > X. Conversely, given f : GxX > X we have f* : R —> R»k[G] such that R is k[G]—comodule ( i . e . f* is co—associative) and (1»e)of* = 1, where e i s the augmentation on k[G]. One checks, using these two facts, that i f f*(a) = Ea(a)»o then a = Ia(c) and ( a ( a ) ) ( £ ) = a ( a ) i f a = B and 0 otherwise. Thus R = IR(o) where R(c) = { a e R | a = a(a) }. The remainder of the .chapter is devoted to the task of sharpening some known results (see . [28]) about Cohen—Macaulay rings and f i n i t e D—group actions. I have assumed throughout that a l l rings are Noetherian k—algebras and that k is an alg e b r a i c a l l y closed f i e l d . Let X be an affine scheme over k and l e t G be a f i n i t e D—group scheme such that f : GxX > X is an action of G on X. For example, i f X is an algebraic group and G i s a closed f i n i t e D—subgroup scheme then GxX > X, (g,x) > gx, is an action of G on X. Note that such an action may be n o n — t r i v i a l even i f G(k) (the set of k—rational points) consists of only one point. Let A be the coordinate ring of X. 1.3.2 Lemma: Let X, A, f be as above and let A(0) =•{ x € A | f(x) = x»1 }. Then (i) A(0) is a subalgebra of A. ( i i ) The inclusion, A(0) -—> A i s an integral extension of 19 rings. ( i i i ) A(a) i s an A(0)-module for a l l o e X(G). (iv) If A i s a normal domain then so is A ( 0 ) . Proof: (i) — ( i i i ) are straightforward. ( i v ) . It su f f i c e s to prove that A ( 0 ) is the intersection M, of K and A since A i s normal (here, K is the quotient f i e l d of A ( 0 ) ) . Let L be the quotient f i e l d of A. Now M i s an X(G)—graded subspace of A. Since K = K ( 0 ) = L ( 0 ) , we must have A ( 0 ) = M. 1.3.3 Lemma [6;ch.7,4.8]: Let A > B be a f i n i t e extension of normal in t e g r a l domains of the same dimension. Then B is a refle x i v e A—module. 1.3.4 Lemma[6;ch.7,4.2]: Let A be a normal integral domain and M a f i n i t e l y generated r e f l e x i v e A—module, such that K«M is isomorphic to K (where K i s the quotient f i e l d of A). Then M ' is isomorphic to a d i v i s o r i a l ideal I of A ( i . e . I i s the intersection of height one primary i d e a l s ) . 1.3.5 Lemma: Suppose A is an integral domain such that each l o c a l ring of A is a unique f a c t o r i z a t i o n domain. Let D be a d i v i s o r i a l ideal of A. Then D is a rank—one projective A—module. Proof: Well known. 1.3.6 Lemma: Suppose A i s a f i n i t e l y generated k—algebra and A ( 0 ) > A is as in lemma 1.3.2. Then A ( 0 ) is f i n i t e l y generated over k and A ( 0 ) > A is a f i n i t e extension of rings. Proof: Assume A = k[ x ( l ) , . . . , x ( n ) ] , x (i) homogeneous. Then x ( i ) * * l e A ( 0 ) for a l l i=1,...,n where 1 = |X(G)| = dim(k[G]) ('**' denotes exponentiation). Let B = k[x(1)**1,...,x(n)**l]. Then B—>A ( 0 )—>A. So B—>A ( 0 ) i s f i n i t e because B—>A i s so. Thus A ( 0 ) is f i n i t e l y generated and A ( 0 ) >A i s f i n i t e . 20 1.3.7 Lemma: Let G be a f i n i t e D—group, X a reduced and irreducible a f f i n e variety, and u : GxX > X an action of G on X. Then for a l l a c X(G), k[x](o)»K(0) and K(0) are isomorphic as A(0)—modules (where K(0) i s the quotient f i e l d of A(0)). Proof: Let A = k[X]. So A = IA(o) and without loss of generality A(a) is non—zero for a l l c. Consider the A(0)—bi1inear map A(a)«A(—a) > A(0). Since A i s a domain, i f x e A(—c) i s non—zero, then m : A(a) > A(0), m(z) = zx, is one—to—one and A(0)—linear. Thus A(a) is isomorphic with an ideal of A(0) and hence A(a)«K i s isomorphic to K. D e f i n i t i o n : Let X be an algebraic variety, dim(X) = n. X i s Cohen-Macaulay i f for a l l l o c a l rings 0(x), x e X, there exists a system of parameters {x(1),...,x(n)} of 0(x) which forms a regular sequence (see [12]). 1.3.8 D-group coverings: Suppose GxX > X i s an action of the D—group G on the normal af f i n e variety X. If X/G is smooth then X is Cohen—Macaulay. Proof: A = k [ x ] i s normal and A(0) = k[x/G] i s regular. Consider the inclusion A(0) > ZA(a). A(a) is a re f l e x i v e A(0)—module by 1.3.3 and A(a) is a rank one A(0)—module by 1.3.7. Thus, A(a) is isomorphic to a d i v i s o r i a l ideal by 1.3.4. Hence, A(o) i s a rank—one projective A(0)—module by 1.3.5. Thus, A(0) > A is a f l a t morphism. So A is Cohen—Macaulay. 1.3.9 D-group quotients: Suppose X is an irre d u c i b l e af f i n e Cohen—Macaulay variety and GxX > X i s an action of the f i n i t e D—group on X. Then X/G i s Cohen—Macaulay. Proof: Let A = k[X] and A(0) = k[X/G]. A is Cohen-Macaulay as an A—module, and thus as an A(0)-module. But A i s the dire c t sum of 21 A ( 0 ) and A(+) as an A-module (A(+) is the dire c t sum of the A(a) as o varies over a l l the n o n - t r i v i a l characters). Thus, A ( 0 ) is a Cohen-Macaulay A(0)-module. 22 II AFFINE ALGEBRAIC MONOIDS 2.1 Preliminaries  2.1.1 Def in i t ion: An af f i n e algebraic monoid E is an t r i p l e (E,m,1) such that (i) E is an a f f i n e algebraic variety over k. ( i i ) 1 e E(k). ( i i i ) m : ExE > E is a morphism of algebraic v a r i e t i e s such that mo(mxl) = mo(lxm) (associative). (iv) If p : E > E, p(x) = 1, then mo(p,l) = mo(l,p) = 1 (two—sided u n i t ) . In categorical terminology, an af f i n e algebraic monoid is a representab-le functor from the category of a f f i n e v a r i e t i e s to the category of monoids. An af f i n e variety i s completely determined by i t s a f f i n e algebra. So we can reformulate the d e f i n i t i o n above in terms of commutative algebra. Let E be an algebraic monoid and l e t A = k[E] be i t s coordinate ring. It follows from ( i i i ) and (iv) above that i f d m*, e : {1} > E i s the inclusion, and n : k > k[E] i s the unit of the algebra structure, then (d»1)od = (1 »d)od -and 1 = (noe,l)od = (1,noe)od. A is thus an augmented k—bigebra. A morphism f of algebraic monoids f : E > E' i s a morphism of algebraic v a r i e t i e s such that fom = m'o(fxf), where m and m' are the mult i p l i c a t i o n s on E and E' respectively, and f ( l ) = 1. Unlike the case for groups the la s t condition does not follow from the f i r s t unless f is dominant. 2 3 2.1.2 Translation of functions Def i n i t ion: Let E be an algebraic monoid. A ra t i o n a l E-module (V,p) is a morphism of monoids p : E > End(V) such that (i) For a l l v « V there exists a f i n i t e dimensional subspace V(v) of V such that v e V(v) and p(x)(V(v)) i s contained in V(v) for a l l x in E. ( i i ) If W i s a finite—dimensional subspace of V which i s E—stable then p|W : E > End(W) i s a morphism of algebraic monoids. Def i n i t i o n : p* : E > End(k[E]). For x e E l e t p(x) : E > E be defined by />(x)(y) = yx. Let />*(x) be the induced endomorphism on k[E]. 2.1.3 Proposition: (k[E],p*) is a ra t i o n a l E—module. Proof: See [ I3;p.62]. The proposition is. there stated for groups but the proof i s v a l i d for monoids as well. 2.1.4 Proposition: Suppose V is a subspace of k[E] which s a t i s f i e s (i) V is f i n i t e dimensional. ( i i ) />*(x)(V) i s contained in V for a l l x in E. ( i i i ) V generates k[E] as a k-algebra. Then p*|V : E > End(V) i s a closed imbedding. Proof: See [13,-p.63]. Remark: Putting 2.1.3 and 2.1.4 together we obtain that any af f i n e algebraic monoid E i s isomorphic to a closed submonoid of End(V) for some f i n i t e dimensional vector space V. 24 2.2 Elements 2.2.1 Lemma: Suppose G is an algebraic group and X i s a Zariski - closed subset of G. Then N(X) = { g € G | Xg is contained in X } is a closed subgroup of G. Proof: One checks that N(X) = { g e G | p*(l(X)) i s contained in I(X) } where I(X) = { f c k[G] | f(x) = 0 for a l l x in X }. Thus i f g e N(X) we have fi*{q) I (X) > I (X) v G] v k[X] (g) -> k (g) G] V k[X] But p*(g) : k[G] > k[G] i s an isomorphism and />*(g) acts r a t i o n a l l y on k[G]. Thus, p*(g) acts r a t i o n a l l y on I(X), so P*(g) : I(X) > I(X) is an isomorphism. Hence ( p * ( g ) ) " 1 ( I ( X ) ) = I(X) and thus g £ N(X) , since p*(g)-} = />*(g"1). 2.2.2 Corollary: Let G be an algebraic group, S a Zariski closed, m u l t i p l i c a t i v e l y closed subset of G. Then S i s a closed subgroup of G. Proof: S is a subset of N(S) by assumption so i f s e S then s _ 1 , s" 2 e N(S) by 2.2.1. Thus 1 = .ss' 1 t S and s s " 2 = s" 1 e S. Thus S i s a subgroup. 2.2.3 Corollary: Let E be an algebraic monoid, p : E > End(V) a' closed imbedding. Then G(E), the set of elements of E in 25 G1(V), is precisely the set of in v e r t i b l e elements of E. Furthermore, G(E) is an algebraic group and there is a morphism o : E > k = End(k) such that G(E) = c " 1 ( k * ) . Proof: By 2.1.4 there exists a closed imbedding p : E > End(V) for some V. Consider, p det S > Aut(V) > k* V V V E > End(V) > k where S i s the intersection of E and Aut(V). S is a closed subset of Aut(V) since p(E) is closed in End(V). Thus, by 2.2.2 S i s an algebraic subgroup of Aut(V). Clearly, S = G(E). Furthermore, i f c = deto/> then a" 1(k*) = G(E). 2.2.4 Lemma: Let x £ End(V), where V is a f i n i t e dimensional vector space over k. Then there i s an idempotent e(x) € End(V) such that (i) e(x) i s in the Zariski closure of { x, x 2, x 3,... } ( i i ) For a l l idempotents f in the Zar i s k i closure of { x, x 2, x 3,... }, fe(x) = e(x)f = f. Clearly, e(x) i s unique. Proof: For any endomorphism x, there exists a decomposition x = A .+ N, where A is in v e r t i b l e when r e s t r i c t e d to i t s image W, and N i s nilpotent (N and A commute to zero). Let e(x) be the idempotent with kernel = kernel(A) and image = image(A) and l e t X be the Zariski closure of { x, x 2,..., } in End(V). By 2.2.2 the intersection S of X and Aut(W) i s an algebraic subgroup of Aut(W). Thus e(x) £ Aut(W). (i) above i s s a t i s f i e d by d e f i n i t i o n and ( i i ) i s s a t i s f i e d because e(x) i s the identity element of 26 Aut(W). Remark: Lemma 2.2.4 may be regarded as a generalization of F i t t i n g ' s lemma. 2.2.5 Corollary: Let E be an algebraic monoid and l e t x e E. Then there exists e(x) e E such that (i) e ( x ) 2 = e(x) ( i i ) e(x) i s in the closure of { x, x 2,..., } ( i i i ) e(x)f = fe(x) for a l l other idempotents s a t i s f y i n g ( i i ) . Proof: By 2.1.4 there exists a closed imbedding p : E > End(V). If x e E then the closure of { x, x 2,..., } in End(V) i s contained in E. Thus apply 2.2.4. 2.2.6 Corollary: Suppose g : E > E' i s a morphism of algebraic monoids. If e 2 = e e g(E) then there exists f 2 = f e E such that g(f) =' e. Proof: g(x) = e so g(e(x)) = e. Note: Let E be an algebraic monoid. Then there exists k > 0 such that for a l l x in E, ye(x) = e(x)y = y, where y i s the k—th power of x. This i s true for End(V), with k = dim(V), by the proof of 2.2.4, and in general by 2.1.4. Notation: Let E be an algebraic monoid. I (E) = { e e E | e = e 2 } . 2.2.7 Corollary: Suppose E i s an algebraic monoid. Then the following are equivalent. (i) G(E) = E. ( i i ) 1(E) = { 1 }. Proof: If x e E-G then e(x) e E-G. Notat ion: Let E be an algebraic monoid, e e E an idempotent. Then eEe i s a closed submonoid of E with identity element e. Let 27 G(e) be the group of units of eEe. 2.2.8 Proposition: G(e) = { x e eEe | e(x) = e } = { x e E | xe(x) = x = e(x)x and e(x) = e }. Proof: Clear. 2.2.9 Corollary: Some power of every element of E i s in G(e) for some e e I ( E ) . Proof: Apply 2.2.8 to the note preceding 2.2.7. 2.2.10 Jordan Decomposition: Suppose x e G(e). Then there are unique elements x(u) and x(s) in G(e) such that (i) x = x(u)x(s) = x(s)x(u). ( i i ) x(u) is unipotent and x(s) is semi—simple in the group G(e). ( i i i ) For any morphism f : E > E', f(x(s)) = f(x)(s) and f(x(u)) = f ( x ) ( u ) . Proof: (i) and ( i i ) are clear; for ( i i i ) , i t s u f f i c e s to prove that f(e(x)) = e ( f ( x ) ) . But f(x)f(e(x)) = f ( e ( x ) ) f ( x ) = f ( x ) . So by 2.2.8, f(e(x ) ) = e ( f ( x ) ) . 2.3 Examples 2.3.1 Algebras: Let E be a f i n i t e dimensional associative algebra. Then E i s a linear algebraic variety and the mu l t i p l i c a t i o n map is b i l i n e a r . Thus E i s an algebraic monoid. 2.3.2 F i n i t e monoids: Let E be a f i n i t e (set valued) monoid, 1 e E and m : ExE > E the m u l t i p l i c a t i o n map. Let k[E] = Hom(E,k) ('Horn' in the category of sets). Then e : k[E] > k, e(f) f ( l ) , and d :k[E] >k[E]»k[E], d(f) = fom induce on E the structure of an algebraic monoid. Corollary: Let E be a f i n i t e monoid, x e E. Then there is an 28 integer n such that the n—th power of x is an idempotent. Proof: By 2.2.9 the k—th power of x is in G(e(x)) for some k. But G(e(x)) i s a f i n i t e group. 2.3.3 D-monoids: Let S be a f i n i t e l y generated submonoid of Z(n), the free abelian group of rank n and let k[S] be the monoid algebra of S over k. Then E(S) = Hom(S,k) (as monoids) i s a D—monoid (diagonalizable). D—monoids are characterized by the property of being isomorphic with a closed submonoid of some monoid of diagonal matrices. 2.3.4 Algebraic structures: Let V be a f i n i t e dimensional vector space over k and l e t V(m) denote the m—th tensor product of V over k. Suppose S is a subset of the union of the Hom(V(n),V(m)) as m and n vary. Define End'(V) = { f « End(V) | f(m)os = sof(n) for a l l s t S }. Then End'(V) i s an algebraic monoid. This example w i l l be discussed in chapter 9 ( see also [8]). 2.3.5 Let V be any af f i n e variety defined over k. Let E be the d i s j o i n t union of V and a point 1 . For x, y e E define xy = x i f x and y are elements of V or y = 1 , and xy = y i f x = 1 . Then E, with t h i s m u l t i p l i c a t i o n , is an a f f i n e algebraic monoid. 29 III IRREDUCIBLE ALGEBRAIC MONOIDS 3 . 1 F i r s t P r i n c i p l e s  Def in i t ion: An algebraic monoid E is i rreduc ible i f i t is so as an algebraic variety. Unlike the case of algebraic groups, arbitrary algebraic monoids are vastly more general than irreducible algebraic monoids. Example 2.3.5 suggests that algebraic monoids in complete generality are not suitable for axiomatic study. 3.1.1 Proposition: Let E be an algebraic monoid. Then there exists a unique maximal irreducible component E(0) of E such that 1 « E(0). Proof: Let (E(i)} be the set of maximal irreducible components of E. Suppose 1 i s an element of both E(0) and E(1). Then E(0)E(1) i s irreducible and contains both E(0) and E(1). Thus, by maximality, E(0) = E(1). Let E° = E(0). 3.1.2 Proposition: Let E be an algebraic monoid, and l e t G be the group of units of E. Then E° i s the Zariski closure of G° in E. Proof: 1 e G° and G° is i r r e d u c i b l e . Thus, G° is a subset of E°. But G° is open in E°. Thus, G° is dense in E°. 3.1.3 Proposition: E° i s an algebraic submonoid of E. Proof: E°E° i s an irreducible subset of E which contains 1. Thus E°E° = E° by 3.1 . 1. 3.1.4 Proposition: Let E be an algebraic monoid and l e t e be an idempotent of E which is in the Zariski closure of G(E). Then -e e E°. 30 Proof: Let f : E > E be the morphism of v a r i e t i e s which maps each element to i t s n—th power. Then for some n, f maps G(E) to G(E)° (since the later group has f i n i t e index in""the former). But every idempotent i s a fixed point of such a morphism. 3.2 Integral Closure And Normalization 3.2.1 Proposition: Suppose we have the following commutative diagram where A and B are integral domains. j A > B v v A»A > B*B If x € B ,is integral over A then d(x) e B»B is integral over A»A ('•' denotes tensor product of vector spaces). Proof: Clear. 3.2.2 Proposition: Suppose we have morphisms A — > A [ l / f ] —> B where B i s a normal k—domain and the second morphism is f i n i t e and dominant. Then A'«A' > B»B i s i n t e g r a l l y closed, where A' is the integral closure of A in B. Proof: A ' [ l / f ] = B since l o c a l i z a t i o n commutes with integral closure. Further, A'»A' i s normal because A' i s . Thus, A'»A' > B»B i s i n t e g r a l l y closed because B»B = A'»A'[1/f•f]. 3.2.3 Theorem: Suppose E is a normal irreducible algebraic monoid and p : G > G(E) i s a f i n i t e dominant morphism of algebraic groups. Then the following diagram can be f i l l e d in uniquely 31 j ' G > E' I I v v G(E) > E j in such a way that (i) E' i s normal and i r r e d u c i b l e . ( i i ) />' i s a f i n i t e morphism of algebraic monoids. ( i i i ) j ' is an open imbedding. Proof: Let R be the integral closure of k[E] in k[G]. Then R i s normal, />'* i s f i n i t e and j ' * is an open imbedding (p' * : k[E] > R and j ' * : R > k[G(E)]). By 3.2.1 and 3.2.2 the comultiplication d, of k[G] r e s t r i c t e d to R is a comultiplication on R. Further, the augmentation of k[G] r e s t r i c t e d to R i s an augmentation on R. Thus, (R,d|R,e|R) is a normal bigebra, f i n i t e l y generated over k. Hence, the diagram of algebras dualizes to y i e l d the diagram of monoids advertised in the assertion of the theorem with R = k[E']. This diagram i s already uniquely determined by the underlying geometry. Remarks: Let E be an irreducible algebraic monoid. Then there i s a unique ir r e d u c i b l e monoid E' and a morphism n : E' > E such that (i) n is f i n i t e , dominant and b i r a t i o n a l . ( i i ) E' i s a normal algebraic variety. The d e t a i l s w i l l be l e f t to the reader. The construction in Theorem 3.2.3 is an important ingredient in the existence results of the next section. 32 3.3 E x i s t e n c e Of A l g e b r a i c Monoids The purpose of t h i s s e c t i o n i s t o c h a r a c t e r i z e the i r r e d u c i b l e a l g e b r a i c groups G f o r which t h e r e i s an a l g e b r a i c monoid E w i t h G(E) = G ( n o n — t r i v i a l l y ) . I t t u r n s out t h a t i n case t h i s i s p o s s i b l e , E may be chosen w i t h 0. O b s e r v a t i o n : L e t G be an i r r e d u c i b l e a l g e b r a i c group and suppose X(G) = Hom(G,k*) i s t r i v i a l . I f G = G(E) f o r some i r r e d u c i b l e monoid E, then G = E. P r o o f : There e x i s t s p : E > End(V) a c l o s e d imbedding. F u r t h e r , p(G(E)) i s c o n t a i n e d i n G1(V). By assumsion, the composite G(E) > G1(V) > k* i s t r i v i a l f o r any c h a r a c t e r of G1(V). Thus G(E) i s c o n t a i n e d i n S l ( V ) . T h i s f o r c e s G(E) = E because S1(V) i s c l o s e d i n End(V). The remainder of t h i s s e c t i o n i s devoted t o a pro o f of the c o n v e r s e . 5 Lemmas 3.3.1 Lemma; L e t S be a f i n i t e l y g e n e r a t e d submonoid of some f r e e a b e l i a n group and suppose p : S > N i s a monoid map such t h a t p - 1 d ) = { 1 } (where N = { 0, 1, 2,.. } ) . Then S* = { 0 } where S* = { s e S | -s e S }. P r o o f : p " 1 ( 0 ) c o n t a i n s S*. 3.3.2 Lemma: L e t Z(n) be a f r e e a b e l i a n group of rank n and l e t <a(1),...,a(n)> be the submonoid of Z(n) ge n e r a t e d by { a ( i ) }. Le t u e Z(n) be non-zero. Then < mu+a(1),...,mu+a(n) >* = { 1 } f o r a l l s u f f i c i e n t l y l a r g e m. P r o o f : Choose p : Z(n) > Z such t h a t />(u) > 0. Then m p ( u ) + p ( a ( i ) ) > 0 f o r a l l i i f m i s s u f f i c i e n t l y l a r g e . Thus 33 Lemma 3.3.1 applies. 3.3.3 Lemma: Suppose E i s a D—monoid and j : E > End(V) is a morphism such that (i) V = EV(a) (a € X(E)) ( i i ) V(0) = { v £ V | j ( t ) ( v ) = v for a l l t £ E } = (0) ( i i i ) 0 £ E. Then j(0) = 0 (the zero endomorphism of V). Proof: One checks that V(0) = { v e V | j(0)(v) = v }. 3.3.4 Lemma: Suppose T is a D-group and p : T > G1(V) is a morphism. Let i : G1(V) > End(V) be the canonical inclusion and suppose V = IV(a), a e X(T). Then the image of k[End(V)] under p*o\* in k[T] is k[o;V(c) is non—zero]. Furthermore, k[c; V(c) i s non—zero] i s thereby i d e n t i f i e d with the coordinate ring of the closure of io/>(T) in End(V). Proof: Straight forward. 3.3.5 Lemma: Suppose there exist morphisms u : G > k* ZG1(V) and j : G > Sl(V) (viewed as morphisms to G1(V)). Let T be a maximal torus of G and suppose V = EV(a) (direct sum decomposition) via j . Consider g(m) : G > G1(V), the morphism obtained by multiplying j and mu. Then, via g(m), V = EV'(c+mu) where V(o+mu) = V i a ) . Proof: Via) = { v € V | j ( t ) ( v ) = a(t)v for a l l t e T }. So i f v £ V(c) then g(m)(t)(v) = (c+mu)(t)v for a l l t £ T . Note: Thus, by 3.3.2 and 3.3.3, i f m i s s u f f i c i e n t l y large and u is n o n — t r i v i a l then 0 i s an element of the closure of g(m)(T) in End(V) . Conclusion: (putting 3.3.1 — 3.3.5 together) Assume X(G) is n o n - t r i v i a l . Let j : G > Sl(V) be an imbedding 34 and l e t u : G > k* be a n o n — t r i v i a l character. Then for a l l 1 > 0, g(m) : G > G l ( V ) i s f i n i t e . Furthermore, by 3.3.2, 3.3.4 and 3.3.5, i f 1 is large enough then the closure of g(m) has a zero of i t s own for T a maximal torus of G. By 3.3.3 we can choose 1 large enough so that the zero of End(V) i s in the closure of g(m). Hence, l e t t i n g p = g(m), we have G >G'—>G1(V)—>End(V) p such that /> is a f i n i t e and dominant and 0 i s an element of the closure of G' in End(V). Let E' be the normalization of the closure of G in End(V). Then we have, 0 e E' and E' is irreducible and normal. Consider the following.diagram. G v G' > E' By Theorem 3.2.3 the diagram can be f i l l e d in uniquely to y i e l d j G > E v v G' > E ' such that j i s an open imbedding and p' i s f i n i t e and dominant. It follows that E also has a zero. In summary, we have established the following r e s u l t . 3.3.6 Theorem: Let G be an irreducible 'algebraic group. Then the following are equivalent. 35 (i) There exists an irreducible algebraic monoid E such that G(E) = G and E i s not a group. ( i i ) There exists an irreducible monoid E such that G(E) = G and 0 e E (with 0 not equal to 1). ( i i i ) X(G) i s a n o n — t r i v i a l group. (iv) rank(R(G)) > 0. 3.4 Closure A useful strategy in the theory of algebraic monoids is to apply the structure theory of algebraic groups and v a r i e t i e s in studying closure properties of various subgroups T of G in E. The purpose of t h i s section i s to assemble some of these techniques. I s h a l l assume throughout that E i s an irreducible  a f f i n e algebraic monoid. 3.4.1 Proposition: Let E be an irreducible algebraic monoid with group of units G and l e t B be a Borel subgroup of G = G(E). Then (i) E i s the union of gZg" 1 as g varies over G, where Z i s the closure of B in E. ( i i ) E i s the union of gZ as g varies over G. Proof: G/B i s complete and B»Z is contained in Z, where '•' denotes either conjugation or l e f t t r a n s l a t i o n . Thus, by 1.2.3 ( i ) , G«Z i s closed in E. Recall from 1.2.2 (iv) that i f B(G) is the set of Borel subgroups of G, then we may regard B(G) as a complete algebraic variety in such a way that GxB(G) > B(G), (g,B) — > gBg~ 1, i s a morphic group action. The purpose of the next two lemmas is to prove that i f x € E then B(x) = { B € B(G) | x i s an element of the closure of B } is a closed non-empty subset of B(G). 36 3.4.2 Lemma: Suppose p : X > Y is an surjective open map of topological spaces and further, that V i s a closed and saturated subset of X ( i . e . V = p - 1 ( p ( V ) ) ) . Then p(V) is a closed subset of Y. Proof: />(X-V) is open in Y and />(X-V) = piX)-p(v) = Y-p(V) since p is saturated. Thus p i V ) i s closed in Y. 3.4.3 Lemma: Let x e E and suppose x e Z, the closure of B(0) 6 B(G). Let V = { g € G | g" 1xg e Z }. Then p { V ) i s a closed subset of G/B(0), where p : G > G/B(0) i s defined by p i g ) = gB(0). Proof: V i s closed since i t is a transporter with closed target. So i f g e V then x e gZg" 1. Hence x e gbZ(gb)" 1 and thus, gb « V for a l l b e B(0). Thus, V is saturated with respect to p. By Lemma 3.4.2, />(V) i s closed in G/B(0) . 3.4.4 Proposition: Let x e E. Then B(x) = { B c B(G) | x i s in the closure of B } is closed in B(G). Proof: B(G) may be regarded as a complete variety under the i d e n t i f i c a t i o n given in 1.2.2 ( i v ) . Under this i d e n t i f i c a t i o n { gB(0) | g" 1xg € Z(0), the closure of B(0) } corresponds to { B e B(G) | x i s in the closure of B }. 3.4.5 Proposition: Let E be an irreducible algebraic monoid and le t T be a maximal torus of G(E). Suppose x e. E and xt = tx for a l l t € T. Then there i s a Borel subgroup B e B(G) such that T is contained in B and x i s in the closure of B. Proof: B(x) is closed in B(G) and B(x) is stable under conjugation by T since T centralizes x. By 1.2.2 ( i i i ) T has a fixed point in B(x). Thus there exists B e B(x) such that T normalizes B. Hence, by 1.2.2 ( i i ) , T i s contained in B. 37 3.4.6 Proposition: Suppose x € E i s semi—simple and T is a maximal torus such that xt = tx for a l l t e T. Then x e X, the closure of T. Proof: By 3.4.5 there exists a Borel subgroup B of G such that x is in the closure Z, of B and T i s contained in B. Now there exists a representation Z > T(V) (upper t r i a n g u l a r ) . From linear algebra we can assume that both T and x are contained in D(V) the diagonal matrices of T(V). So we have { x, X } —> Z > T(V) > D(V) where the la s t map i s the quotient of T(V) modulo i t s ideal of nilpotent elements. The composite of a l l these maps is one—to—one since both X and { x } are contained in D(V). But the image of x is in the image of X because X is the closure of a maximal torus. Thus x 6 X. The following fundamental result i s due to M. Putcha [21;Theorem 1]. 3.4.7 Proposition: Let E be an irreducible algebraic monoid and le t e e 1(E) (idempotents). Then (i) There exists a closed irreducible submonoid E' of E such that Ee i s contained in E' and eE' = eEe. ( i i ) There exists a closed irreducible submonoid E" of E such that eEe is contained in E" and E" is contained in the ce n t r a l i z e r of e in E. 3.4.8 Corollary[21 ] : Let E be an irr e d u c i b l e algebraic monoid and l e t e t 1(E). Let C(e) = { x e E | xe = ex }. Then eEe i s contained in C(e)°, the identit y component of C(e). Proof: eEe is contained in E" as in 3.4.7, and E" is contained in C(e). Thus E" i s contained in C(e)° since E" is i r r e d u c i b l e . 3.4.9 Proposition[21]: Let E be an irreducible algebraic monoid 38 and l e t e « 1(E). Let CG(e) = { g e G(E) | ge = eg }. Then CG(e) > G(e), g — > ge = eg, i s a surjective morphism of algebraic groups. Proof: Consider C(e) > eEe, x — > ex = xe. By 3.4.8, this i s a dominant morphism when r e s t r i c t e d to C(e)°. Thus CG(e)) > G(e) i s dominant. But a dominant morphism of algebraic groups i s surjective. 3.4.10 Proposition: Let x € E be semi—simple. Then x e X, the closure of some maximal torus T of G(E). Proof: Let e = e(x). Then x e G(e). By 3.4.9 CG(e) > G(e), g — > ge, is surjective. Thus there exists a maximal torus T in CG(e) such that x £ eT. By 3.4.6 e € X, the closure of T. Thus x t eT, which i s contained in the closure of T. 3.4.11 Proposition: Let x e E be semi—simple. Then Cl(x), the conjugacy class of x, is closed in E. Proof: By 3.4.10 there exists a maximal torus T of G(E) such that x £ X, the closure of T. So t x t " 1 = x for a l l t £ T since X is a commutative monoid. Thus by 1.2.3 ( i i ) Cl(x) i s closed in E. 3.4.12 Proposition: Let E be an irre d u c i b l e algebraic monoid. Then there are a f i n i t e number of conjugacy classes of idempotents in E. Proof: Let e e 1(E) and let T be a maximal torus of G(E). By Proposition 3.4.10 there exists g £ G(E) such that geg" 1 e X, the closure of T in E. But the number of idempotents in X i s f i n i t e since X is isomorphic to a closed submonoid of the diagonal matrices, D(V) for some V. In subsequent chapters i t w i l l be necessary to know .that 39 under certain conditions the image of an algebraic monoid i s a closed subset of the target monoid. It is easy to construct examples which demonstrate that f a i r l y s t r i c t conditions are required. 3.4.13 Proposition: Suppose E and E' are irreducible algebraic monoids with zeros 0 and 0' respectively. If p : E > E' is a morphism such that /o~1(0') = { 0 } then p i s a f i n i t e morphism. In p a r t i c u l a r , p(E) i s a closed submonoid of E'. Proof: Choose a 1—p.s.g. a : k* — > G(E) such that a extends to a : k —> E with a(0) = 0. This y i e l d s an action of k on E by l e f t t r a n slation and s i m i l a r i l y on E' via p. On the l e v e l of coordinate algebras t h i s translates to: p : k[E'] > k[E] is a morphism of N—graded algebras (where N = {0,1,2,...}). Since a converges to zero we obtain k[E'](0) = k = k[E](0). Since p'MO') = { 0 } we have that k[E] is f i n i t e dimensional modulo k [ E ' ] + . Thus, by 1.1.4, k[E] i s a f i n i t e k[E']-module. 40 IV TYPES OF MONOIDS In this chapter I discuss some of the special properties associated with the five most important types of monoids. These are D—monoids, nilpotent monoids, solvable monoids regular monoids, and reductive monoids. 4.1 D-monoids Def i n i t i o n : An irreducible D-monoid E is an irreducible algebraic monoid such that G(E) i s a torus. D—monoids are quite varied and have been studied extensively from a geometric point of view in [14]. M. Hochster [12] has proven that normal D—monoids are Cohen—Macaulay. D—monoids have also been studied as algebraic monoids by M. Putcha in [19] and [20]. It i s - hoped that ultimately the c l a s s i f i c a t i o n of reductive monoids can be reduced to problems concerned with D—monoids. This section i s mostly summary. Pertinent d e t a i l s not mentioned here are well recorded in [14], [19] and [20]. 4.1.1 Proposition: There i s a categorical equivalence between the category D of irreducible D—monoids and the category M of f i n i t e l y .generated commutative monoids which can be imbedded in a free abelian group. The equivalence i s given by functors, E > X(E) = Hom(E,k) (D—monoid morphisms) S > E(S) = Hom(S,k) (monoid morphisms) X(E) i s the character monoid associated with E. 4.1.2 Proposition: Let E be an irr e d u c i b l e D—monoid and l e t 1(E) = { e e E | e 2 = e } . Then (i) 1(E) is f i n i t e . ( i i ) If 0 € E and i f 0 < e d ) < ... < e(k) = 1 is a saturated 41 chain in 1(E), then dim(E) = k. ( i i i ) If dim(E) = 2 and 0 e E then 1(E) = { 1 , e, f, ef = 0 }. (iv) If e e 1(E) then e is the product a l l the maximal idempotents which are larger than or equal to e. 4 . 1 . 3 Proposition[14;p.12]: There are canonical one—to—one correspondences among { U | U is an open affine G(E)—equivariant subset of E }, 1(E) and { X | X i s a G(E)-orbit in E }. If U i s open a f f i n e and G(E)—equivariant then there i s a unique minimal idempotent e(U) e U. 4 .1.4 Proposition[193: For a l l maximal idempotents e e 1(E)— {1} there i s a unique one—dimensional subgroup G(e) of G(E) such that e is an element of the closure of G(e) in E. 4 . 1 . 5 Proposition[1 4]: Let E be a D—monoid. Then E i s a normal variety i f and only i f for a l l x € X(G(E)), nx e X(E) implies that x e X(E). 4 . 1 . 6 Proposi t ion: Let E and E' be irreducible D—monoids with zeros 0 and 0'. Suppose p : E > E' i s a morphism such that p{0) = 0'. Then the following are equivalent. (i) p is a f i n i t e morphism. ( i i ) p\l(E) : 1(E) > I(E') i s one-to-one. ( i i i ) There exists n e N such that nX(E) i s contained in P*(X(E')). Proof: (i) => ( i i ) . If /.(e) = pit) then />(ef) = p(e) = pit). If e i s not f then ef < e and thus efE i s a proper subset of eE, whereas p(efE) = pieE) . Thus dim(eE) > dim(/>(eE) ) . So p is not f i n i t e . ( i i ) => ( i ) . If p : 1(E) > I(E') i s one-to-one then p'MO1') is the union of the G(E) orbits of the idempotents that i t 42 contains. Thus by 3.4.13 p i s f i n i t e . (i) => ( i i i ) ; With an irrelevant loss of generality we may assume that p is dominant. Thus we have p* X (E ' ) > X (E ) v p* v X(G') > X(G) If p is f i n i t e then mX(G) is contained in />*(X(G')) for some m e N. But by assumption, each element of mX(E) is integral over k[X(E')]. So, by 4.1.5, there exists n e N such that nX(E) is contained in p*(X(E')). ( i i i ) => (i) i s cl e a r . 4.1.7 Proposition: Let E a normal D—monoid with 0. Then there exists F(1),...,F(m) e X(E) such that (i) m i s the number of minimal non—zero idempotents of E. ( i i ) <F(1),...,F(m)> > X(E) i s a f i n i t e morphism (where < ... > denotes "the monoid generated by"). ( i i i ) { F ( i ) } is a l i n e a r l y independent set modulo m2, the square of the ideal of functions that vanish at 0. Furthermore, { F ( i ) } i s the only such subset of characters s a t i s f y i n g a l l these properties. F = { F ( i ) } i s the set of fundamental generators of X(E). Proof: Let e e 1(E) be a minimal non—zero idempotent. Then by 4.1.2 ( i i ) , dim eE = 1. Since E i s normal, eE is normal, because i t i s a retract of E. Thus eE i s isomorphic to k. So k[eE] = k[F(e)], where F(e) i s the unique character which generates k[eE]. E — > eE, x —> ex is a morphism of D—monoids so X(eE) — > X(E) i s a morphism of monoids. Consider, p : E > Z, />(x) (ex),, e minimal, where Z is the direct product of a l l the eE 43 as e ranges over the minimal idempotents of E. Then p is a morphism of D—monoids such that p(e) i s non—zero for every minimal idempotent e in E. Thus, 1(0) = { 0 }, because otherwise i t contains a minimal idempotent. Hence p is f i n i t e by 3^4.13. On the l e v e l of characters we have <F(1),...,F(m)> > X(E) (where the F(e) have been relabled). This proves (i) and ( i i ) . Each F ( i ) i s non-zero modulo m2 because eE is a retract of E. Thus { F ( i ) } i s a l i n e a r l y independent subset of m/m2. This proves ( i i i ) . If { x ( i ) } is a subset of X(E) such that <x(1),...,x(m)> > X(E) i s f i n i t e then i t follows that each x(i) i s some power of one of the F ( i ) . Thus { x(i) } s a t i s f i e s ( i i i ) i f and only i f { x(i) } = { F ( i ) }. 4.2 Nilpotent Monoids  D e f i n i t i o n : Let E be an irreducible algebraic monoid. E is nilpotent i f G(E) i s a nilpotent algebraic group. A well known structure theorem asserts that i f G i s a nilpotent algebraic group then G(u) = { u e G | u i s unipotent } and G(s) = { s e G | s is semi—simple } are closed subgroups of G and G is isomorphic to the dire c t product of G(u) and G(s). The purpose of this section i s to characterize the class of nilpotent monoids for which a generalization of this theorem is possible. M. Putcha has obtained similar results for commutative monoids in [19]. 4.2.1 Lemma: Suppose E i s an irre d u c i b l e nilpotent monoid. Let T be the maximal torus of G(E). Then every semi—simple element of E i s in the closure of T. Proof: By 3.4.10 every semi—simple element of E is in the 44 closure of a torus. Def in i t ion; Let E be an algebraic monoid. If E i s the union of the G(e) as e varies over a l l idempotents, then E i s a C l i f f o r d  monoid (see 2.2.7 — 2.2.10 for a discussion of G(e)). 4.2.2 Lemma: Suppose E i s a nilpotent C l i f f o r d monoid. Let e t 1(E) be a maximal idempotent. Then E(e) = { x £ E | xe = ex = e }° is a one—dimensional D—submonoid of E. Proof: l(E(e)) = { 1, e } and e i s the zero element of E(e). Suppose x £ E(e) and x i s not e. Then e(x) is not equal to e either, because i f e = e(x) then x £ G(e) since E i s C l i f f o r d . But then e is the only element common to both G(e) and E(e). Thus, e(x) = 1, since 1(E) = { 1, e }. Hence, x £ G(E(e)). So E(e) i s the union of G(E(e)) and { e }. Thus, dimE(e) = 1, since by 2.2.3, dim(E-G(E)) = dim(G)-1. Furthermore, G(E(e)) is a D—group, because irreducible unipotent monoids are groups. 4.2.3 Lemma: Let E and e be as in 4.2.2. Then G(u) > eG(u), u — > eu, is a f i n i t e morphism. Proof: Let K = { v e G(u) | ev = e }. By 4.2.2, K° i s irred u c i b l e , and semi—simple. Thus, K° = { 1 } and so K i s f i n i t e . 4.2.4 Lemma: Let E be as in 4.2.2 and suppose e £ 1(E) i s any idempotent. Then G(u) > eG(u) is f i n i t e . Proof: Let 1 > e(l) > ... > e(k) = e be a saturated chain of idempotents. Then e(i)G(u) —> e(i+l)G(u) i s f i n i t e for each i by 4.2.3 applied to the C l i f f o r d monoid e ( i ) E e ( i ) . Since ex e ( k) . . . e (1 ) x , G(u) —> eG(u) i s the composite of f i n i t e morphisms. Thus G(u) —> eG(u) is f i n i t e . 4.2.5 Lemma: Let E be as in 4.2.2. Then m : E(s)xG(u) > E, 45 (x,u) > xu is a f i n i t e b i r a t i o n a l morphism of algebraic monoids. Proof: m is b i r a t i o n a l by the well known result for algebraic groups. Suppose (x,u) s a t i s f i e s xu = e (so e(x) = e). Then eu = x*xu =x*e = x* (for some x* e G(e)). But then x* is a semi—simple element of eG(u). Hence, x* = e and thus x = e. It follows that m i s one—to—one and b i r a t i o n a l . m is onto because image(m) contains G(E) and a l l idempotents. The same i s true for the normalization of E. Thus, by 1.1.2, m i s f i n i t e . 4.2.6 Theorem: Suppose E i s irreducible and nilpotent. Then the following are equivalent. (i ) E is C l i f f o r d . ( i i ) The morphism m : E(s)xG(u) > E, (x,u) > xu is f i n i t e and dominant. If, in addition, E is normal, then m is an isomorphism. Proof: (i) => ( i i ) . Lemma 4.2.5. ( i i ) => ( i ) . Both groups and D—monoids are C l i f f o r d monoids. It follows that E(s)xG(u) i s C l i f f o r d . Thus E i s C l i f f o r d as i t i s the image of a C l i f f o r d monoid. If E i s normal then i t follows from 4.2.5 and 1.1.2 that m is an isomorphism. 4.3 Solvable Monoids  Def i n i t i o n : Let E be an irreducible algebraic monoid. Then E i s solvable i f G(E) is a solvable algebraic group. In t h i s section I prove two general results about solvable algebraic monoids. 4.3.1 Theorem: Let E be a solvable irreducible algebraic monoid. Then there exists an irreducible D-monoid X and a morphism 46 p : E > X such that for any morphism f : E > Y where Y i s a D—monoid there is a unique morphism f* : X > Y, such that t*op = f. Furthermore, for a l l maximal t o r i T of G(E) the composite Z > E > X is an isomorphism, where Z i s the closure of T in E. 4.3.2 Theorem: Let E be an ir r e d u c i b l e algebraic monoid with zero. Then the following are equivalent. (i) E is solvable. ( i i ) N = { x e E | x i s nilpotent } i s a two sided ideal of E. Proof of 4.3.1: Let X(E) be the characters of E. Then X(E) i s a l i n e a r l y independent m u l t i p l i c a t i v e subset of k[E]. Let R = k[X(E)], the monoid algebra of X(E) over k, and let T be a maximal torus of G(E). Let Z be the closure of T in E. Thus, the composite R — > k[E] —> k[Z] is one—to—one, since the same is true when r e s t r i c t e d to G(E), and G(E) is dense in E. Claim: R > k[Z] i s an isomorphism. Proof of claim: There exists a closed imbedding E > T(V) (upper triangular matrices) for some V. We may assume also that Z i s contained in D(V), the diagonal matrices. Suppose we have a character p : Z > k. Then p can be l i f t e d to a character o : D(V) > k since Z > D(V) i s a closed imbedding. But o l i f t s to T(V) because the inclusion D(V) > T(V) s p l i t s . R e s t r i c t i n g t h i s l i f t i n g to E y i e l d s : Every character on Z l i f t s to E. This proves the claim. Now suppose f : E > Y i s a morphism, where Y i s a D—monoid. On the l e v e l of characters t h i s y i e l d s f* : X(Y) > X(E). But X(E) is contained in R. So f factors through X Spec(R). 47 Proof of 4.3.2: (i) => ( i i ) . By 4.3.1 there i s a morphism p : E > X such that for every maximal torus T in G(E) the composite Z > E > X is an isomorphism. Thus, since every semi—simple element is in the closure of a torus, 0 is the only semi—simple element of p~ 1(0). Now />~1(0) is a closed ideal of E; so i f x e />"1(0), then e(x) e />"1(0). Thus / T 1 ( 0 ) is the set of nilpotent elements of E. ( i i ) => ( i ) . Suppose E is not solvable. Then C(T) i s a proper subset of N(T) ( i ^ e . the Weyl group i s n o n - t r i v i a l ) . Let a e W = N(T)/C(T) be a n o n - t r i v i a l element. So int(o) : l(Z) > l(Z) i s n o n — t r i v i a l , since by 4.1.7, the automorphisms of a D—monoid with 0 are f a i t h f u l l y represented on the idempotents. It follows that int(c) acts n o n — t r i v i a l l y on the minimal idempotents of Z. So l e t e, f € I(Z) be minimal non—zero idempotents such that ae = fo. Thus (ae)2 = aefo = aOa = 0 since e and f are d i s t i n c t minimal idempotents of Z. But then ae i s nilpotent, yet oeo*' is not. Hence the nilpotent elements do not form an i d e a l . Much more can be said about the structure of solvable algebraic monoids. In [24;Theorem 23] M. Putcha gives numerical and semigroup characterizations of solvable algebraic monoids. 4.3.3 Proposition: Let E be a solvable i r r e d u c i b l e algebraic monoid and l e t x, y e E(s). Suppose xy = yx and p(x) = />(y) where p is the universal morphism to a D—monoid. Then x = y. Proof: By 4.3.1, i t s u f f i c e s to prove this for E = T(V) (upper triangular matrices). Let C(x) be the c e n t r a l i z e r of x in E. Then C(x) and ZC(x) are both i r r e d u c i b l e , since they are both lin e a r subspaces of T(V). Let T be a maximal torus of'C(x) such that y e X, the closure of T in C(x). But we also have x € X 48 since x i s a central semi—simple element. Thus x = y since />|X is one—to—one. 4.4 Reductive And Regular Algebraic Monoids  De f i n i t i o n : Let E be an irre d u c i b l e algebraic monoid. E is reductive i f G(E) is a reductive algebraic group. D e f i n i t i o n : (a) A monoid E is regular i f for a l l x e E there exists g € G(E) and e e 1(E) such that gx = e. (b) A monoid E is von Neumann regular i f for a l l x e E there exists a e E such that xax = x. Let E be an irreducible algebraic monoid. It i s then a consequence of [21/Theorem 13] that E i s regular i f and only i f E is von Neumann regular. Thus, I have often taken the l i b e r t y of using the d e f i n i t i o n that i s most convenient. The main result of this section (see Theorems 4.4.14 and 4.4.15) asserts that a l l reductive monoids are regular. Let us r e c a l l some properties concerning conjugacy classes in semi—simple algebraic groups. 4.4.1 Proposition[29;p.87-92]: Let G be a semi—simple algebraic group. (i) If x € G then Cl(x) i s closed in G i f and only i f x is semi—simple. ( i i ) Let cl[G] = f f e k[G] | f(xy) = f(yx) for a l l x, y e G }. Then cl[G] is the ring of invariant functions under the induced action of conjugation. ( i i i ) Let T be a maximal torus of G, W the Weyl group. Then 49 cl[G] •> k[G] v v k[T] ' > k[T] commutes, where k[T]' is the the ring of invariant functions in k[T] under the induced action of W. Remark: If G i s an irreducible reductive group then (G,G) = G' is semi—simple and G i s commensurable with the product of G' and Z(G)°. It follows from th i s and 4.4.1 above, that 4.4.1 is also v a l i d for reductive groups. Let c l f E ] = { f c k[E] | f(xy) = f(yx) for a l l x,y e E }. 4.4.2 Proposition: Let E be an irreducible algebraic monoid, T a maximal torus. Suppose for x, y t X, the closure of T, there exists g e G such that gxg" 1 = y. Then there exists w e N(T) (normalizer) such that wxw" ' = y. Proof: g~ 1Tg and T are contained in CG(x)°, the identity component of the c e n t r a l i z e r of x in G(E). Thus there exists z e CG(x)° such that zg" 1Tgz" 1 = T. But then zg" 1 £ N(T), yet (zg" 1)~ 1x(zg~ 1 ) = gxg" 1 = y. If x e E is semi—simple then by 3.4.10, x is in the closure of some maximal torus. Thus, by 4.4.2 the semi—simple conjugacy classes are canonically parametrized by X/W. 4.4.3 Theorem: Let E be a reductive algebraic monoid and l e t T be a maximal torus of G. Let X be the closure of T in E. Then the inclusion c l [ E ] > k[E] followed by the projection k[E] >> k[X] induces an isomorphism of c l [ E ] onto k [ X ] ' , the ring of invariant functions under the induced action of W on k [ X ] . Proof: c l [ E ] i s the intersection in k[G] of k[E] and c l [ G ] . It 50 follows from this that (when r e s t r i c t e d to T) c l [ E ] is the intersection in k[T] of k[X]' and c l [ G ] . Thus, by the remark following 4.4.1," cl [ E ] is i d e n t i f i e d with k[X]'. Remark: From general p r i n c i p l e s (1.2.4) we have, (i) c l [ E ] i s a f i n i t e l y generated k—algebra since G i s reductive and c l [ E ] is a ring of invariants of G. ( i i ) The morphism c l : E > E(cl) = Spec(cl[E]), induced from the inclusion c l [ E ] > k[E] s a t i s f i e s : each f i b r e of the morphism c l contains precisely one closed conjugacy c l a s s . 4.4.4 Theorem: Let E be a reductive algebraic monoid and l e t x 6 E. Then (i) x i s semi—simple i f and only i f Cl(x) (the conjugacy class of x in E) is closed in E. ( i i ) If T is a maximal torus of G then the c e n t r a l i z e r of T in E is equal to the closure of T in E. Proof: Let X be the closure of T in E. From 4.4.3 we have X/W is canonically isomorphic with E ( c l ) . If x e E is semi—simple then by 3.4.11 Cl(x) i s closed in E. Conversely, i f Cl(x) i s closed in E then by the remark above Cl(x) i s the only closed conjugacy class in c l ~ 1 ( c l ( x ) ) . But from our i d e n t i f i c a t i o n of X/W with E(cl) we obtain that X intersects every closed conjugacy c l a s s . This proves ( i ) . If x e C(T) then Cl(x) is closed by 1.2.3 ( i i ) . Thus by (i) above x i s semi—simple. Hence, x i s in the closure of T by 3.4.6. 4.4.5 Proposition: l e t E be reductive and l e t U(E) = { x e E | dim Cl(x) = dim G - rk G }. Then U(E) is a non-empty open subset of E. 51 Proof: By 1.2.1 (iv) U'(E) = { x 6 E | dimCl(x) > dimG - rkG } is open in E. Since E is i r r e d u c i b l e , and U(G) = { x e G | dimCl(x) = dimG - rkG } is non-empty and open (see [29]), i t follows that U(E) = U'(E). 4.4.6 Proposition: { x € U(E) | x is semi—simple } is open in E. Proof: This follows from 1.2.4 ( i i i ) and 4.4.4 ( i ) . Remark: By 4.4.5 we have dim Cl(x) < dim G - rk G for a l l x e E. 4.4.7 Lemma: Suppose G i s an algebraic group and p : G > G1(V) i s a rat i o n a l representation such that V i s completely reducible. Then the unipotent r a d i c a l of G i s contained in the kernel of />. Proof: Without loss of generality assume V is a simple G—module. Let UR(G) be the unipotent r a d i c a l of G and l e t W be the invariants of UR(G) in V. Since UR(G) is unipotent, W i s non—zero and since UR(G) i s normal, W i s a G—submodule of V. Thus W = V. 4.4.8 Theorem: Suppose E is an irreducible algebraic monoid. Then there exists an irreducible reductive algebraic monoid E' and a morphism p : E > E' such that (i) p is dominant ( i i ) kernel(p) = UR(G), the unipotent r a d i c a l of G. ( i i i ) If T i s a maximal torus of G(E) such that p(T) = T' then p : X > X' i s an isomorphism, where X and X' are the closures of T and T' respectively. Proof: There exists a representation c : E > End(V) such that c i s a closed imbedding. Let F be a composition series of the 52 E—module V. Thus, F i s a l i n e a r l y ordered c o l l e c t i o n of subspaces { V(i) } of V such that V(i+1)/V(i) is a simple E-module for a l l i . Let End(V,F) = { f e End(V) | f ( V ( i ) ) i s contained in V(i) for a l l i }. Thus, by d e f i n i t i o n of F, o factors through the inclusion End(V,F) > End(V). There i s a canonical morphism q : End(V,F) > End(Gr(V)), where Gr(v) is the graded object associated with the f i l t r a t i o n F of V. By 4.4.7 UR(G) i s in the kernel of q o a . Thus UR(G) = ker ( q o a ) since k e r ( q o a ) is unipotent. Since q is a morphism of algebras with nilpotent kernel, q r e s t r i c t s to an isomorphism on the le v e l of maximal D—submonoids. Thus, i t follows that the closure of q o c(E) in End(Gr(V),F) s a t i s f i e s conclusions ( i ) - ( i i i ) of the theorem. Remark: Theorem 4.4.8. is useful in the discussion of prime ideals in Chapter 5 and in the proof that reductive monoids are regular. The following application is inspired by close analogy with a well—known result from c l a s s i c a l ring theory. Let R be a f i n i t e dimensional associative algebra over an a l g e b r a i c a l l y closed f i e l d k. If R i s von Neumann regular then R i s a semi—simple ring. 4.4.9 Corollary: Suppose E i s a regular algebraic monoid with 0. Then E is reductive. Proof: If E i s not reductive, l e t />: E > E' be as in 4.4.8. Then by 1.1.1 dim p _ 1(0') > 0. So l e t x fc p- 1(0') be non-zero. If E i s regular then there exists a c E such that xax = x. But then xa = (xa) 2 i s non—zero. So p" 1(0') contains non—zero idempotents. This is impossible by 4.4.8 ( i i i ) since by 3.4.10 53 every idempotent is in the closure of a maximal torus. This contradiction implies that E is not regular. The remainder of thi s section is devoted to the proof that a l l reductive irreducible monoids are regular. This result has been proved by M. Putcha in c h a r a c t e r i s t i c zero using Weyl's theorem on the complete r e d u c i b i l i t y of rational representations. The main ideas of Putcha's proof have survived in my treatment, even though Weyl's theorem i s not true in general. It i s curious that Haboush's theorem (1.2.4 (i ) ) is not required in the proof. The proof requires the following result of M. Putcha. 4.4.10 Proposition[23;Theorem 1.4]: Let E be an irreducible algebraic monoid with group of units G. Let e e 1(E) and l e t E(e) = { x e E | xe = ex = e ]°. Then GE(e)G { a e E | e € EaE }. 4.4.11 Lemma: Let E be a reductive monoid and l e t E(e) be as above. Then E(e) is reductive. Proof: CG(e) = { g e G | ge = eg } is reductive since the conjugacy class of e i s closed. By 3.4.9 CG(e) > G(e), g —> eg, is a morphism of algebraic groups. Thus G(E(e)) i s reductive since i t i s the identity component of the kernel of thi s map. 4.4.12 Lemma: Suppose E i s a regular irreducible algebraic monoid with 0. Let N = { x € E | x i s nilpotent }. Then N is a closed subset of codimension larger than or equal to two. Proof: Clearly, N is closed. Since E i s regular i t has no ideals consisting e n t i r e l y of nilpotent elements. But every closed irreducible subset of E—G of codimension one in E i s a maximal irreducible component of E—G. Furthermore, each maximal 54 irreducible component of E—G i s an ideal because E is i r r e d u c i b l e . 4.4.13 Lemma: Suppose p : E > E' i s a f i n i t e dominant morphism of irreducible monoids with 0. If E' has no non-zero nilpotent ideals then E has no non—zero nilpotent ideals. Proof: If V is a nilpotent ideal of E then X, the closure of p(V) in E', i s a nilpotent ideal (since p i s dominant),. Thus, by assumption, V i s contained in p _ 1 ( 0 ) , which i s f i n i t e . It follows that V = { 0 }. 4.4.14 Theorem: Suppose that E i s an irreducible reductive algebraic monoid. Then E is regular. Proof: We may assume that E is a normal variety since the image of a regular monoid is regular. Assume also, for the moment, that E has a zero, and inductively that a l l reductive monoids of dimension less than dim(E) are regular. Now, as in the proof of 4.4.8, there exists a morphism p : E > E" such that p is generically finite—to—one and dominant, and E" has a f a i t h f u l completely reducible representation. Further, i f X and X" are the closures of respective maximal t o r i , then p : X > X" i s an isomorphism. Let E' be the monoid associated with the integral closure of k[E"] in k[E] (see 3.2.3). Thus we have a : E > E' b i r a t i o n a l and B : E' > E" f i n i t e and dominant with Boa = p. Let f : E" > End(V) be a f a i t h f u l completely reducible representation. Assume that there exists a nilpotent ideal N of E" and l e t t > 0 be i t s index of nilpotency. Let W be the subspace of'V spanned by NV. Clearly the index of nilpotency of N r e s t r i c t e d to W is t - 1. Thus W is a proper subspace of V. 5 5 Further, W i s E"—invariant since EN i s contained in N. Since V is completely reducible, there exists a subspace U of V such that V i s the direct sum of U and W, and U is E"—stable. But by d e f i n i t i o n of W, NU i s contained in W. Thus NU = { 0 } since U and W are complementary. Thus N has index of nilpotency t - 1. This contradiction proves that E" has no non—zero nilpotent ideals. It follows e a s i l y that E" can have no ideals consisting e n t i r e l y of nilpotent elements. Let z t E" be an ar b i t r a r y element. Then E"zE" contains non—nilpotent elements. Thus there exists a non—zero idempotent e € E"zE". By 4.4.10 there exists g,h e G such that gzh e E"(e). But E"(e) is reductive (by 4.4.11) and of dimension s t r i c t l y less than dim E". Hence inductively E"(e) is regular, so there exists u,v e G(E(e)) such that ugzhv = f = f 2 . Thus E" is regular. Now p : E' > E" i s f i n i t e and dominant so by 4.4.13 E' has no non—zero nilpotent ideals. Thus E' i s regular as well by the same arguement. Consider the morphism c : E ——> E'. Recall that a i s b i r a t i o n a l and induces an isomorphism a : X > X' on maximal irreducible D—submonoids. Clearly, i f N and N' are the respective sets of nilpotent elements of E and E', then o" 1(N') = N. Thus, o : E — N > E' - N'. Let e e E - N be an idempotent. Then e is in the closure of C(e)° = C(e). Now o r e s t r i c t s to a morphism a : CG(e) —r—> CG ' ( a(e)). Restri c t i n g a to the closure of respective maximal t o r i y i e l d s an isomorphism. Thus, o also induces an isomorphism on Weyl groups (of CG(e) and CG'(c(e))). It follows that a : CG(e) > CG'(o(e)) i s b i j e c t i v e and hence, a : Cl(e) > Cl(c(e)) is b i j e c t i v e as well (since Cl(e) = G/CG(e)). Thus, a is one-to-one when 56 r e s t r i c t e d to idempotents, since by 4.4.2, a preserves the conjugacy classes. Now suppose that aix) = o(e) for some x e E—N. Thus, c(e(x)) = e(c(x)) = o(e). So, e(x) = e, since a is one—to—one when r e s t r i c t e d to idempotents. By 4.4.10, and the induction hypothesis, x e E is a regular element. So, there exists g,h e G(E) such that gxh f is an idempotent. But then, c(g)o(e)a(h) = a(g)a(x)a(h) = ait). Hence, by [21] c(e) and o(f) are conjugate. Thus, e and f are conjugate, since a preserves conjugacy. Thus, we may assume that gxh = e = e(x). It follows that x c G(e) because in any representation of E, rank(x) = rank(e ) . By the proof of 4.4.11., eEe is reductive. Thus we have a : eEe >.o(e)E'c(e) such that (i) a is dominant. ( i i ) a is one—to—one when r e s t r i c t e d to the closure of a maximal torus. ( i i i ) eEe i s reductive. It follows from the induction hypothesis that a|eEe is finite—to—one. Thus, since a" 1(o(e)) is contained in G(e), a _ 1 ( o ( e ) ) is f i n i t e . Since every element of E'—N' is a unit times an idempotent, i t follows that a : E—N > E'—N' i s finite—to—one. Hence, a : E — N —> E' — N' is onto and finite—to—one because E' is regular. By 1.1.2, a induces an isomorphism of E - N onto E' - N'. Let U' = E' - N'. Identifying U' with E — N via a we have a morphism U' — > E. Thus by 1.1.5 there i s a unique morphism f : E' > E extending U' > E ( r e c a l l that the codimension of N' in E' is larger than one). 57 Thus o i s an isomorphism because oof = 1. So E i s regular. Now assume that E i s reductive but does not necessarily have a zero. Let e e 1(E) be a minimal idempotent and l e t x e E. Then without loss of generality xe = ex = ke for some k € G(E). But then by 4.4.10 there exists g, h e G(E) such that gxh e E(e). By d e f i n i t i o n , e e E(e) i s the zero of E(e). Further, by 4.4.11 E(e) i s reductive. Thus, by the above arguement, E(e) i s regular. Hence, there exist u,v e G(E(e)) such that ugxhv i s an idempotent of E(e). But then E i s regular. We thus have a s i g n i f i c a n t generalization of a fundamental theorem of modern algebra. 4.4.15 Theorem: Let E be an irreducible algebraic monoid with zero. Then the following are equivalent. (i) E is regular. ( i i ) E is reductive. ( i i i ) E has no n o n — t r i v i a l nilpotent ideals. Proof: 4.4.9 and 4.4.14. 4.5 Connected Monoids With Zero  D e f i n i t i o n : Let E be an algebraic monoid with 0. Then E i s connected i f E i s connected in the Zariski topology (equivalently, i f k[E] has no n o n — t r i v i a l idempotents, since any n o n - t r i v i a l idempotent yi e l d s a di r e c t product decomposition). Let E be a connected monoid with 0 and l e t E° be the irreducible component of 1. Let T be a maximal torus of G° and le t e e X be a minimal idempotent of X, the closure of T in E°. e i s not the identity element of E since t h i s would imply that G° i s closed in E, thereby contradicting the connectedness of E. Let e(l) = e, and E(1) = eEe. Then E(1) i s an algebraic monoid 5 8 w i t h 0 a n d i d e n t i t y e l e m e n t e . I f e ' i s a n o t h e r m i n i m a l i d e m p o t e n t t h e n E ' ( 1 ) i s i s o m o r p h i c w i t h E ( 1 ) s i n c e , i n a n y i r r e d u c i b l e m o n o i d a l l m i n i m a l i d e m p o t e n t s a r e c o n j u g a t e . A s s u m i n g e ( l ) i s n o n — z e r o t h e p r o c e d u r e c a n be a p p l i e d t o E ( l ) . T h u s , we o b t a i n a s e q u e n c e o f i d e m p o t e n t s 1 = e ( 0 ) > e ( 1 ) > . . . > e ( k ) = 0 a n d m o n o i d s E ( l ) = e ( l ) E ( l - 1 ) e ( l ) , 1 = 1 , . . . , k , s u c h t h a t , f o r a l l 1 , e ( l ) i s a m i n i m a l i d e m p o t e n t o f E ( l — 1 ) ° . I f a n o t h e r s u c h s e q u e n c e 1 = f ( 0 ) > f ( 1 ) > . . . > f ( m ) = 0 i s c h o s e n , t h e n k = m a n d E ( l ) i s i s o m o r p h i c w i t h E ' ( l ) f o r a l l 1 . T h e r e i s a c o n v e r s e t o t h i s r e s u l t . F o r t h i s we n e e d a l e m m a . 4 . 5 . 1 Lemma: S u p p o s e k* a c t s on t h e a f f i n e v a r i e t y X i n s u c h a way t h a t t h e a c t i o n e x t e n d s t o k . S u p p o s e t h a t F ( X , k ) , t h e f i x e d p o i n t s e t o f t h i s a c t i o n , i s c o n n e c t e d . T h e n X i s c o n n e c t e d . P r o o f : The a c t i o n o f k on X i n d u c e s a d i r e c t sum d e c o m p o s i t i o n k [ E ] = I k [ E ] ( o ) w h e r e a r a n g e s o v e r N = X ( k ) . T h u s , t h e c o m p o s i t e , k [ E ] ( 0 ) > k[X] > k [ F ( X , k ) ] i s a n i s o m o r p h i s m ( F ( X , k ) i s t h e f i x e d p o i n t s e t o f t h e a c t i o n ) . Now I , t h e s e t o f i d e m p o t e n t s o f k [ X ] , i s f i n i t e . So I i s c o n t a i n e d i n k [ X ] ( 0 ) . T h u s , t h e c o - o r d i n a t e r i n g o f X h a s no n o n — t r i v i a l i d e m p o t e n t s a n d X i s t h u s c o n n e c t e d . 4 . 5 . 2 T h e o r e m : L e t E be a n a l g e b r a i c m o n o i d w i t h z e r o . T h e n t h e f o l l o w i n g a r e e q u i v a l e n t . ( i ) E i s c o n n e c t e d i n t h e Z a r i s k i t o p o l o g y . ( i i ) T h e r e i s a c h a i n o f i d e m p o t e n t s 1 = e ( 0 ) > e ( l ) > . . . > e ( k ) = 0 s u c h t h a t e ( i + 1 ) e e ( i ) E e ( i ) ° f o r i = 0 , . . . , k - 1 . 59 ( i i i ) For a l l non-zero idempotents e € 1(E), G(eEe),the group of units of eEe, i s not closed in eEe. Proof; (i) => ( i i ) . Already given, ( i i ) => ( i ) . Inductively we may assume that e ( l)Ee(1) i s connected. Since e ( l ) t E° there exists a 1—p.s.g. p : k* > G° such that p extends to p : k > E° with p(0) = e ( l ) . Thus, e ( l ) E is connected since the fixed point set of the action f(t,x) = xpit) on e(1)E is e(1)Ee ( l ) (so 4.5.1 app l i e s ) . But then E is connected (again by 4.5.1) since e ( l ) E is the fixed point set of the action g(t,x) = />(t)x on E. (i) => ( i i i ) . This follows from the fact i f E is connected then eEe is connected. ( i i i ) => ( i i ) . If G is not closed in E then by 2.2.7 there exists a n o n - t r i v i a l idempotent e e E°. Thus inductively we can construct a chain of idempotents a s i n ( i i ) . 60 V IDEALS The purpose of t h i s c h a p t e r i s t o r e c o r d some of the g e n e r a l p r o p e r t i e s of i d e a l s . The main r e s u l t i s a s t r u c t u r e theorem f o r prime i d e a l s ( 5 . 2 . 1 ) . I have assumed throughout t h a t E i s an i r r e d u c i b l e a l g e b r a i c monoid. An i d e a l of E i s a subset J , of E such t h a t EJE i s c o n t a i n e d i n J . 5.1 P r e l i m i n a r y R e s u l t s 5.1.1 P r o p o s i t i o n : L et E be s o l v a b l e and l e t I be an i d e a l of E. Then the f o l l o w i n g are e q u i v a l e n t . ( i ) I f some power of x i s i n I then x i s i n I (I i s r a d i c a l ). ( i i ) I = />" 1(/o(I)) where p : E > X i s the u n i v e r s a l morphism t o a D—monoid ( 4 . 3 . 1 ) . P r o o f : ( i i ) => ( i ) . Any i d e a l i n a D—monoid i s r a d i c a l as i s any p u l l b a c k of a r a d i c a l i d e a l . ( i ) => ( i i ) . p i s onto, so p(I) i s an i d e a l of X. Thus p(I) i s the union of a f i n i t e number of o r b i t s of idempotents (under the a c t i o n of r i g h t t r a n s l a t i o n ) . L e t eG(X) be an o r b i t of p{I) . Now />" 1(eG(X)) = { x e E | some power of x i s i n G ( e ( x ) ) , />(e(x)) = e }. So p " 1 ( p ( l ) ) = { x 6 E | some power of x i s i n I }, s i n c e , i f x e I then e ( x ) t I . 5.1.2 C o r o l l a r y : L e t E be i r r e d u c i b l e and s o l v a b l e . Then t h e r e i s a c a n o n i c a l one—to—one c o r r e s p o n d e n c e between r a d i c a l i d e a l s of E and r a d i c a l i d e a l s of X, where p : 'E > X i s the u n i v e r s a l D—monoid a s s o c i a t e d w i t h E. 5.1.3 P r o p o s i t i o n : L et E be i r r e d u c i b l e and suppose t h a t I i s an i d e a l of E. L e t Z be the c l o s u r e i n E of some B o r e l subgroup B of G ( E ) . I f the i n t e r s e c t i o n 1(B) of I w i t h Z i s a r a d i c a l i d e a l then I i s c l o s e d i n E. 61 Proof: If 1(B) is rad i c a l i t is closed by 5.1.2. But I i s the union of a l l the conjugates of 1(B). Thus I is closed by 1.2.3 ( i ) . 5.1.4 Corollary: Suppose P is a prime ideal of E ( i . e . P i s an ideal such that E — P i s m u l t i p l i c a t i v e l y closed). Then P i s a closed subset of E. Proof: Any prime ideal i s r a d i c a l . 5.1.5 Corollary: Suppose P and Q are prime ideals of E such that P(s) = Q(s) (they contain the same semi—simple elements). Then P = Q. Proof: If P(s) = Q(s) then the same i s true of P(B) and Q(B) (the intersections of the closure of B with P and Q respectively), where B i s a Borel subgroup. Thus />(P(B)) p(Q(B)) where p is the universal morphism from the closure of B to a D-monoid. Thus, by 5.1.1 P(B) = Q(B). So P = Q. 5.1.6 Proposition: Let E be i r r e d u c i b l e , T a maximal torus of G(E), and W i t s Weyl group. Suppose I is a W—invariant prime ideal of Z, the closure of T in E. Then there exists a W—invariant character z e X(Z) such that I = z"'(0). Proof: Assume Z i s normal. Then Z — I i s a normal algebraic monoid variety. Let e e Z — I be the minimal idempotent. By Sumihiro's theorem [31;Corollary 2] there exists a T—invariant a f f i n e open subset U of Z — I with e e U. Thus U = Z — I since any open subset of Z — I with e e U intersects every other T-orbit. So Z - I is a f f i n e and Z - I > Z is an open imbedding. Consider k[Z] > k[Z - I ] . If A = { f € k[Z] | f ( l ) = 0 } then Ak[Z - I] = k[Z - I ] . Since A = ( x ( i ) ) , for some x(i) e X(Z), we have E a ( i ) x ( i ) •= 1 for some { a ( i ) } in 62 k[Z - I ] . Let a(i) = E b ( i , j ) y ( i , j ) , b ( i , j ) e k[Z - I ] , y ( i , j ) e X(Z - I ) . Then E b ( i , j ) x ( i ) y ( i , j ) = 1. After c o l l e c t i n g terms we have 1 = E n ( i , j ) x ( i ) y ( i , j ) where a l l the x ( i ) y ( i , j ) are d i s t i n c t . Thus x ( i ) y ( i , j ) = 1 for some i and j because characters are l i n e a r l y independent. Let x = x(i) and y = y ( i , j ) . Then x e k[Z - I] is a unit. Thus, k[Z - I] = k[Z.][l/x] and consequently A = r((x)) (r((x)) denotes the radi c a l of (x)). Let w e W. Then A = r(w*(x)) since A i s W—invariant. Thus A = r((z)) where z is the product of a l l w*(x) as w ranges over W. Further, z i s W—invariant and I = z ~ 1 ( 0 ) . Hence, i t remains to find z in case Z is not necessarily normal. Let n : Z' > Z be the normalization of Z. The Weyl group acts on Z' as well; thus, choose z e k[Z] as above, so that z _ 1 ( 0 ) = I' = n _ 1 ( l ) and z is W—invariant. By 4.5.1 some power, say y, of z is an element of X(Z). But z - ' ( 0 ) = v ' ( 0 ) in Z'. Hence the inverse image in Z' of v - 1 ( 0 ) i s equal to n ~ 1 ( I ) . Thus v ~ 1 ( 0 ) = I since n i s onto. 5.2 The Structure Of Prime Ideals  5.2.1 Theorem: Let E be an ir r e d u c i b l e algebraic monoid, T a maximal torus of G(E) , W the Weyl group of T. Then (i) If P i s a prime ideal of E there exists a character x e X(E) such that P = x" 1 ( 0 ) . ( i i ) There are canonical b i j e c t i o n s among the set of primes of E, the set of W—invariant primes of X (the closure of T in E) and the set of W—invariant idempotents of X. Proof: Assume E i s reductive. Let P be a prime ideal of E and le t X be the closure of a maximal torus T of G(E). Consider P(T), the intersection of P and X. P(T) is a W-invariant prime ideal of X. Thus, by 5.1.6 there exists a W-invariant character 63 x on X such that P(T) = x ~ 1 ( 0 ) . By 4.4.3, x l i f t s uniquely from kfX] to x e c l [ E ] . I f S i s another maximal torus then S = gTg~ 1 so i t follows that x is a character on E and that x~ 1(0) has the same semi-simple elements as P. Thus, by 5,1.5 P = x ~ 1 ( 0 ) . Now assume that E is not necessarily reductive. By 4.4.8 there exists a morphism p : E > E' such that E' is reductive, p : X > X' is an isomorphism, and p~ 1(1) •= UR(G), the unipotent r a d i c a l of G. Then p(P(T)) = p{P)(T), since i f x e p(P)(T) then there exists s e P semi—simple such that pis) = x. But then there exists u € UR(G) such that usu" 1 e P(T). Thus, /o(usu" ' ) = pis) = x. Hence, by the remark following 4.4.1, there exists v e c l [ E ] such that v~ 1(0) intersected with X is equal to P(T) (since c l [ E ' ] is contained in k[E]). Hence, by 5.1.5, P = v _ 1 ( 0 ) . This proves ( i ) . Proof of ( i i ) . If P is a prime ideal of E then P(T) is W-invariant. If P(T) = Q_(T) then P = Q by 5.1.5. Conversely, i f I is a W—invariant prime then by 5.1.6 there exists a W-invariant character v e X(T) such that I = v " 1 ( 0 ) . If E — > E 1 is the morphism of 4.4.8 to the reductive monoid E', then v can be l i f t e d to a class function on E' which i s a p r i o r i a class function on E. It follows that v i s actually a character on E' and thus, on E. Since v~ 1(0)(T) = I(T) we see that every W—invariant prime of X occurs in t h i s fashion. If I i s a W-invariant prime then e ( l ) , the minimal idempotent of X — I, i s a W—invariant idempotent. Conversely, i f e e X i s a W-invariant idempotent then i t follows from 4.1.2 ( i v ) , that 1(e), the union of a l l f x as f ranges over a l l maximal idempotents not s t r i c t l y larger than e, i s a W—invariant 64 p r i m e i d e a l o f X s u c h t h a t e e X — I i s t h e m i n i m a l i d e m p o t e n t . R e m a r k : T h e o r e m 5 . 2 . 1 d e m o n s t r a t e s an i m p o r t a n t m o t i f i n t h e t h e o r y o f a l g e b r a i c m o n o i d s . One h o p e s t h a t u l t i m a t e l y much o f t h e t h e o r y o f r e d u c t i v e m o n o i d s c a n be r e d u c e d t o p r o b l e m s c o n c e r n i n g D — m o n o i d s a n d t h e i r s y m m e t r i e s . 65 VI TWO-DIMENSIONAL REGULAR MONOIDS The c l a s s i f i c a t i o n and structure theory of semi—simple rank one monoids, according to the next chapter, requires a deeper understanding of lower—dimensional monoids. The purpose of this chapter is to expose the properties of two—dimensional monoids which are relevant to these developments. For completeness I have also included the case (case 2 below) which i s not needed in subsequent chapters. 6.1 Structural Properties  Def i n i t i o n : A monoid E is regular i f for a l l x e E there i s an idempotent e and a unit g such that gx = e. Note that any two—dimensional irreducible monoid is solvable. 6.1.1 Proposition: Let E be a two—dimensional ir r e d u c i b l e (non—trivial) algebraic monoid. Then the following are equivalent. (i) E i s regular. ( i i ) Either E i s a D—monoid or i t does not have a zero. ( i i i ) E is C l i f f o r d (see 4.2). Proof: Let p : E > X be the universal D—monoid associated with E (as discussed in 4.3). (i) => ( i i ) . P l a i n l y , a D—monoid i s regular. So assume p i s not an isomorphism. Since E i s n o n — t r i v i a l , X is not 0—dimensional. Thus dim X = 1. If 0 e E then 1(E) ={ 0, 1 } since I(X) = { 0, 1 }. Thus p'HO) = { x e E | x i s nilpotent }. Further, by 1.1.1, dim(/>"1(0)) > dim(p). Thus E cannot be regular since the set of nilpotent elements is a two—sided ideal of E. Hence, i f E is regular E cannot have a zero. 66 (i) => ( i i i ) . D—monoids are C l i f f o r d so again we may assume that E does not have a zero and that dim X = 1 . (a) Assume E is commutative. Then 1(E) = { 1, e } and e i s non—zero. Since by 4.2.1 es = se = e for a l l semi-simple elements s, we must have that eG(u) = G(u)e is one dimensional. Hence, the morphism G(u)xE(s) —-> E i s finite—to—one. The image is open and m u l t i p l i c a t i v e l y closed and contains a l l semi—simple elements. Thus, by 5.1.5, i t is onto. Hence, E is C l i f f o r d . (b) Assume E i s non—commutative. Then i f e € E is an idempotent dim Cl(e) = 1, since otherwise, Cl(e) = { e }. So, e would be an element of the closure of every 1—p.s.g. of G. This i s absurd since the union of the 1—p.s.g.'s is dense in G (this would force e to be the zero of E). Let V be a component of E. — G. Then V is a two—sided ideal of E since i t s codimension is one. Thus, V contains an idempotent by 2.2.5. Thus, Cl(e) is a subset of V. But the dimensions are the same so Cl(e) = V. Hence E — G = V is irreducible of dimension one and E — G = C l ( e ) . Clearly E is thus C l i f f o r d . ( i i i ) => ( i ) . Any C l i f f o r d monoid i s regular. Remark: From the proof of 6.1.1 we have the following r e s u l t . Suppose E i s non—commutative, dim E = 2 and e e E — G i s an idempotent. Then E is the union of G(E) and C l ( e ) . Case 1: E non—commutative, dim E = 2. Then either (a) eE = Cl(e) and Ee = { e } or (b) Ee = Cl(e) and eE = { e }. 67 Proof: If e, f c Cl(e), then ef = fe implies e = f by 4.3.3. Thus, eEe = { e }. Since e i s not the zero of E, either eE = Cl(e) or Ee = C l ( e ) . Thus the conclusion follows. For the remainder of case 1 I s h a l l assume that eE = Cl(e) and Ee = { e } (the other case is s i m i l a r ) . The example to keep in mind i s the set of two—by—two upper—triangular matrices (a(i , j ) ) such that a(1,1) = 1. Note that for a l l x, y e E, xey = ey. 6.1.3 Theorem: Let E be as above and l e t e e E be a n o n — t r i v i a l idempotent. Let p : k* —> G be a 1—p.s.g. such that e is in the closure of p(k*). Then the action (g,x) > g _ 1xg of k* on E extends to an action of k on E. Proof: g _ 1xg = xg. So the action c l e a r l y extends. Case 2: E i s commutative. As in the proof of 6.1.1 the morphism m : G(u)xE(s) > E, m(u,s) = us, i s f i n i t e and b i r a t i o n a l . The remainder of t h i s section i s pre—occupied with several results concerning low—dimensional monoids. They are a l l ingredients in the structure theory of the next chapter. 6.1.5 Proposition: Let E be a reductive monoid with 0, one—dimensional center and semi—simple rank one. Let e be an idempotent not equal to 0 or 1. Let R(e) = { g e G | eg = ege } and L(e) = { g e G | ge = ege }. Then R(e) and L(e) are opposite Borel subgroups. Proof: The intersection of R(e) and L(e) i s the c e n t r a l i z e r of e in G which i s a maximal torus. Thus R(e) and L(e) are 'opposite'. We prove R(e) i s Borel. There exists g t G unipotent, such that eg i s not equal to 68 e . T h i s f o l l o w s f r o m t h e f o r m u l a i m m e d i a t e l y p r e c e d i n g 6.1.3 a p p l i e d t o e i t h e r t h e c l o s u r e o f k * B ( u ) o r k * B ( u ) ~ , w h e r e B a n d B" a r e t h e B o r e l s u b g r o u p s c o n t a i n i n g t h e c e n t f a l i z e r o f e i n G , a n d k* i s t h e 1 — p . s . g . whose c l o s u r e c o n t a i n s e . S i m i l a r i l y , t h e r e e x i s t s m e G u n i p o t e n t s u c h t h a t me i s n o t e q u a l t o e . By 3 . 4 . 7 Ee i s c o n t a i n e d i n t h e c l o s u r e o f R ( e ) . B e c a u s e o f m a b o v e d i m Ee > 2 , a n d b e c a u s e o f g , d i m R ( e ) < 3 . T h u s d i m R ( e ) = 3 a n d d i m Ee = 2 i s t h e o n l y p o s s i b i l i t y . T h u s R ( e ) i s B o r e l . 6.1.6 C o r o l l a r y : L e t E be a s i n 6.1.5. T h e n d i m E e = d i m eE = 2 . 69 VII SEMI-SIMPLE RANK ONE, REDUCTIVE MONOIDS This chapter i s an exposition of the main computations of the thesis. These results include a c l a s s i f i c a t i o n of a l l normal reductive monoids with zero and one—dimensional center, in case the semi—simple rank is one. 7.1 is a discussion of the possible groups and the possible monoid types which a r i s e in thi s way. 7.2 is a record of some of the immediate c o r o l l a r i e s that result from the von Neumann regularity of the underlying monoid. In 7.3 a procedure is devised whereby f i n i t e morphisms between certain monoids can be replaced by morphisms with D—group kernels. 7.4 contains more technical preliminaries and a proof that normal, reductive monoids with zero and one—dimensional center are Cohen—Macaulay as algebraic v a r i e t i e s in case the semi—simple rank i s one. In 7.5 and 7.6 two c l a s s i f i c a t i o n theorems are established. The f i r s t makes use of certain bicartesian squares associated with the monoids in question and the second i s based on a computation of the characters of a maximal irreducible D—submonoid. 7.1 Rank Two, Semi-simple Rank One, Reductive Groups  7.1.1 Proposition: Suppose G is a non—abelian reductive group, rk G = 2, and rkss G = 1. Then G is isomorphic to one of G l ( k 2 ) , Sl(k 2)xk*, or PGl(k 2)xk*. Proof: Case 1. (G,G) = S l ( k 2 ) . Consider the morphism m : Sl(k 2)xk* > G, m(x,t) = xt (here k* i s the identity component of the center of G). If the kernel of m is n o n — t r i v i a l 70 (scheme th e o r e t i c a l l y ) i t follows that ker(m) = { ([a,a],a) e Sl(k 2 ) x k * | a 2 = 1 }, where [x,y] denotes the diagonal matrix with given entries. Hence, G = SI(k 2)xk*/ker(m) = G l ( k 2 ) . Case 2. (G,G) = PGl(k 2). In this case the kernel of m has to be t r i v i a l , since PGl(k 2) has no n o n — t r i v i a l f i n i t e normal D—subgroups. Thus G is isomorphic with PGl(k 2)xk*. 7.1.2 Proposition: Suppose E is an irreducible algebraic monoid such that G = G(E) i s as in 7.1.1 and l e t T be a maximal torus of G(E). Then there are three p o s s i b i l i t i e s for the closure X, of T in E. (i) I(X) = { 1 }, in which case G(E) = E. ( i i ) I(X) = { 1, e }, in which case there exists a morphism G'xk > E which i s f i n i t e and dominant (G' = (G,G)). ( i i i ) I(X) = { 1, e, f, 0 }, in which case 0 i s the zero of E as well. Proof: (i) follows from 2.2.7. ( i i ) Suppose I(X) = { 1, e }. Then i f w is an element of the normalizer of T, wew~1 = e. Hence, e i s contained in the closure of a W—invariant irreducible torus, S. But then S is ce n t r a l , because G i s reductive. Thus, e i s central and hence, e i s in the closure of every maximal torus. Therefore, I(E) = { 1, e }. Consider E > eE, x —> ex. Since e is not the zero, eg is not equal to e for some g e G'. Thus G' > eE is f i n i t e to one and dominant since G has no n o n — t r i v i a l normal subgroups of positive dimension. Hence m : G'xk > E, (x,y) > xy, is finite—to—one and dominant. Thus i t i s also onto because the complement of the image is an ideal with no semi—simple elements. Thus m is f i n i t e by 1.1.2. 71 ( i i i ) I(T) = { 1 , e, f, 0 }. I f w i s a n o n - t r i v i a l element of the normalizer of T, then the fixed idempotents of w are 0 and 1. Thus 0 i s a central idempotent of E and by the conjugacy of maximal t o r i , 0 i s the 0 of every maximal torus. Thus 0 i s the 0 of E since the semi—simple elements of G are dense in E. 7.2 Properties Of Semi-simple Rank One Monoids The purpose of this section is to record some of the geometric properties of semi—simple rank one monoids. Throughout I have assumed without further mention that E is i r r e d u c i b l e , 0 e E, dim Z(G(E)) = 1, and rkss G(E) = 1.-Let G = G(E). 7.2.1 Proposition: N, the set of nilpotent elements of E, is irreducible of dimension two. Proof: It follows e a s i l y , since E is regular, that the set of nilpotent elements of the closure of a Borel subgroup, is irreducible of dimension one. Thus, since Borel subgroups are a l l conjugate and of codimension one, N i s irreducible of dimension two. 7.2.2 Proposition: dim(E — G) = 3 and E — G is i r r e d u c i b l e . Proof: E — G = x" 1(0) for some character x : E > k by 5.2.1. So dim(E — G) 3 by K r u l l ' s p r i n c i p a l ideal theorem. Let z c E — G. Since E is regular, there exists g e G such that gz = e, a non—zero idempotent. Since a l l idempotents not equal to 0 or 1 are conjugate, GzG = (E - G) - { 0 }. Thus E - G i s equal to the closure of GzG, which is i r r e d u c i b l e . 7.2.3 Corollary: The action GxGxE > E, given by (g,h,x) > gxh" 1 has three o r b i t s , { 0 }, (E - G) - { 0 } and G. 7.2.4 Corollary: E — G i s the only n o n — t r i v i a l two—sided i d e a l . 7.2.5 Proposition: Let T be a maximal torus and l e t X be i t s 72 closure in E. Then E — { 0 } and X — { 0 } are smooth algebraic v a r i e t i e s , assuming E i s normal. Proof: Let Sing(E) be the singular locus of E. By Rrull's characterization of normality, codim(Sing(E)) > 2. Thus, by 7.2.4, Sing(E) i s contained in { 0 } because Sing(E) is a two—sided ideal of E. By 4.4.4 ( i i ) X - { 0 } is the fixed point set of the action of T on E - { 0 } by inner automorphisms. Thus, X — { 0 } is smooth since E — { 0 } is smooth and T is l i n e a r l y reductive. In the next section we s h a l l see that i f E is normal then both E and T are Cohen—Macaulay. 7.2.6 Construction: Big C e l l . Let B and B" be opposite Borel subgroups and l e t e 2 = e e X, the closure of the maximal torus associated with B and B". Now T(e) = X — fx is the unique open submonoid of X such that I(T(e)) = { 1, e } (where f i s the other n o n — t r i v i a l idempotent of X). Further, T(e) is a f f i n e and T(e)-T = ZG°e. Let Z and Z" be the closures of B and B" in E and l e t k* be the 1—p.s.g of T which converges to e. Notice that e i s in the closure of k*B(u) which is a two—dimensional regular monoid. Thus, the results of chapter 6 may be applied. Assume that Ze = Ee and eZ" = eE (as in 3.4.7 and 6.1.6). Consider the morphism of v a r i e t i e s m : B(u)xT(e)xB"(u) > E, m(x,y,z) = xyz. m is b i r a t i o n a l by the well known construction from group theory. To show that m is finite—to—one i t su f f i c e s 73 to show that m~ 1(e) i s a f i n i t e set since B(u)x(Te)xB~(u) i s an orbit under the action B(u)x(ZG)°xB-(u)x(B(u)x(T(e))xB"(u)) > (B (u)x(T(e ) )xB" (u)) (u,t,v)*(x,y,z) = (ux,ty,zv~ 1). Suppose that xyz = e, x e B(u), y = te, t e Z(G)° and z e B~(u). So, etz = x" 1e and thus, x" 1e commutes with e. By the remark following 6.1.1 (applied to the closure of B(u)k*), x"'e = e. S i m i l a r i l y , ez = e. Thus, te = e as well. But, { (x,t,z) t B(u)x(ZG)°xB"(u) | xe = te = ez =e } is f i n i t e since, by assumption, dim(B(u)e) = dim(ZG°e) = dim(eB~(u)) = 1. 7.2.7 Proposition: Assume E is normal. Then m : BxT(e)xB~ > E is an open imbedding. Proof: m i s finite—to—one and b i r a t i o n a l . Since E i s normal, m is an open imbedding by 1.1.2. 7.2.8 Corollary: Suppose E is normal and E' is another algebraic monoid. Let T be a maximal torus of G(E). Suppose we have morphisms p : G(E) > E' and o : T(e) > E' such that p \T = a|T. Then there exists a unique morphism p : E > E' such that fi|G(E) = p and fi|T(e) = c. Proof: Let U = B(u)T(e)B"(u) be as in 7.2.7. Define fi' : U > E*, £'(x,y,z) = p(x)o(y)p(z). Thus fi1 agrees with p on G(E) and with a on T(e). Thus there exists fi" : V > E' extending both p and p (where V is the union of U and G(E)). But the codimension of E — V is greater than or equal to two since V intersects E — G and E — G i s ir r e d u c i b l e . Thus, by 1.1.5 there exists a unique morphism p : E > E' extending fi". 74 7.3 Constructing Morphisms And Applications Let k be an a l g e b r a i c a l l y closed f i e l d of c h a r a c t e r i s t i c p > 0. If E i s an algebraic monoid defined over k, with group of units G, then there exist n o n — t r i v i a l purely inseparable morphisms p : E > E. The purpose of this section is to c l a s s i f y these morphism in case the group G is isomorphic to S l ( k 2 ) x k * . Since the results recorded here are elementary in nature, proofs w i l l often be omitted or sketched. 7.3.1 Proposition: Let B be the subgroup of upper—triangular matrices of S l ( k 2 ) and l e t p : B > B be a b i j e c t i v e morphism. Then there exists n e N and g e B such that qp(a(i,j))g"1 = ( F ( n ) ( a ( i , j ) ) ) for a l l ( a ( i , j ) ) e B, where F(n) : k > k is the Frobenius morphism composed with i t s e l f n times. 7.3.2 Proposition: Let p : S l ( k 2 ) > S l ( k 2 ) be a b i j e c t i v e algebraic group homomorphism. Then there exists g e S l ( k 2 ) and n c N such that qp((a(i,j)))g"1 = ( F ( n ) ( a ( i , j ) ) ) for a l l ( a ( i , j ) ) t S l ( k 2 ) . Proof: There exists g £ S l ( k 2 ) such that p' = qpq' 1 s a t i s f i e s p'(B) = B (B as above). Thus, by 7.3.1, p'|B = F(n)|B for some n £ N. Thus, by 1.2.2 ( i ) , />' = F(n). 7.3.3 Proposition: Let p : G(1) > G(2) be a morphism of algebraic groups, where G(1) and G(2) are each isomorphic to one of G l ( k 2 ) , S l ( k 2 ) or P G l ( k 2 ) . Let o(i) : Sl(k 2)xk* > G ( i ) , i = 1,2, be given by a(g,t) = U ( i ) ( g ) ) t , where e (i) : S l ( k 2 ) -> (G(i),G(i)) i s the universal covering map and k* > G(i) i s the identity component of the center of G ( i ) . Then there exists a unique morphism p' : S l ( k 2 ) > S l ( k 2 ) such that poa(.) -a(2)cV . 75 Proof: p l i f t s to />'. on the l e v e l of Borel subgroups. The morphism p' is as in Proposition 7.3.1. This morphism extends to a l l of S l ( k 2 ) x k * . Thus, by 1.2.2 ( i ) , pood) = o(2)o/>*. 7.3.4 Proposition: Suppose we have the following s o l i d arrow diagram in the category of algebraic monoids, where G ( i ) , i=1,2, i s the group of units of E ( i ) , i = 1,2 and E d ) * , i = 1,2, i s constructed in accordance with 3.2.3 applied to the f i n i t e morphisms a and fi. Assume that a l l horizontal morphisms are f i n i t e and dominant. Assume further, that E d ) i s normal. Then the dotted arrow can be f i l l e d in uniquely. E d ) > E(2) Sl(k G(1 ) -> G(2) a fi )xk* > Sl( )xk* v E( 1 )*-v -> E(2)* Proof: n*(k[E(2)]) is contained in the intersection (in k[Sl(k 2)xk*]) of k[Gd)] and k [ E ( l ) * ] . But k [ E ( 1 ) ] i s equal to the intersection of k[G(l)] and k [ E ( l ) * ] , since E(1) i s normal. Thus the arrow exists. 7.3.5 Proposition: Suppose p : E(1) > E(2) is a f i n i t e dominant morphism of normal algebraic monoids (G(i) = G(E(i)) as in 7.3.3). Then either (i) There exists a f i n i t e dominant morphism a : E(1) > E(2) such that kernel(o) is a f i n i t e D-group; or ( i i ) There exists a commutative diagram 76 E(3) -> E ( 1 ) v E(2) — v -> E' such that every morphism i s f i n i t e and dominant and every kernel i s a f i n i t e D—group. Proof: By 7.3.3 and 7.3.4 i t s u f f i c e s to prove th i s i f G(1) G(2) = S l ( k 2 ) . So we have p : S l ( k 2 ) x k * > Sl(k 2)xk*. Since p is b i j e c t i v e , we may assume, by 7.3.2, that p = (F(n),sF(m)), where F(n) i s as in 7.3.1 and (s,p) = 1. Case 1: n < m. Consider the diagram, (F(n),sF(m)) S l ( k 2 ) x k * -> Sl( k 2 ) x k * -> Sl( k 2 ) x k * (F(n),F(n) ) (1,sF(m-n)) V B V V E( 1 ) > E(1) > E(2) > B exists by 7.2.8 since on the l e v e l of characters F(n) : X —-> X, F(n) as in 7.3.1, is the desired extension. Thus, on the l e v e l of characters, we have X(1) <- X(1) <- X(2) X( 1 ) '<-<-X( 1 ) '< X(2) ' The dotted arrow exists because the diagram A is a pullback. Hence, again, by 7.2.8, we can f i l l in the dotted arrow (of case 1) to a morphism E(1) > E(2). 77 Case 2: n > m. Let E' = E(2)/K, where K = { x e G(2) | (F(n-m))(x) = 1 }. Then we have (F(n),sF(m)) (1,F(n-m)) E ( 1 ) > E ( 2 ) > E' f = (l,F(n-m)) is the desired morphism E(2) > E'. Composing (l,F(n—m)) and (F(n),sF(m)) we obtain g = (F(n),sF(n)). Noting that E' = E ( l ) / k e r ( g ) , we also obtain the following diagram, (F(n),sF(n)) > Sl(k 2)xk* > Sl(k 2)xk* > Sl(k 2)xk* ( 1 ,s) (F(n),F(n) ) v E(2) v > E' v -> E' e exists (by 7.2.8) just as in case 1 above, On the l e v e l of characters we have X ( 1 ) < X' < X' X( 1 )*< X' *<-< X' * where * denotes the characters of the closure of the torus in question. The dotted arrow exists because the image of X'* in X ( 1 ) i s f i n i t e over the image of X'* in X ( 1 ) * . Applying 7.2.8 again we obtain, in t h i s case E ( 1 ) > E' . From above, we also have a morphism f : E(2) > E'. Thus, taking the pullback of these two morphisms and r e s t r i c t i n g the resulting diagram to (normalized) identity components, we obtain 78 the diagram advertised in ( i i ) above. 7.4 Cohen-Macaulay Monoids Let E be a reductive algebraic monoid such that dim Z(G(E)) = 1 , rkss G(E) = 1 and 0 € E. 7.4.1 Lemma: There exists a representation p : E > End(V) such that (i) p i s an irreducible representation. ( i i ) p is a f i n i t e morphism. Proof: By the proof of 4.4.8 there exists an irreducible representation p : E > End(W) of E such that p{0) = 0 and no idempotent of E is sent to 0. Let V be an E—simple summand of W such that some idempotent e of E i s non—zero on V. By 7.2.3, p\V : E > End(V) has a t r i v i a l kernel. It follows from 3.4.13 that p\V i s a f i n i t e morphism. Let p : E > End(V) be f i n i t e and irreducible as in 7.4.1. Then we have the following commutative diagram: F(n) S l ( k 2 ) > End(k 2) m v v E > End(V) = End(m(K 2)) p where m denotes the m—th symmetric power and F(n) i s as in 7.3.1. This follows from the fact that every irreducible representation of (G(E),G(E)) is isogenous to a symmetric power of the canonical two—dimensional representation of S l ( k 2 ) . Clearly, (i) m : End(k 2) > End(m(k 2)) i s f i n i t e . ( i i ) p(Z(E)°) = Z(End(m(k 2)) = m(Z(End(k 2))). 79 Thus, fi(E) = m(End(k 2)). Hence, i f we let E(1) = image(m), then we obtain, m : End(k 2) > E(1) and p : E > E(1) such that (i) both m and p are f i n i t e morphisms. ( i i ) kernel(m) i s a f i n i t e D—group (X(kernel(m)) = Z/mZ). By 7.3.5 applied to p above, we have; 7.4.2 Proposition: Let E be a reductive, normal algebraic monoid with 0, such that dim ZG(E) = 1 and rkss G(E) = 1. Then there exists either (i) a morphism p : E > m(End(k 2)) such that p i s f i n i t e and dominant ('m' denotes m—th symmetric power) and kernel(/>) is a f i n i t e D—group, or ( i i ) morphisms e : E > E' and a : m(End(k 2)) > E' such that both B and a are f i n i t e and dominant and have f i n i t e D—group kernels. 7.4.3 Note: In case ( i i ) we may assume that E' is normal. Then there exists an isomorphism E' > ml(End(k 2) such that with th i s i d e n t i f i c a t i o n , com = ml (the ml—th symmetric power), where 1 = degree(o). Thus in either case (7.4.2 (i) or ( i i ) ) we have morphisms m : End(k 2) > m(End(k 2)) and p : E > m(End(k 2)) such that both m and p are f i n i t e and dominant and have f i n i t e D—group kernels. 7.4.4 Theorem: Let E be as in 7.4.2. Then E is Cohen—Macaulay. Proof: We have from 7.4.3, the following diagram, where R = End(k 2). 8 0 X -> R v E m v -> m(R) Here X is the normalization of the identity component of the pull—back of m and p. A l l morphisms have f i n i t e D—group kernels and R is a smooth variety. By 1 . 3 . 8 , X is Cohen—Macaulay and thus, by 1 . 3 . 9 , E is Cohen—Macaulay. 7 . 4 . 5 Theorem: Let E be as in 7 . 4 . 2 and l e t T be a maximal torus of G(E). Then the closure of T in E is a normal algebraic variety. Proof: Again from 7 . 4 . 3 we have o X > R v E m v -> m(R) where a l l morphisms are f i n i t e and dominant and R i s a regular variety. It s u f f i c e s to prove that i f T is a maximal torus of X then Z , i t s closure in X , is normal. This follows from the fact that B ( Z ) is isomorphic to Z / p ~ 1 ( 1 ) , so, by 1 . 3 . 2 ( i v ) , p ( Z ) is normal i f Z i s . Now let W be a maximal irreducible D—submonoid of R and l e t Z = a'1(W) (a p r i o r i non—reduced). Let n = degc = d i m k [ p ~ 1 ( 1 ) ] . Since R i s regular and X is Cohen—Macaulay, a i s a f l a t morphism. Thus, o | Z : Z > W i s f l a t of degree n. We have inclusions Z*(red) > Z(red) and j : Z(red) > Z , where Z(red) is the reduced variety associated with Z and Z*(red) = />~1(T) (T is the group of units of W) . Now Z*(red) is -a 81 commutative subgroup of G(X) consisting e n t i r e l y of semi—simple elements. Thus Z*(red) i s actually a maximal torus. Since p " 1 ( l ) is contained on Z*(red), p|T is f l a t of degree = h = dim k [ p " 1 ( l ) ] . Thus j is an isomorphism because otherwise deg p|T = deg p < deg a = n. P l a i n l y , Z is then equal to the closure of Z*. Hence, a|Z : Z > W is f l a t and thus Z is Cohen-Macaulay. But then Z i s normal because by 7.2.5 the singular locus has codimension larger than or equal to two (by 7.2.5). 7.5 C l a s s i f i c a t i o n I By 7.4.3 and the proof of 7.4.4 we have, for E normal, reductive with 0 e E, rk G(E) = 2 and rkss G(E) = 1, the following commutative diagram in the category of algebraic monoids; 7.5.1 Diagram f • E' > End(k 2) v v E > E" such that a l l morphisms are f i n i t e and dominant and a l l kernels are f i n i t e D—groups. We would l i k e to have as r i g i d a diagram as i s possible, so as to maximize i t s technical e f f i c i e n c y in further developments. To this end, we may assume that K, the Intersection of ker(f) and ker(g), is scheme t h e o r e t i c a l l y t r i v i a l because both f and g factor uniquely through E' > E'/K. Thus, the composite, ker(g) > E' > End(k 2), is a closed imbedding. Further, f(ker(g)) is contained in ker(a) since diagram 7.5.1 commutes. Letting H = f(ker(g)) we see that s factors through 82 -> E" since g is the universal morphism vanishing on End(k 2)/H • ker(g) . Thus, summing up, we may assume that, diagram 7.5.1 s a t i s f i e s the following properties. (i) Every kernel i s c e n t r a l . ( i i ) ker(f) and ker(g) have t r i v i a l scheme theoretic intersect ion. ( i i i ) f : ker(g) > ker(a) is an isomorphism. (iv) g : ker(f) > ker(fl) i s an isomorphism. (v) The diagram i s bicartesian. If Z' i s the closure of some maximal torus in E' and Z = g(Z'), Z* = f(Z') and Z" = a(Z*) = *(Z) then we have the following commutative diagram in the category of algebraic monoids. 7.5.2 Diagram: v Z -> z* v -> Z" such that every morphism is f i n i t e and dominant and the diagram is bicartesian on the group l e v e l ( i . e . It is both a pull—back and a push—out). G(E) = G l ( k 2 ) Re s t r i c t i n g the diagram 7.5.2 to the centers of each group we have the following commutative diagram in the category of algebraic groups. 83 7.5.3 Diagram: ZG' v k* -> k* v -> k* Further, 7.5.3 is bicartesian because a l l the kernels in diagram 7.5.1 are c e n t r a l . Since G' = G l ( k 2 ) or Sl(k 2)xk*, ZG' = k* or (Z/2Z)xk*. If G' = G l ( k 2 ) then 7.5.3 becomes, f k* > k* v k* v -> k* (9 Thus, degrees and degrees are both odd, since i f degreec is even then G(E") (in 7.5.1) is isomorphic to PGl(k 2)xk* and thus, degree? is even as well. But then the pull—back (k* at upper l e f t ) could not be irreducible (the number of irreducible components of the pull—back is equal to the greatest common div i s o r of deg(c) and deg(>s)). S i m i l a r i l y , degrees is odd. Furthermore, degreec and degrees are r e l a t i v e l y prime for the same reason. If G' = Sl(k 2)xk* then 7.5.3 becomes f (Z/2)xk* 9 v k* -> k* v -> k* Since f and g are the r e s t r i c t i o n s of morphisms Sl(k 2 ) x k * 84 Gl(k 2 ) with f i n i t e D—group kernels, f ( i , x ) = i+mx and g(i,x) = i+nx for some m and n (where i e Z/2 is the n o n — t r i v i a l element viewed as an element of k*. Here, I have written k* a d d i t i v e l y ) . Thus, (m,n) = 1 because by assumption the intersection of ker(f) and ker(g) is t r i v i a l . But m and n cannot both be odd because i+mi = i+ni = 0 for m and n odd (and by assumption, ker(f) and ker(g) have t r i v i a l i n t e r s e c t i o n ) . Thus, (m,n) = 1 and one of m and n is even. Conversely, i f (m,n) = 1 and one of m and n i s odd then i+mx = i+nx has no solution for x (because t h i s implies that x has order 2 and no element of order 2 s a t i s f i e s the equation). Let us summarize these results as follows: 7.5.4 Proposition: Let f ZG' g v k* -> k* v -> k* fi be the diagram of 7.5.3. (a) Then there are two p o s s i b i l i t i e s . (i) G' = G l ( k 2 ) Then ZG' •= k*, o(x) = nx, fi(x) = mx, (m,n) = 1 and mn i s odd. ( i i ) G' = Sl(k 2 ) x k * Then ZG' = (Z/2Z)xk*, a(x) = 2nx, p(x) = 2mx, (m,n) = 1 and mn is even. (b) Furthermore, a l l diagrams defined in (a) occur as the r e s t r i c t i o n of the appropriate diagram 7.5.1 to the centers of the various groups. 85 Proof: It remains to v e r i f y (b). Case ( i ) . Define a : G l ( k 2 ) > G l ( k 2 ) , c(x) [det(x)**m,det(x)**m]x and <s : G l ( k 2 ) > G l ( k 2 ) , *(x) = [det(x)**n,det(x)**n]x. On the l e v e l of the center, o(x) x**(2m+l) and s(x) = x**(2n+l). So choose m and n such that (2m+1,2n+l) = 1 (here '**' denotes exponentiation and [u,v] denotes the diagonal matrix with given e n t r i e s ) . Case ( i i ) . Define o : G l ( k 2 ) > PGl(k 2)xk*, o(x) = ([x],det(x)**m) and p : G l ( k 2 ) > PGl(k 2)xk*, p(x) = ([x],det(x)**n). Then on the l e v e l of the center .a(x) = x**2m and B ( X ) = x**2n. So choose m and n so that (m,n) = 1 and mn i s even. The procedure I have adopted in the c l a s s i f i c a t i o n i s to follow the diagram 7.5.2 from Z* to Z" to Z, keeping track of the induced map on the level of characters. 7.5.5 X(Z"). Z" as in 7.5.2; degree a odd. Notation; The diagonal two—by—two matrix ( a ( i , j ) ) , w i l l be written as [ a (1 , 1 ) , a ( 2 , 2 ) ] and the characters of a D—monoid w i l l always be written a d d i t i v e l y . Z* = { [a,b] | a,b e k } and ker(a) = { [x,x] | x**n = 0 } for some odd value of n ('**' denotes exponentiation). Thus, i f u : [a,b] > a, and v : [a,b] > b are the generators of X(Z*), we have a short exact sequence X(Z") > X(Z*) > Z/nZ a* j where j(u)=j(v) i s the generator of Z/nZ. Thus, by observation, X(Z") = ((n-1)u/2+(n+1)v/2,(n+1)u/2+(n-1)v/2) = (z,w). Since Z" is normal (see '7.4.5 and 4.1.5), X(Z") is equal to the intersection in X(T*) of X(Z*) and X(T"). Thus, i t follows that 8 6 X(Z") = { x € (z,w) | lx c <(n+1)z/2+(1-n)w/2,(n+1)w/2+(1-n)z/2> for some 1 }. (n+1)z/2+(1-n)w/2 and (n+1)w/2+(1-n)z/2 e X(Z") are c a l l e d the fundamental generators of X(Z") (see 4.1.7). 7.5.6 Summing up, we have a* : X(Z") > X(Z*) with c*(z) = (n-1)u/2+(n+1)v/2 a*(w) = (n+1)u/2+(n-1)v/2 The fundamental generators of X(Z") are (n+1)z/2+(1—n)w/2 and (n+1)w/2+(1-n)z/2. 7.5.7 X(Z"); Z" as in 7.5.2; degree o even. Z* = { [a,b] | a,b e k } and ker(o) = { [a,a] | a**k = 1 }. Here, k = 2n. From the proof of 7.5.4 (or d i r e c t l y ) , we have a : T* > T", o([a,b]) = (ab~ 1,(ab)**n). Hence, i f X(T") = (z,w), where z and w are the projections onto the f i r s t and second factors,'we have, a* : X(Z") > X(Z*) c*(z) = U—V a* (w ) = n (u+v) . In t h i s case the fundamental generators are w+nz and w—nz. Hence, X(Z") = { x e (z,w) | lx € <w+nz,w—nz> for some 1 } Note that we could compute a presentation of X(Z") d i r e c t l y from thi s. 7.5.8 Summing up, we have a* : X(Z") > X(Z*), with a*(z) = u—v o*(w) = n(u+v). The fundamental generators of X(Z") are w+nz and w—nz. 7.5.9 X(Z) ; degr.eeg odd. 87 From the proof of 7.5.4, B : Gl ( k 2 ) > Gl( k 2 ) is given by [a,b] > [a**((m+1)/2),b**((m-1)/2)] when r e s t r i c t e d to T, the set of diagonal matrices (here,'**' denotes exponentiation). Thus, i f T = { [a,b] | a,b e k* }, and u and v denote the characters u : [a,b] > a, v : [a,b] > b, then a*(z) = (m+1)u/2+(m-1)v/2 and B * ( W ) = (m-1)u/2+(m+1)/2. So i f X(T") = (z,w) and X(Z") i s as in 7.5.5 we have 7.5.10 B*((n+1)z/2+(1-n)w/2) = (m+n)u/2+(m-n)v/2 B*((n+1)w/2+(1-n)z/2) = (m+n)v/2+(m-n)u/2 Since B * is f i n i t e and Z is normal, we obtain X(Z) = { x e (u,v) | lx e <(m+n)u/2+(m-n)v/2,(m+n)v/2+(m-n)u/2> for some 1 }, and F = { (m+n)u/2+(m-n)v/2, (m+n)v/2+(m-n)u/2 } is the set of fundamental generators of X(Z). 7.5.11 X( Z) ; degree B even. From the proof of 7.5.4 B : G l ( k 2 ) > PGl(k 2)xk* is given by a([a,b]) = (ab" 1,(ab)**m) for some m, when r e s t r i c t e d to the diagonal group T of G l ( k 2 ) . Thus i f u([a,b]) = a and v([a,b]) b then we have B *(z) = u—v B*(w) = m(u+v). So i f X(T) = (z,w) and X(Z") i s as in 7.5.7 we have 7.5.12 «*(w+nz) = (m+n)u/2+(m-n)v/2 s*(w-nz) = (m+n)v/2+(m-n)u/2 Since B * is f i n i t e and Z is normal, we obtain X(Z) = { x € (u,v) | lx £ <(m+n)u+(m—n)v,(m+n)v+(m—n)u> for some 8 8 1 } and F = { (m+n)u+(m-n)v, (m+n)v+(m-n)u } i s the set of fundamental generators of X(Z) because (m+n,m—n) = 1 whenever (m,n) = 1 and mn is even. 7.5.13 Construction of X(Z), G(E) = G l ( k 2 ) . Summary. Given. E, there is a bicartesian diagram in the category of algebraic monoids. f E' > End(k 2) c V E v -> E" Let Z be the closure in E of some maximal torus T of G(E) and let X(T) = (u,v). Case ( i ) : degreeo = n is odd. Then degreep = m is odd and (n,m) = 1. Further, X(Z) = { x e (u,v) | lx e <(m+n)u/2+(m-n)v/2,(m-n)u/2+(m+n)/2> for some 1 } Case ( i i) : degreeo = 2n i s even. Then degreeo = 2m is even, (m,n) = 1 and mn i s even. Further, X(Z) = { x t (u,v) | lx e <(m+n)u+(m—n)v,(m+n)v+(m—n)u> some 1 } In both cases, w(u) = v w (v) = u for the n o n — t r i v i a l element w e W, the Weyl group of T. Thus, 0 1 1 0 re l a t i v e to the basis { u, v } of X(T). To construct a l l possible character monoids (4.1.1), that w = 8 9 o c c u r i n t h i s f a s h i o n , l e t a , B e Z a > \B\ (O,B)=1 I f a+B i s o d d t h e n X.(O,B) = { x e ( u , v ) | l x e < a u + £ v , av + au> some 1 }. T h e s e a r e t h e c h a r a c t e r m o n o i d s o f c a s e ( i ) w h e r e m = a+B a n d n = a—B. I f a+B i s e v e n t h e n X ( c , p ) = { x c ( u , v ) | l x e < c u + i 3 v , av+Bu> some 1 }. T h e s e a r e t h e c h a r a c t e r m o n o i d s o f c a s e ( i i ) , w h e r e m = (O+B)/2 a n d n = (a—B)/2. G ( E ) = S l ( k 2 ) x k * To c l a s s i f y t h e m o n o i d s w i t h , g r o u p S l ( k 2 ) x k * I h a v e u s e d t h e r e s u l t s c o n c e r n i n g G l ( k 2 ) a n d some g e n e r a l r e s u l t s a b o u t D — g r o u p a c t i o n s . L e t E be a s i n 7.5.1 a n d s u p p o s e G ( E ) = S l ( k 2 ) x k * . T h e r e i s a c a n o n i c a l m o r p h i s m , m : S l ( k 2 ) x k * > G l ( k 2 ) , m{x,B) = x [B , B ] . On t h e t o r i c l e v e l , m ( [ o , a ' 1 ] , B ) = [ o p , c ~ 1 p ] . m i s i s o m o r p h i c t o t h e q u o t i e n t m o r p h i s m o f S l ( k 2 ) x k * by t h e s u b g r o u p K = { {[a,a],a) | c 2 = 1 }. T h u s we h a v e m S l ( k 2 ) x k * > G l ( k 2 ) j v v E > E/K H e r e , k [ E / K ] = k [ E](0)- ( w h e r e 0 i s t h e t r i v i a l c h a r a c t e r on K ) , a s i n 1.3.1 a p p l i e d t o t h e a c t i o n K x E > E , ' ( x , y ) > x y . By 9 0 1.3.2 ( i v ) , E/K i s a normal algebraic monoid, and further, j : Gl(k 2) > G(E/K) is an isomorphism. Hence, E/K = E' is a monoid of the type just c l a s s i f i e d . It follows from the d e f i n i t i o n of m, that (on the le v e l of maximal t o r i ) m|T : T > T' induces m* : X(T') > X(T), m*(u) = a+b m*(v) = a—b where X(T) = (a,b) and X(T') = (u,v). So i f Z' i s the closure of T' in E', then X(Z') = { x e (u,v) | kx e <OU+BV,OV+BU> for some k } as summarized in 7.5.13. Thus, since m*(u) = a+b and m*(v) a—b, we have m*(au+0v) = (a+p)a+(a—p)b m*{cv+0u) = (a+p)a+(p—a)b. Case ( i ) : a+p i s odd. Then as in 7.5.13 ( i ) , (a+p,a—p) = 1 and hence F = { (a+p)a+(a—p)b, (a+p)a+(p—a)b } i s the set of fundamental generators. Thus, to construct the possible character monoids (4.1.1) that occur in t h i s fashion, l e t m,n > 0 , (m,n) = 1, mn odd (here, m=a+p and n=a—p). Then X(T(m,n)) = { x e (a,b) | kx e <ma+nb,ma—nb> for some k }. If w c W is the n o n — t r i v i a l element of the Weyl group of T, then w(a) = a and w(b) = —b, so 1 0 0 -1 r e l a t i v e to the basis { a, b } of X(T). w = 91 w = Case (i i ) : a+B i s even. Then as in 7.5.13 ( i ) , ((a+B)/2,(O-B)/2) = 1 and hence, F = { (a+B)a/2+(a-B)b/2, (a+B)a/2+(s-a)b/2 } i s the set of fundamental generators. Thus to construct the possible character monoids that occur in this fashion, l e t m,n > 0, (m,n) = 1, mn even (here m=(c+*)/2, n=(a - e)/2). Then X(Z(m,n)) = { x e (a,b) | kx c < ma+nb, ma-nb > for some k }. If w e W i s the n o n — t r i v i a l element of the Weyl group of T, then w(a) =- a and w(b) = —b, so 1 0 0 -1 re l a t i v e to the basis { a, b } of X(T). 7.5.14 X(Z); G(E) = S l ( k 2 ) x k * . Summary. Given E, there exists m : E > E' such that G(E') = Gl(k 2) and m : Sl(k 2 ) x k * > Gl(k 2) i s given by mix,B) = x [ B , B] . Let T be a maximal torus of G(E) and l e t Z be the closure in E. Using the morphism m, and our c l a s s i f i c a t i o n of the monoids E', we obtain: Z = Z(m,n) for some m,n > 0, (m,n) = 1, where X(Z(m,n)) = { x e (a,b) | kx e < ma+nb, ma-nb > for some k }. If w e Aut(Z) is the n o n — t r i v i a l element, then 1 0 0 -1 re l a t i v e to the basis { a, b } of X(T) = X(T(m,n)) G(E) = PGl(k 2)xk* w = 92 Let E be as in 7.5.1 and suppose G(E) = PGl(k 2)xk*. There is a canonical morphism, c : G l ( k 2 ) > PGl(k 2)xk*, c(x) = ([x],det(x)). If T' = { [a,b] | ab e k* } and T = c(T'), then the sequence u(2) > T' > T is exact, where u(2) = { [a,a] | a 2 = 1 }. Thus on the l e v e l of characters, we have c* : X(T) > X(T'), c*(x) = u+v c * (x ) = u—v where X(T). = (x,y) and X(T') = (u,v). By Theorem 3.2.3, the diagram Gl(k 2) > PGl(k 2)xk* > E can be completed uniquely to a diagram, c Gl(k 2) > PGl(k 2)xk* j v v E ' > E f such that f is f i n i t e and dominant, j : G l ( k 2 ) > G(E') i s an isomorphism, and E' is normal. Thus, we have c* : X(Z) > X(Z'), where Z' and Z are the closures of respective maximal t o r i . By the results of 7.5.13, X(Z') = { x e (u,v) | kx 6 <ou+i?v, av + 0U> for some k }. Thus, since c*(x) = u+v and c*(y) = u—v, we have c*(ax + 0y) = (a+p)u+(a-js)v c*(ay + i 3 x ) = ( a+B ) u+ ( B — a ) v. Note: The image of X(T) in X(T') is the subset of a l l elements ru+sv such that r+s is even. Case ( i ) : a+B is odd. 9 3 By the note, ou+pv and ov + pu are not elements of X(Z). By 4.1.7 the fundamental generators of X(Z) are multiples of ou+pv and ov + pu since c* : X(Z) > X(Z') is a f i n i t e morphism. Thus, again by the note, F = { 2(ou+pv), 2(ov+pu) } is the set of fundamental generators. Since a+p i s odd we may write (uniquely) a = (m+n)/2, p = (m-n)/2 where m,n > 0, (m,n) = 1 and mn i s odd. Thus, to construct the possible character monoids that occur in this fashion, let m,n > 0, (m,n) = 1, mn odd. Then X(T(m,n)) = { v e (x,y) | kv € <mx+ny,mx—ny> for some k }. If w is the .non—trivial element of W, the Weyl group of T, then w(x) = x and w(y) = —y. So 1 0 0 -1 r e l a t i v e to the basis { x, y } of X(T). Case ( i i ) ; a+p i s even. In t h i s case (by the note preceding case ( i ) ) we have ou+pv, ov+pu e X(Z). Since these are the fundamental generators for X(Z'), F = { au+pv, ov+pu } is the set of fundamental generators of X(Z). Since a+p i s even, we can write o=m+n and p=m—n, where m,n > 0 (m,n)=1 and mn is even. Thus, F(1) = (m+n)u+(m—n)v = mx+ny and F(2) = (m+n)v+(m—n)u = mx—ny •Thus to construct the possible character monoids that occur in t h i s fashion, l e t m,n > 0, (m,n) = 1, mn even (here o=m+n, p=m—n). Then w = 94 w = X(Z(m,n)) = { v e (a,b) | kv e < mx+ny, mx-ny > for some k }. If w i s the n o n - t r i v i a l element of W, the Weyl group of T, then w(x) = x and w(y) = —y. So 1 0 0 -1 rel a t i v e to the basis { x, y } of X(T). 7.5.15 X(Z); G(E) = PGl(k 2)xk*. Summary. Given E, there exists c : E' > E such that G(E') = Gl ( k 2 ) and c : Gl ( k 2 ) > PGl(k 2)xk* is given by c(x) = ([x] ,det(x)). Let T be a maximal torus of G(E) and let Z be the closure of T in E. Using the morphism c, and our c l a s s i f i c a t i o n of the monoids E', we obtain; Z = Z(m,n) for some m,n > 0, (m,n) = 1, where X(Z(m,n)) = { v e (x,y) | kv e < mx+ny, mx-ny > for some k }. If w e Aut(Z) is the n o n — t r i v i a l element, then 1 0 0 -1 rel a t i v e to the basis { x, y } of X(T) = X(T(m,n)). 7.5.16 Remark: A comparison of 7.5.14 and 7.5.15 demonstrates that the data c o l l e c t e d from the monoids with group Sl( k 2 ) x k * is ide n t i c a l to the data col l e c t e d from the monoids with group PGl(k 2)xk*. Thus our description i s only c h a r a c t e r i s t i c i f we know the unit groups. However, when we embellish this description by f i t t i n g in the root systems, the resulting numerical data w i l l completely d i s t i n g u i s h the monoids from one another. w = 9 5 The following theorem i s a summary of the results obtained from 7.5.1 to 7.5.15. 7.5.17 C l a s s i f i c a t i o n I; Le't G be one of the groups Sl(k 2)xk*, G l ( k 2 ) or PGl(k 2)xk* and let Q+ denote the set of posi t i v e r a t i o n a l numbers. Then there is a canonical one—to—one correspondence. Q* <—> E(G) = { E | E as in (*), E normal, G(E) = G }. For G = Gl{k 2) the correspondence is as follows. Given E there i s a unique bicartesian diagram, E(m"1) > End(k 2) a v v E > E(n) P such that a l l morphisms are f i n i t e and dominant and each kernel is a f i n i t e D—group. If degree a = n is odd then degree A = m i s odd and (m,n) = 1. If degree a = 2n is even then degree p = 2m is even, (m,n) = 1 and one of m and n is even. In any case, the map E(G) > Q+ given by E > deg(o)/deg(p) is well defined and one—to—one. Conversely, given r e Q+, r = m/n, where m,n > 0 and (m,n) =1. It i s then possible to construct a bicartesian diagram as above such that deg c = n and deg e = m i f mn is odd, or deg a = 2n and deg fi = 2m i f mn i s even. Thus we obtain the inverse map Q+ > E(G), r > E ( r ) . A l l normal monoids with group. SI ( k 2) xk* are constructed 96 from the monoids with group Gl( k 2 ) using integral closure and the morphism m : Sl(k 2)xk* > G l ( k 2 ) , m(x,t) = xt. A l l normal monoids with group PGl(k 2)xk* are constructed from the monoids with group Gl( k 2 ) using f i n i t e D—group scheme quotients and the morphism c : Gl(k 2) > PGl(k 2)xk*, c(x) = ([x],det(x)). 7.6 Polyhedral Root Systems And C l a s s i f i c a t i o n II In 7.5.16 we observed that the correspondence E > X(Z) is not a complete invariant unless G(E) = G l ( k 2 ) . The purpose of this section is to find the root system *, amidst the characters X(T), and to see how i t relates to the set of fundamental generators F of X(Z). This w i l l lead to a complete numerical invariant, E > (X(T),*(T),F), the polyhedral root system. 7.6.1 Lemma: Suppose f : G > G' i s an epimorphism of reductive algebraic groups such that ker(f) is contained in the center of G. Let T and T' be maximal t o r i of G and G', respectively, such that f(T) = T'. Let * and #' be the roots (weights of the adjoint representation). Then (f|T)*(#') = 4>. Proof; Let B + and B" be opposite Borel subgroups containing T, so that T i s the intersection of B + and B". Let g = T(G), the tangent space of G at 1. Then g = g++t+g~ (direct sum) where g + = T(B +(u)), g~ = T(B _(u)) and t = T(T) (the tangent spaces at the i d e n t i t y ) . Further, df : g > g' preserves these direct summands since f(B +(u)) = B' +(u) and f(B'(u)) = B'"(u). Now, df : g + > g' + and df : g~ > g'~ are isomorphisms since ker(f) is c e n t r a l . Thus, i t follows that f*(#') = *. 97 7.6.2 Polyhedral Root System for E; G(E) = G l ( k 2 ) . From 7.5.1 we have E' > End(k 2) a v v E > E" fi Since ker(o) and ker(fl) are cen t r a l , we apply 7.6.1 to follow the roots around the diagram from End(k 2) to E. Let Z* and Z" be the closures of maximal t o r i in End(k 2) and E", respectively, such that o(Z*) = Z". c induces o* : X(Z") > X(Z*). Case ( i ) : degreeo odd. From 7.5.5, we have o*(z) = (n-1)a/2+(n+1)a/2 c*(w) = (n+1)a/2+(n-1)b/2 (where X(Z") = (z,w) and X(Z*) = (a,b)). By 7.6.1, o*U") = ** = ( a-b, b-a }. Thus *" = { a-b, b-a } = { w—z , z—w } . If Z is the closure of the maximal torus T such that B ( T ) = T", we have fi : X(Z") > X(Z) = (u,v). From 7.5.9 we obtain, p*(z) = (m+1)u/2+(m-1)v/2 and fi*(w) = (m+1)v/2+(m-1)u/2. Thus fi*(z—w) = u—v and B*(W—z) = v—u. Hence, # - { u-v, v-u }, so, gathering a l l the relevant data, 9 8 X = (u, v) a, p e Z * = { u-v, v-u } o > | p | > 0 F = { cu+pv, OV+BU } (c,p) = 1 Here, o=(m+n)/2 and p=(m—n)/2. Case ( i i ) : degreeo even. From 7.5.7 we have a* : X(Z") > X(Z*) and thus, o*(z) = a-b c*(w) = n(a+b). Thus <*>" = { z, -z }. If Z is the closure of the maximal torus T such that p(T) = T", we have p* : X(Z") > X(Z) = (u,v). From 7.5.11 we obtain, p*(z) - u—v and p*(w) = m(u+v). Hence, $ = { u—v, v—u }. So in this case again we obtain, X = ( u , v ) a,B e Z * = { u-v, v-u } o >|B|> 0 F = { au+pv, cv+pu } (a,p) = 1 Here a=m+n and p=m—n. 7.6.3 Polyhedral Root System for E, G(E) = Sl(k 2 ) x k * . The morphism m : Sl(k 2 ) x k * > G l ( k 2 ) , m(x,p) = x[p,p], induces m* : X(T') > X(T), m*(u) = a+b, m*(v) = a-b, where T' and T are maximal t o r i of Gl(k 2 ) and S l ( k 2 ) , respectively, and X(Z") = (u,v), X(Z) = (a,b). Thus, since m*($') = *, we have * = { 2b, -2b }. ; So, gathering a l l the relevant data (see 7.5.14) we have, 9 9 X = (a, b) m, n € N # = { 2b, -2b } m,n > 0 F = { ma+nb, ma—nb } (m,n) = 1 7.6.4 Polyhedral Root System for E; G(E) = PGl(k 2)xk*. The morphism c : Gl(k 2) > PGl(k 2)xk*, x > ([x],det(x)), induces c* : X(T) > X(T'), c*(x) = u+v and c*(y) = u—v, where T' and T are maximal t o r i of Gl(k 2 ) and PGl(k 2)xk*, respectively, X(T') = (u,v), andX(T) = (x,y). Thus, since c*($) = we have * = .{ y > -y }• So, gathering a l l the relevant data (see 7.5.15) we have, X = (x, y) m,n e N * = { y, -y } m,n>0 F = { mx+ny, mx—ny } (m,n) = 1 Def i n i t i o n ; Let E be an irreducible algebraic monoid with 0 and l e t T be a maximal torus of G(E). Let X denote the characters of T, * the roots, and F the fundamental generators (see 4.1.7). Then (X,*,F) i s the polyhedral root system of (E,T). 7.6.5 C l a s s i f i c a t i o n I I : Let G be one of the groups G l ( k 2 ) , S l ( k 2 ) x k * or PGl(k 2)xk*. Then any normal algebraic monoid E with 0 and group of units G i s uniquely determined by i t s polyhedral root system (X(T),#(E),F(E)). The following i s a l i s t of a l l possible polyhedral root systems for each group G ((u,v) denotes the free abelian group on the generators u and v). 1 00 (i) G = Gl ( k 2 ) X = (u,v) a, s e Z * = { u-v, v-u } a > | fi | > 0 F = { au+fiv, av + fiu } =' 1 ( i i ) G = Sl( k 2 ) x k * X = (a,b) m, n e N * = { 2b, -2b } m,n > 0 F = { ma+nb, ma—nb } (m,n) = 1 ( i i i ) G = PGl(k 2)xk* X = (x,y) m,n e N * = { y r -y-} m,n>0 F = { mx+ny, mx—ny } (m,n) = 1 7.6.6 Remark: It is interesting to note that ( i ) , ( i i ) and ( i i i ) above exhaust a l l the reasonable p o s s i b i l i t i e s among two—dimensional normal D—monoids. Precisely, l e t X be a two—dimensional i r r e d u c i b l e normal D—monoid with 0. Then X i s isomorphic to the maximal irreducible D—submonoid of some E as in 7.6.5 i f and only i f X has a n o n — t r i v i a l automorphism. 101 VIII IRREDUCIBLE MONOID VARIETIES The purpose of this chapter i s to i n i t i a t e the study of more general monoid v a r i e t i e s . D e f i n i t i o n ; An algebraic monoid variety is an irreducible (not necessarily affine) algebraic variety E, defined over the a l g e b r a i c a l l y closed f i e l d k, such that ( i ) 1 e E ( i i ) m : ExE > E is an associative morphism of algebraic v a r i e t i e s with 1 as two—sided unit. On the two extremes we have the quasi—affine monoids and the projective monoids. This chapter is devoted to the proofs of the following r e s u l t s . (1) If E is irreducible and quasi—affine then E i s a f f i n e . (2) If E is irreducible and projective then E is an abelian va r i e t y . 8.1 Quasi-affine Monoids Let U be a quasi—affine variety defined over k and let k [u] denote the set of global sections of the structure sheaf O(U). U is not determined by k[U], For example, i f U = k 2 — { 0 }, then U is quasi—affine and k[U] = k[k 2] (see also 1.1.5). Let j : U > X be an open imbedding, where X is a f f i n e and l e t J = (f(1 ),...,f(n)) be the ideal of regular functions on X which vanish on X — j(U). The induced morphism s a t i s f i e s (i) k [ X ] [ l / f ( i ) ] > k'[U] [ 1/f (i) ] is an isomorphism for a l l i = 1,...,n. ( i i ) k[U] is the intersection of the k [ U ] [ l / f ( i ) ] as i varies from 1 to n. ( i i i ) U is isomorphic to the union of the X(i) in X, where 1 02 X(i) = { x t X | f ( i ) ( x ) is non-zero }. The a f f i n e variety X i s somewhat a r b i t r a r y . Any k—algebra R, contained in k [u] such that (i) R i s f i n i t e l y generated over k. ( i i ) { f ( i ) } i s contained in R. ( i i i ) R [ 1 / f ( i ) ] = k [ U ] [ l / f ( i ) ] , i = l,...,n. induces an isomorphism of U onto the union of the Y ( i ) , i = 1,...,n, where Y is the a f f i n e variety associated with R. 8.1.1: U is uniquely determined up to isomorphism by (k[U],{ f ( i ) }). Remarks: U i s a f f i n e i f and only i f { f ( i ) } generates the unit ideal of k[U] (see [1], chapter 3, exercise 24). Any f i n i t e subset { g(i) } of k [u] such that r ( ( f ( i ) ) ) = r ( ( g ( i ) ) ) (radical) works equally well. Suppose now that E is a quasi—affine irreducible algebraic monoid and l e t (k[E],{ f ( i ) }) be as above. The morphism m : ExE > E induces d : k[E] > k[E]»k[E] = k[ExE] in such a way that n : k > k[E] (k—algebra structure) e : k[E] > k (unit) d : k[E] > k[E]»k[E] (multiplication) induces on k[E] the structure of a bigebra. By a well—known result, 8.1.2: k[E] = c o l i m i t ( k [ E ] ( a ) ) , where each k[E](o) is a f i n i t e l y generated' sub-bigebra of k[E] (see [32] p.24. The proof there is stated for Hopf algebras, but works equally well for bigebras). 8.1.3 Proposition: Suppose E is an irreducible quasi—affine 1 03 algebraic monoid. Then there exists an irreducible a f f i n e algebraic monoid E' and a morphism g : E > E' such that g is an open imbedding. Proof: Let (k[E],{ f ( i ) }) be as . in 8.1.1. By 8.1.2 there exists a bigebra R, contained in k[E] such that { f ( i ) } i s a subset of R and R [ l / f ( i ) ] = k [ E ] [ l / f ( i ) ] for a l l i . But this is the same as being given an open imbedding g : E > E' of algebraic monoids where E' i s the af f i n e monoid associated with R. 8.1.4 Theorem: Suppose E is an irreducible quasi—affine algebraic monoid. Then E is a f f i n e . Proof: By 8.1.3 there exists g : E > E' an open imbedding, where E' is a f f i n e . Thus, g induces an isomorphism on unit groups. Hence, E' - g(E) is a prime ideal of E', because g(E) is m u l t i p l i c a t i v e l y closed. Thus by 5.2.1 (i) there exists a character x e k[E'] such that E' - E = x" 1 ( 0 ) . Thus, E = E' - x" 1(0) i s a f f i n e . 8.2 Projective Monoids Let E be an irreducible projective monoid variety, and l e t m : ExE > E be the mu l t i p l i c a t i o n morphism. Consider m~1(1) = { (x,y) c ExE | xy = 1 }. Suppose (x,y) c nr 1 (1 ) and l e t x : E > E, x(z) = xz. Thus xoy = 1. Hence yE is dense in E, because they have the same dimension. But yE = {z.« E | yxz -z}. So yE is closed in E and thus yE = E. But then there exists z e E such that yz = 1. So, x = x(yz) = (xy)z = z. Thus xy = 1 i f and only i f yx = 1. Therefore, the morphism g : nr 1 (1 ) > E, g(x,y) = x, i s one—to—one. Since the dimension of every component of nr 1(1) is larger than or equal to dim E (see 1.1.1), g is dominant. It follows that m"1(1) is irred u c i b l e . 1 04 Furthermore, g is b i j e c t i v e since g(m " 1 ( 0 ) is closed in E ( n r 1 ( l ) i s a complete v a r i e t y ) . Thus, 8.2.1: If E is projective then every element x e E i s invert i b l e . Remark: It does not follow automatically from 8.2.1 that E i s a group scheme. Even though i : E > E, i(x) = x" 1, is well defined as a set map, we have no a p r i o r i guarantee that i is a morphism of v a r i e t i e s . Now, i t follows easily that m : ExE > E i s separable and since E is homogeneous, m is smooth. Thus m"1(1) is irreducible and smooth. Define i : nr 1 ( 1 ) > m" 1(1), i(x,y) = (y,x) and u : n r 1 ( 1 ) x n r 1 ( 1 ) > m" 1(1), u((x,y),(u,v)) = (xu,vy). P l a i n l y , ( n r 1 (1 ) , u, i , (1 , 1 )) defines on n r 1('1) the structure of ah algebraic group, with m u l t i p l i c a t i o n u, inverse i , and unit (1,1). Furthermore, g : n r 1 (1 ) > E, g(x,y) = x, is a b i j e c t i v e morphism of algebraic monoids. Thus, to complete the discussion, i t s u f f i c e s to demonstrate that g i s separable. 8.2.2 Lemma: Let Z = n r 1 ( 1 ) and l e t g be as above. Then g i s separable. Proof: Let j : Z > ExE be the inclusion and l e t p(i) : ExE > E, i = 1,2, be the projection morphisms. Let m : ExE > E be the m u l t i p l i c a t i o n morphism. Using the projections p(1) and p(2), the tangent space of ExE at (1,1) is i d e n t i f i e d with the d i r e c t sum, TE+TE of the tangent space at 1 of E with i t s e l f . Further, i f TZ is the tangent space of Z at (1 , 1 ) , 1 05 TZ > TE+TE > TE d j dm is exact, because m is a smooth morphism. Using the bigebra structure of the l o c a l ring 0(1,E), i t follows that dm(x,y) = x + y. Thus dj(TZ) = { (x,-x) | x e TX }. Hence, dg : TZ > TE is an isomorphism since g = p ( l ) o j . Thus, by 1.1.3, g is separable. Def i n i t i o n : An abelian variety i s an irreducible projective algebraic group. 8.2.3 Theorem:1 Suppose E i s an i r r e d u c i b l e projective algebraic monoid. Then E = G(E) is an abelian variety. Proof: nr 1(1) i s a projective algebraic group and thus an abelian variety. g : nr 1 ( 1 ) > E, g(x,y) = x, i s a b i j e c t i v e separable morphism of smooth algebraic monoids and thus an isomorphism. Remark: Abelian v a r i e t i e s are much studied in algebraic geometry. It i s a remarkable fact that every projective algebraic group i s commutative. D. Mumford has obtained similar, more general results (see Abelian Varieties, Tata, Bombay, page 44); accordingly, our associativity assumption on E is superfluous. 106 IX APPLICATIONS TO RATIONAL HOMOTOPY THEORY There have been some recent applications involving algebraic monoids to problems in topology and algebra. Although such matters may be considered digressive from the themes of this thesis, they have been, at least for myself, ingressive to many of the problems in the theory of algebraic monoids. The purpose of t h i s chapter i s to describe, in general terms how algebraic monoids are related to several n o n — t r i v i a l problems in rat i o n a l homotopy theory. 9.1 Algebraic Categories And Positive Weights Spaces In [30] Sullivan establishes a complete and algebraic description of uniquely d i v i s i b l e homotopy invariants. He then observes that i f X is a simply—connected C.W.—space then Aut(X(0)) i s an algebraic group defined over Q, where X(0) i s the 0 — l o c a l i z a t i o n of X. Furthermore, there i s a d i f f e r e n t i a l graded algebra, M(X) such that End(X(0)) is isomorphic to End(M(X)) modulo d.g.a. homotopy. In [3] R. Body and D. Sullivan consider the following class of ' s u f f i c i e n t l y d i v i s i b l e ' simply—connected C.W.—spaces. Let Z(p) = { r e Q | r=m/n, (n,p)='1 }. X is s u f f i c i e n t l y p - d i v i s i b l e i f for any map f : X > Y such that f* : H*(Y;Z(p)) > H*(X;Z(p)) i s an isomorphism there exists a map g : Y > X such that g* : H*(X;Z(p)) > H*(Y;Z(p)) is an isomorphism. They es t a b l i s h the following fundamental results about t h i s class of spaces. 9.1.1 Theorem[3]: Let X be a simply—connected C.W.—space. Then the following are equivalent. 1 07 ( i ) X i s s u f f i c i e n t l y p — d i v i s i b l e f o r some p . ( i i ) X i s s u f f i c i e n t l y p — d i v i s i b l e f o r a l l p . ( i i i ) X i s s u f f i c i e n t l y 0 - d i v i s i b l e ( Z ( 0 ) = Q). ( i v ) 0 , t h e b a s e p o i n t m o r p h i s m o f M ( X ) , i s i n t h e Z a r i s k i c l o s u r e o f A u t ( M ( X ) ) i n E n d ( m ( X ) ) . ( v ) M ( X ) = M h a s p o s i t i v e w e i g h t s , i . e . T h e r e e x i s t s a d i r e c t sum d e c o m p o s i t i o n ( c o m p a t i b l e w i t h t h e u s u a l g r a d i n g on M ) , M IM(c) s u c h t h a t d(M(o)) i s c o n t a i n e d i n M(o) f o r a l l a, M ( O ) M ( B ) i s c o n t a i n e d i n M ( a + f i ) f o r a l l a a n d p a n d , a n d M(o) = 0 f o r a l l a < 0 . The i m p o r t a n t o b s e r v a t i o n h e r e i s t h a t we now h a v e a c o m p l e t e l y a l g e b r a i c d e f i n i t i o n o f ' s u f f i c i e n t l y d i v i s i b l e ' . I n [ 2 ] , [ 8 ] a n d [ 9 ] t h e f o l l o w i n g q u e s t i o n i s c o n s i d e r e d f o r X a s i m p l y — c o n n e c t e d C . W . — s p a c e ( w i t h some m i l d f i n i t e n e s s c o n d i t i o n s ) . D o e s X ( 0 ) s a t i s f y u n i q u e f a c t o r i z a t i o n i n t h e h o m o t o p y c a t e g o r y , w i t h r e s p e c t t o t h e f o r m a t i o n o f p r o d u c t s ? B e c a u s e o f S u l l i v a n ' s r a t i o n a l h o m o t o p y t h e o r y , t h i s i s now a q u e s t i o n o f p u r e a l g e b r a . D o e s M ( X ) s a t i s f y u n i q u e f a c t o r i z a t i o n w i t h r e s p e c t t o t h e f o r m a t i o n o f g r a d e d t e n s o r p r o d u c t ? I n [ 9 ] t h i s q u e s t i o n i s a n s w e r e d a f f i r m a t i v e l y i n c a s e X i s s u f f i c i e n t l y d i v i s i b l e i n t h e a b o v e s e n s e ( t h e d u a l q u e s t i o n r e g a r d i n g c o p r o d u c t s i s a l s o c o n s i d e r e d ) . S i n c e we a r e h e r e c o n c e r n e d w i t h how a l g e b r a i c m o n o i d s a r e i n v o l v e d , I w i l l g e n e r a l i z e a n d m o d i f y t h e c o n t e x t a c c o r d i n g l y . So l e t u s c o n s i d e r t h e f o l l o w i n g c a t e g o r i e s , c a l l e d a l g e b r a i c c a t e g o r i e s ( s e e [ 8 ] ) . 108 Let V(k) be the category of vector spaces over the f i e l d k and let S be a category whose objects are in one—to—one correspondence with the non—negative integers, N = { 0, 1, 2,.... }. Associated with S is the category n(S). If C i s a category l e t |C| denote the class of objects of C The objects of n(S) are pairs (V , c(V)) (or just (V,a)) where V e |V(k)| and c(V) i s a functor from S to V(k) such that c(0) = k, O(1) = V and aim) = V(m) for a l l m, where V(m) i s the the tensor product of V with i t s e l f m times. So, in p a r t i c u l a r , i f x e hom(m,n) and y e hom(n,p) then o(y ) o o(x) = o(yox). The morphisms of n(S) are the linear maps in V(k) which preserve the S—structure. Horn'(V,W) = { f c Hom(V,W) | f(n)oo(V)(x) = a(W)(x)of(m), for a l l x e hom(m,n), and a l l m,n e |S| }. Here f(n) denotes the tensor product of f with i t s e l f n times. Assume further, that 0 e S i s the zero object. For each m 6 |S|, there are unique morphisms n : 0 —> m and e : m — > 0. We s h a l l also, assume that the morphisms of fi(S) preserve t h i s structure. It follows that k i s the zero object of n(S). ' Remark: The d e f i n i t i o n above i s easiest to apply in practice, i f S can be realized as a subcategory of SETS. The purpose of the d e f i n i t i o n i s to provide a context for abstracting from the idiosyncracies of various algebraic categories in order to display the essence of how an object may be influenced by i t s algebraic monoid of endomorphisms. Let n(S) be an algebraic category and l e t (V,a) be an object, of n(S).- Assume that V s a t i s f i e s some s u f f i c i e n t l y s t r i c t finiteness conditions ( s t i l l very general in p r a c t i c e ) . Then 1 09 G = Aut(V,a) i s the a l g e b r a i c group of u n i t s of the a l g e b r a i c monoid E = End(V,a). P r o o f : f c End(V) i s i n End(V,a) i f and o n l y i f c ( x ) o f ( k ) = f ( l ) o o ( x ) f o r a l l k , l e |S| and a l l x t h o m ( k , l ) . Thus, by our f i n i t e n e s s a s s u m p t i o n , End(V,o) i s an a l g e b r a i c subset of End.(V) . C l e a r l y , Aut(V,o) i s the a s s o c i a t e d a l g e b r a i c group of u n i t s . D e f i n i t i o n : L e t (V,o) be an o b j e c t of n(S). Then (V,o) has p o s i t i v e w e i g h t s i f 0(V) i s an element of the c l o s u r e of Aut(V,a) i n End(V,a). Thus, c o n d i t i o n ( i v ) of 9 . 1 . 1 can be f o r m u l a t e d i n t h i s v e r y g e n e r a l s e t t i n g . 9.1.2 Theorem: Suppose (V,o) has p o s i t i v e w e i g h t s . Then V = • V ( i ) , i = 1,...,m, i n such a way t h a t each V ( i ) i s • - i r r e d u c i b l e i n n(S). Fu r t h e r m o r e , i f V = «W(j) j = 1,...,n, where each W(j) i s »-irreducible i n n(S), then m = n and t h e r e e x i s t s p : {l,...,m} > { l , . . . , n } , b i j e c t i v e , such t h a t V ( i ) and W ( p ( i ) ) a re i s o m o r p h i c . S k e t c h of p r o o f : I f V = »V(j) l e t e ( i ) : V > V be g i v e n by the composite of e» . . . • 1 • . . . •m : V( 1 ) • . ..•V(m) > k»...«V(i)•...»k and n » . . . . » n : k»...»V(i)•...»k > V(1)•...•V(m) = V where e : W > k and n : k > W are the unique morphisms t o and from the z e r o o b j e c t , k. Then e ( i ) 2 = e ( i ) , i = 1,...,m and e ( i ) o e ( j ) = e ( j ) o e ( i ) = 0(V) i f i i s not e q u a l t o j . Because V has p o s i t i v e w e i g h t s , we can c o n s t r u c t a maximal k — s p l i t t o r u s T of Aut'(V) such t h a t { e ( i ) | i = 1,...,m } i s 1 10 contained in the closure of T in End'(V). If { f ( j ) | j 1,...,n } i s another set of s p l i t t i n g idempotents, then we can assume that { f ( j ) } is in the closure of T as well because maximal k — s p l i t t o r i are a l l conjugate [5]. But then { e(i) } = { f ( j ) }, because by the results of [9], { e ( i ) o f ( j ) | j = 1,...,n } determines a •—product decomposition of V(i) for each i . Since V(i) is •— i r r e d u c i b l e by assumption, e ( i ) o f ( j ) = e(i ) for some j . S i m i l a r i l y , f(j)oe(k) = f ( j ) for some k. But then e(i) = e ( i ) o f ( j ) = e ( i ) o ( f ( j ) o e ( k ) ) = e(i) o e ( k ) . Thus, e(i)oe(k) i s non—zero. So, e(i) = e(k). Thus, i t follows that any two irreducible •—product decompositions are equivalent in the sense advertised. 9.1.3 Examples: Theorem 9.1.2 (applied to the . relevant •—product) applies to any of the following categories. (i) simply—connected minimal d i f f e r e n t i a l algebras and morphisms. ( i i ) simply—connected minimal d i f f e r e n t i a l graded coalgebras and morphisms. ( i i i ) connected minimal d i f f e r e n t i a l graded Lie algebras and morphisms. (iv) connected minimal Lie coalgebras and morphisms. (v) any of (i) — (iv) without the d i f f e r e n t i a l and minimality r e s t r i c t i o n s . Categories (i) — (iv) a l l give r i s e to the same homotopy theory i f the c h a r a c t e r i s t i c of the ground f i e l d i s 0 [16]. Each of the categories defined in (v) may be considered a subcategory of one of the categories defined in (i) — ( i v ) . Furthermore, a l l objects defined in (v) have positive weights. One question, 111 however, is l e f t open by Theorem 9.1.2. How does one generalize the result to the situation of objects without positive weights? No counterexamples are known. 9.2 Homotopy Types With Connected Endomorphism Monoid The main result of [27] is a structure theorem of Sullivan's minimal algebras based on a synthesis of algebraic f i b r a t i o n s and idempotents that adhere to Q—split t o r i . Assuming, for s i m p l i c i t y , that M is a f i n i t e l y generated minimal algebra defined over an a l g e b r a i c a l l y closed f i e l d k, of c h a r a c t e r i s t i c 0, the result i s as follows. 9.2.1 Theorem: There exists a sequence 1 = e(0) > e ( l ) > . . . > e(m) of idempotents in End(M) such that e ( i + l ) is an element of the closure of Aut(e(M)) in End(e(M)) and e ( i + l ) i s a minimal such idempotent. Futhermore, i f F ( i ) i s the quotient d.g.a. of e(i—1 )(M) by the ideal generated e ( i ) ( M ) + , then F ( i ) i s a minimal algebra with positive weights. The series terminates at e(m)(M) because Aut(e(m)(M)) i s closed in End(e(m)(M)). The integer m is uniquely determined and each e(i)(M) i s uniquely determined up to isomorphism. Theorem 9.2.1 f i t s in neatly with the characterization of connected monoids given in Theorem 4.5.2. 9.2.2 Corollary: Let M be as in 9.2.1. Then the following are equivalent. (i) End(M) i s connected in the Zariski topology. ( i i ) The sequence 1 > e ( l ) > ... > e(m) terminates with e(m) = 0(M). ( i i i ) For a l l non—zero idempotents e € End(M), Aut(e(M)) i s not closed in End(e(M)). 1 1 2 The same result may be applied to any of the categories n(S) considered in 9.1. It is not known, however, whether the unique f a c t o r i z a t i o n results for positive weight spaces can be extended to rational homotopy types with connected endomorphism monoid. 1 13 REFERENCES 1 . M. Atiyah and I. Macdonald, Introduction to Commutative  Algebra, Addison Wesley, Massachusetts( 1 9 6 9 ) 2 . R. Body and R. Douglas, "Unique Factorization of Rational Homotopy Types", P a c i f i c Journal of Mathematics 9 0 ( 1 9 8 0 ) , 2 1 - 2 6 . 3 . R. Body and D. Sullivan, "Zariski Dynamics of a Homotopy Type", unpublished n o t e s ( l 9 7 6 ) . 4 . A. Borel, Linear Algebraic Groups, W.A. Benjamin, New Y o r k ( l 9 6 9 ) . 5 . A. Borel and J. T i t s , "Theorems de Structure et de Conjugaison pour les Groupes Algebrique Lineaires", C.R. Acad. Sc. Paris Ser. A 2 8 7 ( 1 9 7 8 ) , 5 5 - 5 7 . 6 . N. Bourbaki, Commutative Algebra, Hermann, P a r i s ( l 9 7 2 ) . 7 . M. Demazure and P. Gabriel, Groupes Algebrique, North Holland, P a r i s d 9 7 0 ) . 8 . . R. Douglas, "Positive Weight Rational Homotopy Types", I l l o n o i s J. Math. (to. appear ) .. 9 . R. Douglas and L. Renner, "Uniqueness of Product and Coproduct ' Decompositions in Rational Homotopy Theory", Trans.A.M.S. 2 6 4 ( 1 9 8 1 ) , 1 6 5 - 1 8 0 . 1 0 . F. Grosshans, "Observable Groups and Hilbert's Fourteenth Problem", Am.J. Math. 9 5 ( 1 9 7 3 ) , 2 2 9 - 2 5 3 . 1 1 . W. Haboush, "Reductive Groups are Geometrically Reductive", Annals of Mathematics 1 0 2 ( 1 9 7 5 ) , 6 7 - 8 3 . 1 2 . M. Hochster, "Rings of Invariants of T o r i , Cohen-Macaulay Rings Generated by Monomials, and Polytopes", Annals of Mathematics 9 6 ( 1 9 7 2 ) , 3 1 8 - 3 3 7 . 1 3 . J. Humphreys, Linear Algebraic Groups, GTM 2 1 , Springer Verlag, B e r l i n ( 1 9 7 3 ) . 1 4 . G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal Imbeddings I, Lecture Notes in Mathematics, v . 3 3 9 , Springer-Verlag, Ber1 i n ( 1 9 7 3 ) . 1 5 . D. Mumford, Introduction to Algebraic Geometry, Harvard N o t e s ( l 9 7 6 ) . 1 6 . J. Neisendorfer, "Lie Algebras, Coalgebras, and Rational Homotopy Theory for Nilpotent Spaces", P a c i f i c Journal of Mathematics 7 4 ( 1 9 7 8 ) , 4 2 9 - 4 6 0 . 1 7 . P. Newstead, Introduction to Moduli Problems and Orbit 1 1 4 Spaces, TATA Institute of Fundamental Research, Bombay(Springer-Verlag)(1978) 18. M. Putcha, "On Linear Algebraic Semi—groups", Trans.A.M.S. 259(1980), 457-469. 19. , "On Linear Algebraic Semi-groups I I " , Trans.A.M.S. 259(1980), 471-491. 20. , "On Linear Algebraic Semi—groups I I I " , Internat. J. Math. And Math. S c i . 4(1981), 667-690. 21. , "Green's Relations on a Connected Algebraic Monoid", Linear and Multilinear Algebra, (to appear). 22. , "The Group of Units of a Connected Algebraic Monoid", Linear and Multilinear Algebra, (to appear). 23. , "Reductive Groups and Regular Semi—groups", Journal of Algebra, (submitted). 24. , "The J—class Structure of Connected Algebraic Monoids", Journal of Algebra 73(1981), 601-612. 25. , "Linear Algebraic Semigroups", Semigroup Forum 22(1981), 287-309. 26. : , "A Semi—group Approach to Linear Algebraic Groups", Journal of Algebra, (submitted). 27. L. Renner, "Automorphism Groups of Minimal Algebras", UBC Thesis, Vancouver(1978). 28. P. Roberts, "Abelian Extensions of Regular Local Rings", Proc.Amer.Math.Soc. 78(1980), 307-310. 29. R. Steinberg, Conjugacy Classes in Algebraic Groups, Lecture Notes in Mathematics v. 366, Springer—Verlag, Berlin(l974). 30. D. Su l l i v a n , "Infinitesimal Computations in Topology", IHES Publications in Mathematics 47(1977), 269-332. 31. H. Sumihiro, "Equivariant Completion", J. Math. Kyoto University 14(1974), 1-28. 32. W. Waterhouse, Introduction to Affine Group Schemes, GTM 66, Springer-Verlag, Berlin(1979). 

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