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Automorphism groups of minimal algebras Renner, Lex Ellery 1978

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AUTOMORPHISM GROUPS OF MINIMAL ALGEBRAS by LEX ELLERY RENNER B . S c , U n i v e r s i t y of Saskatchewan, 1975 A.THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1978 Lex E l l e r y Renner In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date O o f . // / l ? Abstract Rational homotopy theory i s the study of uniquely d i v i s i b l e homotopy inv a r i a n t s . For each nilpotent space X the a s s o c i a t i o n X ——»• minimal"algebra for X i s a complete determination of these i n v a r i a n t s . If X i s a space and M i t s minimal algebra, the algebraic group Aut M and the representation A. Aut M X • Gl(^) have considerable influence on the structure of M . This thesis contains X a systematic study of t h i s i n t e r a c t i o n . Chapter I contains preliminary r e s u l t s from algebraic group theory and general topology. In Chapter II I define and study inverse l i m i t s of algebraic groups. I prove that many of the known s t r u c t u r a l properties of algebraic groups remain v a l i d i n this more general s e t t i n g . Emphasis i s placed on the conjugacy theorems that are p a r t i c u l a r l y u seful for studying minimal algebras'. Chapter III i s the main part of the thesis where I develop a structure theory for minimal algebras which r e l a t e s t o r o i d a l symmetry to r e t r a c t s . P r e c i s e l y , i f M i s a minimal algebra, then there e x i s t s a 1-parameter subgroup A: Q ——> Aut M X-such that X extends to with A(0) = e = e 2: M • M . . X X Further i f e so chosen i s minimal then i t i s uniquely determined up to conjugation by Aut M . A In the i n t e r e s t i n g case where e = 0 I give a p r o - a l g e b r a i c group m t h e o r e t i c proof of uniqueness of coproduct and product decompositions i n the appropriate homotopy category. i v TABLE OF CONTENTS In t r o d u c t i o n . . 1 Chapter I P r e l i m i n a r i e s 6 1.1 k-groups 6 1.2 General Topology 12 Chapter I I P r o - a l g e b r a i c Geometry 22 2.1 P r o - v a r i e t i e s 22 2.2 Pro-k-groups 23 2.3 Conjugacy Theorems 26 2.4 Imbedded Pro-k-groups 32 Chapter I I I Automophism Groups of Minimal Algebras 36 3.1 P r e l i m i n a r i e s 36 3.2 Minimal Algebras 36 3.3 Q-structures on Minimal Algebras 38 3.4 • Weight S p l i t t i n g s of Minimal Algebras 42 3.5 Types of Weight S p l i t t i n g s 44 3.6 1-parameter Subgroups and Weight S p l i t t i n g s 47 3.7 Products , Coproducts and T o r o i d a l Symmetry 53 References 73 INTRODUCTION From S u l l i v a n ' s theory of r a t i o n a l homotopy types [17] one has a w e l l defined correspondence • M : Top —?> 171 which assigns to each n i l p o t e n t space X i n Top i t s minimal algebra M(X) ( [ 8 ] , [10], and [17]). The usefulness of the method space ——> s i m p l i c i a l forms on space — ? model f o r space as e x h i b i t e d i n S u l l i v a n ' s fundamental paper [17] i s t h r e e - f o l d . 1) C a t e g o r i c a l : The usual homotopy category of n i l p o t e n t spaces l o c a l i z e d w i t h respect to 0-equivalence i s equivalent to the homotopy category of n i l p o t e n t d i f f e r e n t i a l graded algebras l o c a l i z e d w i t h respect to i t s f a m i l y of weak equivalences. 2) C l a s s i f i c a t i o n : a) The homotopy type of a n i l p o t e n t 0 - l o c a l space X i s uniquely determined by the isomorphism c l a s s of M(X) , i t s minimal algebra. b) By imposing a d d i t i o n a l a l g e b r a i c s t r u c t u r e s ( i n t e g r a l cohomology r i n g , s t r u c t u r e l a t t i c e , etc.) S u l l i v a n proves c l a s s i f i c a t i o n r e s u l t s f o r i n t e g r a l homotopy theory "up to f i n i t e ambiguity" using p r o p e r t i e s of a r i t h m e t i c groups. 3) Computational: Given a n i l p o t e n t space X , one can compute 1T^'(X)®Q wi t h i t s L i e algebra s t r u c t u r e and H (X;Q) w i t h i t s Massey product s t r u c t u r e by examining a p r e s e n t a t i o n of M(X) a minimal model f o r X. One s t r i k i n g consequence of t h i s general a l g e b r a i c p i c t u r e i s 2 that Horn sets between O-local spaces can be computed a l g e b r a i c a l l y using the notion(s) of algebraic homotopy ([8], [10] and [17]). Under su i t a b l e f i n i t e n e s s conditions S u l l i v a n [17] observes that for a minimal algebra M , Aut(tt)(resp. Aut(rD/^) the group of s e l f -equivalence of M (resp. the group of d.g.a. homotopy classes of s e l f -equivalences) are a f f i n e algebraic groups. We observe here that, more generally, for a minimal algebra M of f i n i t e type, Aut M and Aut ti / ^ are p r o - a f f i n e Q-groups (2.2.2). Under s u i t a b l e f i n i t e n e s s conditions much of the important structure theory of Q-algebraic groups remains v a l i d f o r p r o - a f f i n e Q-groups. In chapter I basic notation i s set up and well known r e s u l t s from algebraic group theory are c o l l e c t e d . Certain types of p r o j e c t i v e systems are also studied and t h e i r usefulness for algebraic groups i s introduced. In chapter II we develop a theory of p r o - a f f i n e k-groups which i s general enough f o r a large c l a s s of minimal algebras and s t i l l y i e l d s "good" r a t i o n a l i t y and s t r u c t u r a l p r o p e r t i e s . A natural a p p l i c a t i o n of t h i s general setup i s achieved i n chapter III where we focus on the r e l a t i o n of t o r o i d a l symmetry to t o p o l o g i c a l closure properties of Aut(M) £ End(tt) (which i s an open imbedding of (pro-) a f f i n e v a r i e t i e s ) . Using the conjugacy theorem of chapter II we i n t e r p r e t the existence of "non-closed t o r i " T £ Aut(M) c End(M) as s t r u c t u r a l information about r e t r a c t s . The following general r e s u l t i s obtained f o r a s u i t a b l e c l a s s of minimal algebras M . Theorem 3.6.2. Let M e M .. Then there e x i s t s M(l) e M. , unique up to d.g.a. isomorphism, such that i ) M(l) -7—> M —>M(1) where . .1 p P ° 1 = ^ (1) * i i ) there e x i s t s A : K >Aut(M) a 1-parameter subgroup which"extends'to A : K > End(M) with A(0) = i ° p . i i i ) N e M s a t i s f i e s i ) and i i ) implies there e x i s t s j : M.(l) " > N and g : N > M (1) such that . q ° j = Thus we have f o r a l l M e M an e s s e n t i a l l y Unique "smallest" • r e t r a c t M(l) e M such that the idempotent i ° p = e : M > M that gives i t to us i s the l i m i t of a 1-parameter subgroup A : K* > Aut(W) I t e r a t i n g t h i s process y i e l d s for a l l M e M an e s s e n t i a l l y unique c o l l e c t i o n ,{M(K)}n £ M such that k— 0 i) M(0) = M i i ) M(k) ——> M(K-l) > M(k) s a t i s f i e s theorem 3.6.2 for a l l k > 0 Among other things (3.6.3) t h i s gives us a good i n s i g h t into the. structure of (weak) 0-equivalence i n the i n t e g r a l homotopy. category. This- can be i l l u s t r a t e d by the following i n t e r e s t i n g s p e c i a l case Let P be the f u l l subcategory of 1-connected f i n i t e CW complexes s a t i s f y i n g the following condition _P : X e P i f whenever Y i s a f i n i t e 1-connected CW complex and f . : X > Y i s a p-equivalence [14] there 4 ex i s t s g : Y > X which i s a p-equivalence. Thus P i s the subcategory of f i n i t e 1-connected CW complexes where p-equivalence i s an equivalence r e l a t i o n . P i s p r e c i s e l y the category of p-universal spaces of Mimura and Toda which have been studied i n [3] and [14]. I t i s then a consequence of [3] that the following are equivalent. 1) X e P 2) Y e P and there e x i s t s Z and maps f : Z > X and q : Z > Y which are p-equivalences. 3) There e x i s t s a prime q ^ 0 and a map f : X > X such that 1^ fq # ® 1 • 7 T*(X) ® Q > TT^(X) ® Q and f ® 1 = 0 : TT (X) ® Z/pZ > IT (X) ® Z/pZ q# . * * 4) The augmentation map iM(v^ '• >'M(X) s a t i s f i e s 6 e Aut(M(x)) ( Z a r i s k i closure) where M(X) i s the minimal model f or X . One s t r i k i n g observation ((4) above) i s that we are dealing with a r a t i o n a l problem. Said d i f f e r e n t l y , the existence of s e l f maps that a n n i h i l a t e p-torsion functors and induce isomorphisms on q-torsion functors i s independent of p and q . Thus Theorem 3.6.2 can be viewed as the r a t i o n a l algebraic p i c t u r e for the general setup. Another a p p l i c a t i o n of the pro-algebraic conjugacy theorems i s given i n the l a t t e r part of chapter 3 where product and coproduct decompositions are studied. Generalizing [1] and [2] we prove that for a s u i t a b l e category P of minimal algebras (those having p o s i t i v e weights and finite-dimensional s p h e r i c a l cohomology) a l l objects s a t i s f y unique f a c t o r i z a t i o n into products and copfoducts. n P r e c i s e l y , i f M e P then M = II M . where each M. i s . 1=1 1 H-irreducible (and n o n - t r i v i a l ) . Furthermore, such a decomposition i s unique up to isomorphism and reordering of the f a c t o r s . m Dually, i f N e P then N = Ji N. where each N. i s 1L -i = l 3 1 ' - i r r e d u c i b l e . Such a decomposition i s unique up to isomorphism and reordering. 6 Chapter 1 - Pre l i m i n a r i e s The purpose of t h i s chapter i s to assemble f or easy reference well known r e s u l t s from algebraic group theory and general topology. Adequate reference f o r the r a t i o n a l i t y theory of algebraic groups may be found i n [4] and [9]. The general topology i s more or les s s e l f contained. 1.1. k-groups Throughout the chapter (and the e n t i r e paper) k denotes a p e r f e c t f i e l d and K an algebraic closure of k . For the app l i c a t i o n s we have i n mind k w i l l be the r a t i o n a l numbers. D e f i n i t i o n ([4] p. 85): a) A k-group consists of the following data i ) a K-affine v a r i e t y G defined over k ! i i ) k-morphisms i : G > G and u : G * G > G (where G x G has the Z a r i s k i product topology). i i i ) e e G a distinguished element, such that G has the structure of a group with: i d e n t i t y •= e . , , -1 i (x) = x u(x,y) = x ° y b) A k-morphism <j> of k-groups G and H i s a set map <J> : G — > H , such that i ) o i s a group morphism . i i ) (f> i s a k-morphism of a f f i n e v a r i e t i e s . c) A k-subgroup H £ G i s a subgroup of G which i s k-closed (closed i n the Z a r i s k i k-topology). Remark: Since we assume that k i s p e r f e c t H c G i s k-closed i f f H i s defined over k . See [4]. The following l i s t of well known r e s u l t s i s taken from [4] and [9]. G denotes an a r b i t r a r y k-group. 1.1.2 ([9] p. 3) (Existence of Quotients) If G i s a k-group and H i s a k-subgroup then there e x i s t s on G/H a unique structure of an algebraic v a r i e t y (not n e c e s s a r i l y a f f i n e ) such that, i ) TT : G ——> G/H the natural p r o j e c t i o n i s an open k-morphism i i ) G acts ,k-morphically on G/H by l e f t t r a n s l a t i o n , i i i ) I f f : G > X i s a k-morphism of v a r i e t i e s then there e x i s t s a unique k-morphism f : G/H > X such that . f • G —-> X TT * , ~ commutes i f f f i s constant on- the f i b r e s . G/H f iv) I f . k £ L £ K where L i s a f i e l d extension of k and £ € (G/H) (L) (= L - r a t i o n a l points of G/H) then TT 1 ( £ ) i s defined (and separable) over L . Remark ([4] p. 181): I f H i s also normal i n G then G/H i s also an a f f i n e k-group with the above mentioned v a r i e t y s t r u c t u r e . 1.1.3 ([9] p. 3) (Existence of r a t i o n a l points) Let G be a connected k-group (G = G°) . Then' G(k) •£ G i s 8 dense i n the Z a r i s k i k-topology.. I f k i s also i n f i n i t e then G(k) £ G i s also dense i n the Z a r i s k i K-topology. Note: This can f a i l i f k i s not p e r f e c t or i f G i s not connected. 1.1.4 ([4] p. 150) (Jordan-Chevalley decomposition) Let G be a k-group. Then there e x i s t s subsets G , G c G u s -(uniquely determined) such that i) G ft G = {e} . s u i i ) G and G are f u n c t o r i a l s u i i i ) for a l l x e G there e x i s t unique x e G and x e G s s u u such that X = x ' x = x * x . s u u s iv) i f x e G(k) then x , x e G(k) . (This can f a i l i f k s u i s not p e r f e c t ) . v) i n any k - r a t i o n a l representation p : G > G£(n,K) of G P ( G S ) consists of semi-simple elements and P. ( G u) consists of unipotent elements. D e f i n i t i o n ([4] p. 200): T £ G i s c a l l e d a torus i f T i s a closed connected abelian subgroup c o n s i s t i n g of semi-simple elements. (Equivalently, * * * i f T = K x ... x K as algebraic groups, where K i s the m u l t i p l i c a t i v e group of units of K) . D e f i n i t i o n ([4] p. 204): A torus T £ G i s c a l l e d k - s p l i t i f i ) T i s defined over k i i ) Horn (T,K*) spans K[T] where K[T] i s the a f f i n e k-group' co-ordinate r i n g of T . 9 i i ) i s equivalent to * * i i ) ' T = K x . . . X K as k-groups (see [4] p. 204). 1.1.5 ([9] p. 4). I f G i s a k-group then G has a maximal torus T £ G defined over k . Note: The n o n - t r i v i a l i t y of t h i s statement i s that T i s defined over k . Maximal t o r i e x i s t f o r dimension reasons. 1.1.6 ([9] p. 11) (Conjugacy of maximal k - s p l i t t o r i ) . I f S, T c G are maximal k - s p l i t t o r i then there e x i s t s g e G(k) such that gTg ^  = S . Remark: 1.1.6 i s also true for other classes of subgroups i n G . See [4] or [9] for d e t a i l s . I t w i l l be useful i n l i m i t considerations to keep track of G(k) (e s p e c i a l l y f o r the conjugacy theorems we have i n mind). More s p e c i f i c a l l y we are i n t e r e s t e d i n maps ' (j> : G —•—> H of k-groups f o r which o(G(k)) = <MG) (k) . D e f i n i t i o n : ([4] p. 357). A connected k-group G i s c a l l e d k-solvable i f there i s a composition s e r i e s G = G- 3 G z> ... o G = {e} 0 — 1 - - n such that i ) G^ i s a connected k-subgroup i i ) G./G. = K* or K \ . i i + l as k-groups, f o r 0 <^  i < n 10 (The terminology i n [4] i s k - s p l i t ) . 1.1.7 ([4] p. 359-62). I f G i s a connected k-group then i) G i s k-solvable i f f G i s t r i g o n i z a b l e over k . i i ) G unipoteht implies G i s k — solvable. 1.1.8 ([-4] p. 363). I f H i s a k-group (not necessarily connected) and N £ H i s a connected k-solvable subgroup, then i n . IT : H > H/N , T r (k) : H (k) > (H/N) (k) i s s u r j e c t i v e . D e f i n i t i o n : A k-grbup morphism <j> : G > H i s c a l l e d . k-proper i f <()(N(k)) = <j>(N) (k) for a l l N £ G k-closed subgroup. Remark: Maps which are not k-proper abound i n algebraic group theory. As an example l e t K be an algebraic closure of Q , the r a t i o n a l numbers and * * * consider p^ : K * K > K , the p r o j e c t i o n onto the f i r s t f a c t o r . * * Let N c K x K be the k-closed subgroup defined as 2 N = {(x,y)/x-y = 1} . Then i t i s easy to v e r i f y that P 1(N(Q)) c P ; L(N) (Q) . * 1.1.8. shows us that t h i s f a i l s only because N n ({l} x K ) i s not connected. Lemma 1.1.9:. Let (j) : G ——> H and i|> : H >' N be morphisms of k-groups. a) If • <(> i s one to one then (j; i s k-proper. b) If <j) and \> are k-proper ip ° <j> i s k-proper. 11 Proof: a) I f L £ G i s a k-subgroup then <j>(L(k)) £ <f>(L)(k) . Conversely, i f x e <jr(L)(k)- then <$> / ( x ) e N i s a k-closed point. Thus <J> ^ (x) e N(k) since k i s p e r f e c t , b) Is obvious. . Theorem 1.1.10: Let <j> : G — > H be a rrtorphism of k-groups such that i ) ker 4> i s unipotent i i ) char k = 0 .' Then <j> i s k-proper. Proof: ker <j> i s unipotent and k-closed. Thus i t i s also connected ([12] p. 101), since char k = 0 . Let N £ G be a k-closed subgroup and l e t ^ = <t>jjq - By 1.1.2 we can f a c t o r <j> as follows N/(NnKer<{>) By 1.1.8 p(N(k)) = (N/(NnK ))(k) and by 1.1.9.a) 9 i((N/(NnK.))) (k) = i(((N/(NnK;).)) (k)) . Thus 4>(N) (k) = i(p(N)) (k) = i((N/(NnK )))(k) = i(((N/NnK ))(k)) <p = i(p(N(k))j = o (N(k)) q.e.d. 12 1.2. General Topology . Limits of Proper Systems . '. I t was hinted i n the introduction that we are i n t e r e s t e d i n a s a t i s f y i n g theory of pro-algebraic Q-groups. This involves f i n d i n g suitable conditions on p r o j e c t i v e systems and algebraic groups that w i l l allow us to prove d e l i c a t e structure theorems about a general c l a s s of l i m i t i n g objects. Roughly speaking what we need i s a system where inverse l i m i t s "behave l i k e images" (we would l i k e to preserve properties l i k e connectedness, compactness, i r r e d u c i b i l i t y and the l i k e . ) The s i n g l e most important r e s u l t from general topology i s the following. Projective L i m i t Theorem 1.2.1: Let {x. , TT . . } be a projective, system J ^ • x ij xel of t o p o l o g i c a l spaces X ' such that i) X^ ^ 0 i s compact and T . ' ' i i ) TT . . i s a closed continuous map for i > j . Then X =.lim X. f 0 «- x Proof: [14 p. 57] . Let S = { {A.} | A^ c x^ i s closed and non-empty TT . . (A . ) £ £ ID i 1 Then 0 since {X^} e S, . Define, a p a r t i a l ordering on S as follows; {A.} < {B.} i f A', C.B. for a l l i . x — x x ~ x Since each X^ i s compact, every chain i n (S, <_) has a lower . bound. (This i s j u s t the f i n i t e i n t e r s e c t i o n property). Thus by Zorn's lemma S has minimal elements. Suppose {A^} e S i s minimal. By i i ) B. = n IT.. (A.) i s closed non-empty and T r. r(B.) c B, i i ] ] i k . x - k for i > k , so {B.} e S . B u t {B.} < {A.} . Thus {B.} = {A.} by — 1 i — i i i minimality. But then A. = n TT . . (A .) and thus n . . (A .) = A. for a l l 1 j>i ?X J D i D i j > i . F i x i € I and choose x. e A. . — l l Let C . = TT . 1 (x. ) n A . i f j > i 3 D 1 1 3 -C. = A. . i f j l i 3 3 . J" By i ) C. i s closed and by construction {C.} < {A.} . Thus {G.} = {A.} 3 3 ~ 3 3 3 again by minimality of {A^} . Since i e I was a r b i t r a r y .{A^} = {x^} . . By d e f i n i t i o n of S (x.) e lim X. . q.e. 1 i d 1 Remark: The conclusion of t h i s theorem i s true under more general assumptions. S p e c i f i c a l l y , i f we require that each TT^_. has compact poi n t inverses•then the r e s u l t i n g proposition i s true f or T^ spaces. Also with our assumptions the l i m i t i n g object i s compact. In any case an inverse system s a t i s f y i n g the assumptions of Theorem 1.2.1 w i l l be c a l l e d a proper  system. Theorem 1.2.. 2: Let {x. , TT . .} be as i n theorem 1.2 .1. Then i f ••—- a i : X = lim X. <- l . . . i) TT. (x) = n TT . . (X.) • 1 - D l i 3 1 3 i i ) X = lim TT . (X) J~ 3 Note: This theorem says that an a r b i t r a r y proper system can be replaced by an equivalent one where a l l the doubly indexed maps are onto.. (An inverse system i s c a l l e d proper i f i t s a t i s f i e s the assumptions of 1.2.1) Proof of 1.2.2: i) C l e a r l y IT . (X) c n TT..(X.) . Conversely, l e t — 1 - j>i ^ > x. e n TT ... (X.) and l e t i w . 3 1 D . F . = Tr ' . - . 1 (x. ) j > i F. = X. j i i D D J - -Then {F , TT } i s a proper system. Thus by theorem 1.2.1 l i m F T ? 0 . But F. = {x.} . Thus there e x i s t s x e l i m F. c x such that TT , (x) = x. l i -<- < ~ i i Thus x. e TT . (x) . Hence n • IT .. (X.) 'c IT . (x) . i i . j i I I :>i i i ) Since TT.-(X) £ X.' we have lim TT; (X) £ X . But i f x e X then x = (x.) and TT . (x) - x. . But then each x. e n TT . . (X.) = IT . (X) 1 iel 1 1 1 j>l 3 1 3 So x e lim TT . (X) . . i D e f i n i t i o n : Let {X., Tr..} and {Y., 0..} be inverse systems of i I D • -r i I D • -r i e l I G I t o p o l o g i c a l spaces. A morphism 6 : {X. , TT. .} > {Y. , 6 . .} i I D i I D i s a c o l l e c t i o n & = of continuous maps <j>. : X. 1 > Y. such that l l l . l 71 . D 1 X > Y. commutes i 6. 1 i whenever D 21 1 • Of course {cf>.} induces d> = l i m <f>. : lim X. > lim Y. as follows: Let (x.) £ lim X. then (<j>.(x.)') e lim Y. since i ^ - i I . I <- l e.. •4>.. = <j>. D I D i Di Thus define (lim <?. ) (x.) = (<(>. (x.)) . l l i i Remark: i) I t i s a well known f a c t about t o p o l o g i c a l spaces that i f each 4K i s continuous then cf> i s continuous i n the inverse l i m i t topologies f o r l i m X and lim Y. . I f the X.'s and Y.'s also have the structure i -t- I i i of a group and the <f>. 's respect t h i s structure as w e l l , then lim X^ and lim Y. have natural group structures f or which <f> = lim <J>. i s a group •;• 1 <- 1 homomorphism. i i ) The above d e f i n i t i o n of morphism i s not the most general one that i s u s e f u l . I t i s however convenient and s u f f i c i e n t for the purposes of t h i s paper' to have a f i x e d index set. Theorem 1.2.3: Let d> : {X. , TT . . } > {Y., 6. .} be a morphism of proper — . i i ] . , . i i ] i t  systems where 9. . and TT . . are onto when they e x i s t . Then <j> = lim <j>. iD iD + i onto i f f 4 > . i s onto for a l l i . Proof: Suppose 9 i s onto. I f y. e Y/ there ex i s t s y e lim Y^ such that 6 (y) = y. (Theorem 1.2.2). But then . 1 1 lim X. < - 1 X. 1 -» lim Y. -> Y. 1 so that. ^ i s onto. Conversely, suppose 9. i s onto for a l l i . Then i f (y.) = y e lim Y for each f i x e d i there e x i s t s x. e X. such that 1 *- i - ( 1 9. (x.) = y. . Thus A. ^  (y'. )' = F, f 0 , and since the appropriate maps 1 1 1 1 1 1 commute TT (F ) £ F. . Since each F i s also closed, F = lim F. / 0 j i 3 1 1 < - 1 (Theorem 1.2.1). But by construction 9(F) = (y )• . q.e (Actually we proved more than advertized, i . e . I f { , ®ij^ """s P r o P e r and 9 i s onto then 9^ i s onto for a l l i , and i f {x^, ^ s proper and onto for a l l i then 9 i s onto.) For the remainder of t h i s s ection we assume that Tr. . i s onto whenever i >_' j . Theorem 1.2.2 assures us that no generality i s l o s t . Lemma 1.2.4: Let 9 : X > Y be a map of proper systems. Then 9 (X) = lim 9 . (X.) 1 1 Proof: C l e a r l y 9(X) £ lim 9.(X.) . Conversely, i f y = (y.) e lim 9.(X.) • . 1 1 . 1 ^ - 1 1 then 9 . ^  (y. ) ^  0 ' i s closed and TT . (9 . (y )) c 9 , 1 (y ) s o we have a • 1 1 1 : 1 1 3 3 1 7 proper.system. Thus lim (|>71(y.) ¥• 0 • But (x) = (x.) e lim <(). 1,(y.. ) implies i 1 x x x that $(x) = y . . q.e.d Lemma 1 . 2 . 5 : i) The closed sets for the inverse l i m i t topology on X are given by {lim F . ITT , , ( F , ) - c F_ ., F . closed} . <- x' X 3 x : x i i ) each TT^ i s a closed map. Proof: i ) Let F . c x. be closed and l e t x - x F . = T T T 1 ( F . ) j > i F . = X . j > i I D ~ Then TT . ( F . ) c F , and we have a proper system. Let F = lim F ^ 0 . j _ k . •«- k Then F c ir/ ( F , ) ' . by d e f i n i t i o n . Conversely, i f x = (x.). e TT , ( F . ) then - X X X 1 X X x. £ F . and j > i implies IT , . (x .) = x. . So x . e TT . ^ ( F . ) .= F . . x x - D 1 D i D ID i D Thus I T . 1 ( F . ) = lim F . . Observe also that i f : F" = lim F ? a e Q some set x x «- D <•• i then n F A = lim n F . . aeQ aeQ Thus to complete the proof of i ) we only have to show that i f F = lim F. and G = lim G. then F u G i s also of t h i s form. <- x «- x To do t h i s we f i r s t observe that i f F = lim F . then as i n the <• x proof of 1 . 2 . 2 TT.(F) = n T T..(F.) i s . closed. Thus F = l i m 7 r . ( F ) ( 1 . 2 . 2 ) X . . J X J • • -6- • 1 J > 1 an d' s i m i l a r i l y G = lim TT. (G) . But then TT , (FuG) = TT. (F) u TT . (G) i s closed Hence yet another a p p l i c a t i o n of 1 . 2 . 2 implies that F u G = lim ir^ (FUG) 18 i i ) This i s now easy since i f F c x i s closed, by i ) , F = lim F. where F. e x . are closed. Then by 1.2.2 < - 1 x i . TT . (F) = n TT . . (F . ) 1 . 3>i D 1 3 . i s closed. q.e.d. Theorem 1.2.6: Let X = lim X. be the l i m i t of a proper system. Then i f X. i s connected for a l l i , X i s connected. i . Proof: If X = A u B where A and B are closed then TT . ( A ) u TT . (B) = X. . i I I and TT. ( A ) and TT.(B) are closed by 1.2.5. Thus TT . ( A ) n TT.(B) ^ 0 . i i i i Let N . = TT . ( A ) nTr. (B) . Then TT . . (N .) c N . since i r . . °. Tr. = TT . . l l l 3 1 : ~ 1 3 1 3 1 Thus we have yet another proper system. So lim N . = N ^ 0 . Thus 1 A n B = lim TT . A n lim TT . B = lim (TT . ( A ) ri TT . (B)) =? N 7^  0 . q.e.d. 1 1 ^ - 1 1 Coset Topologies for Algebraic Groups In order to apply the r e s u l t s of section 1.2 to inverse systems of algebraic groups we must f i r s t contrive an appropriate, topology so that (unlike the Z a r i s k i topology) morphisms are closed maps. Consequently an a r b i t r a r y inverse system of algebraic groups i s a proper system of t o p o l o g i c a l spaces (1.2.1). Since the topology s e l e c t e d (see the remarks preceding 1.2.10) i s i n t r i n s i c i t may be forgotten or resurrected depending upon the whim of the moment. '' Recall that a noetherian t o p o l o g i c a l space i s one i n which any descending chain of closed subsets i s stable. 19 Proposition 1.2.7: L e t X, be a noetherian t o p o l o g i c a l space and l e t F c x be closed a e fi . Then' n F = n . F where F c fi i s f i n i t e . a ~ fi ° F Q V .-. ° ~ • Proof: Obvious from the d e f i n i t i o n . Proposition 1.2.8: Let H, N £ G be subgroups of the group G and l e t x, y e G ... Then, e i t h e r xH n yN i s empty or i t i s a coset of H n N . Proof: Note: xH n yN = x(H n x "'"•yN) and H n gN? 0 i f f g e HN . Thus H n gN ^ 0 implies that H n gN = H (i h ' i i = h-H n h-N = h(H n.N) q.e.d. We now s p e c i a l i z e these simple r e s u l t s to the k-group G . Cor o l l a r y 1.2.9: Let {g H } be cosets of k-closed subgroups H c G (a k-group) . Then e i t h e r n g H = 0 or i t i s a coset of some k-closed a a a subgroup of G . Proof: Since algebraic v a r i e t i e s are noetherian t o p o l o g i c a l spaces use 1.2.7 and 1.2.8 and induction (on 1.2.9). W -Topology K. Using propositions 1.2.8 and 1.2.9 we can now define a topology W (G) on a k-group G that w i l l be su i t a b l e f o r the formation of proper inverse l i m i t s . 20 Re c a l l that t h i s means we want a topology for which i) G i s compact and i i ) . <j> : G > H i s closed and continuous, where <j> i s k-group morphism. This i s best accomplished as follows. Let G be a k-group and n consider the c o l l e c t i o n F = { u g.H.} of a l l f i n i t e unions where ' ' i = l 1 1 H. £ G i s a k-closed subgroup and -g e G are a r b i t r a r y . Then l e t W (G) = F u {0} . Propositions 1.2.7, 1.2.8 and 1.2.9 assure us that W (G) i s a topology on G . Notice that allowing the g^ a r b i t r a r y i s necessary, since i f g, h e G(k) and gH n hN i- 0 for some k-closed subgroups H and N of G then gH n hN = x(H. n N) where x e G . But we do not ne c e s s a r i l y have x e G(k) . This s t i l l y i e l d s the r i g h t topology on G(k) since for a k-closed subgroup H £ G we have gH n G(k) ^ 0 i f f there e x i s t s x e G(k) such that gH = xH . Proposition 1.2.10: Let G,. H be k-groups and <j> : G ——> H be a k-group morphism. Then i) G i s T and compact i n the W -topology, i i ) <j) i s closed and continuous i n the W -topology . K. Proof: i) G i s T by d e f i n i t i o n and the W -topology i s weaker than the I. K. Z a r i s k i K-topology, i i ) I f N £ G i s a k-closed subgroup, then cf>(N) £ H i s k-closed ([4] p. 88). Also <J> preserves, cosets and unions. q.e.d. 21 Remark: In general the W -topology i s neither weaker nor stronger, than the Z a r i s k i k-topology. . C o r o l l a r y 1.2.11: Suppose also that 9 : G > H i s k-proper. Then i f 9 = 9'| > 9, : G(k) > H(k) i s a closed map i n the induced k - l G (k) • K topology. Proof: By d e f i n i t i o n (of k-proper) '<J> (N (k/) ) = <|>(N)(k) f o r N £ G a k-closed subgroup. Now proceed as i n 1.2.10. q.e.d. Homogeneous Spaces Let G be a k-group and H £ G a k-closed subgroup. Then the W -topology on G/H i s the quotient topology induced from the W -topology on G v i a the natural map TT : G : > G/H . A simple argument then shows that i f 9 : G. N i s a morphism of k-groups and 9(H) £ M where H £ G and M £ N .are k-closed subgroups then the induced map 9 : G/H > N/M i s closed and continuous. S i m i l a r l y i f 9 i s also k-proper then 9 : G(k)/H(k) >--N(k)/M(k) i s closed and continuous. Warning: G(k)/H(k) i s not n e c e s s a r i l y the same as (G/H)(k) . (It i s i f N i s k-solvable and connected. See 1.1.8). G(k)/H(k) i s s i n g l e d out because i t i s useful i n our a p p l i c a t i o n s ( E s p e c i a l l y section 2.3). 22 Chapter 2 - Pro-Algebraic Geometry 2.1. Pro-Varieties (Inverse l i m i t s of algebraic v a r i e t i e s ) A simple but useful property of algebraic v a r i e t i e s (in f a c t a defining property) i s that the diagonal i s closed. i . e . I f X i s a k-variety then AX = {(x,x) e X x X> i s k-closed i n the Z a r i s k i product topology. An equally useful f a c t i s that the same i s true for the p r o j e c t i v e l i m i t of algebraic v a r i e t i e s . A pro-algebraic v a r i e t y i s , by d e f i n i t i o n the inverse l i m i t X = lim X. of an inverse system {X. , TT . .} of algebraic, v a r i e t i e s [12]. ' Unless the contrary i s s p e c i f i c a l l y allowed X = lim X. w i l l be given the inverse l i m i t topology induced from the Z a r i s k i topology on the X R e c a l l that (for sets) i f {x. , TT . .}. and {Y., 9..}. are i i ] l e i I i ] i e l inverse systems then so i s {X. x Y., TT . x 0 .} and i i i ] i ] i d lim X. x. Y. = lim X x l i m Y . < i i <• l -c- i Proposition 2.1.1; Let X = lim X. be a pro-algebraic v a r i e t y . Then AX = {(x,y)|x = y} £ x x x i s closed (in the Z a r i s k i topology). Proof.: AX. c x. x x. i s closed, for a l l i , and i - l i Ax = {((x.), (y.)) x. = y. f o r a l l i} i i a. J I = n {((x.), (y.)) | x. = y.} j 1 1 D J = n T T . 1 (Ax.) i s closed . q.e.d. Corol l a r y 2.1.2: Let 9 , lp : X > Y be morphisms of pro-algebraic v a r i e t i e s ( i . e . : 9 = lim 9 . where each 9 . i s a map of v a r i e t i e s and {9^} i s a map of inverse systems) . Then E (A,) = {x e x|<}>(x) = ^ (x) } £ i s closed. Proof: I f .4>ITip : X > Y x y i s the unique map so that commutes, then E(<J>,IJJ) = ( W ) 1(AY) which i s closed by 2.1.1. 2.2. Pro-k-groups We are now i n a p o s i t i o n to develop the elementary properties of pro - a f f i n e k-groups. There w i l l be occasion to d i s t i n g u i s h between a pro-k-group and a k-pro-group (2.2.2 and 2.2.3). The necessity of t h i s d i s t i n c t i o n i s i l l u s t r a t e d i n the following example. * Example 2.2.1: Let G = K for 0 < n e Z where K i s an algebraic * — — n closure of Q , the r a t i o n a l numbers. Then G has the structure of a n Q-group i n the usual way. Define a d i r e c t i o n on Z + = {n e z|n > 0} as, m _< n i f m|n . Then f o r n > m define 9 : G > G as 9 (x) = xn/^m . One — n,m n m n,m 24 e a s i l y v e r i f i e s that {G , d> } i s an inverse system of Q-groups. n n,m n.e.Z • Let G = lim G and consider G(Q) = { (x ) e G|X e Q* for a l l n} . I f <• n n n (x ) e G(Q) then d> (x ) ='xm = x . Thus x has m-th roots i n n mn,n mn mn n n Q for a l l m > 0 . Thus x = 1 and hence G(Q) = {(1)} . n Although such systems are i n t e r e s t i n g , the l i m i t of t h i s one i s sadly lacking i n the s o r t of r a t i o n a l i t y properties that are enjoyed by algebraic groups. We s h a l l see that t h i s cannot happen to inverse systems of k-groups where a l l maps are .k-proper. D e f i n i t i o n 2.2.2: A pro-k-group i s the p r o j e c t i v e .(= inverse) l i m i t of an inverse system ^ G^' 7 1 - j ^ °^ k-groups. A morphism of pro-k-groups i s the l i m i t of a morphism of inverse systems {<}>.} : (G. , TT . ,} > {H , 6. .} such that each i> i s a k-group 1 1 1 3 i i ] I morphism. Thus i f {o>. } : {G. , TT. .} > {H. , 8 . •.} then o> = lim o>. : 1 1 I ] I X ] - ( - 1 lim G. > lim H. i s a morphism of the pro-k-groups G = lim G. and +- i - -H = lim H. D e f i n i t i o n 2.2.3: A k-pro group i s a pro-k-group G = lim G^ where each TT . . : G. > G. i s k-proper. ID i D Recall that a map <j> : G > H of k-groups i s c a l l e d k-proper i f <j>(N(k)) = <j)(N)(k) for each k-closed subgroup N of G . Remark: Since the k-group G has two natural topologies, the Z a r i s k i i . k-topology and the W^-topology, the same can be s a i d of lim G^ . Since both topologies are i n t r i n s i c , e i t h e r may be forgotten f o r s t r a t e g i c 25 purposes. ' In the next proposition we take advantage of the W -topology. Recall that for a k-group morphism 9 : G > H $ is. closed'and continuous i n the W -topology (see 1.2.10). Thus an inverse system K. { G . , T T . .}. of k-groups s a t i s f i e s the condition of the P r o j e c t i v e 1 1 3 • • • Lim it Theorem (1.2.1) and so we may apply the r e s u l t s of section 1.2 to pro-k-groups. 0 0 0 Notation: I f G = lim G. i s a pro-k-group l e t G = lim G. £ G . G •*- 1 -<- 1 i s c a l l e d the connected component of the i d e n t i t y . (See 1.1.1 for a d e f i n i t i o n of G^) . 1 Remark: By 1.2.6 G ° c G i s connected i n the W -topology. I t i s also easy to prove that G® contains a l l other connected pro-k-subgroups of Proposition 2.2.4: Let G = lim G. and H = lim H . be pro-k-groups and — ^ . 1 V 1 l e t 9 : G > H be a morphism of pro-k-groups. Then i) 9(G) i s a pro-k-subgroup of H i i ) 9(G°) = 9(G)° Proof: i ) 9(G) = lim <j>.(G.) by Lemma 1.2.4. i i ) C l e a r l y 9(G^) £ H so we can assume H = H , and by Theorem 1.2.2 we can assume a l l projections are onto. Thus G >> 9 (G) TT . J 8. 3 » 9 . (G.) j 3 3 26 and by 7.4.B of [12] 9.'(G°) = <j>.(G.)0 . Thus <f> (G°) = (() (lim G°) =. lim 9.(G°) = lim 9.(G.)° j J = 9(G)° q.e.d. Theorem 2.2.5: (Jordan-Chevalley decomposition) Let G be a pro-k-group. Then for a l l x e G there e x i s t s x , x e G (uniquely determined) such that s u i ) x e-G(k) implies x , x e G(k) . (By d e f i n i t i o n u s G(k) = lim G. (k) <= lim' G. ) . <- i - <- l i i ) x = x -x: = x -x s u u s i i i ) TT.(X )-TT.(X ) = TT.(X) i s the c l a s s i c a l Jordan-Chevalley i s 1 u 1 . decomposition for a l l i . iv) For any morphism 9 : G > H of pro-k-groups cMxO = 9 (x) and 9(x ) = 9 ( x ) . u u Proof: Do i t co-ordinatewise using the c l a s s i c a l Jordan-Chevalley decomposition (1.1.4). 2.3. Conjugacy Theorems Pre l i m i n a r i e s : I f G i s a k-group. and X i s a k-variety, then a_ k-morphic action of G on X i s a morphism p : G x X > X 2 7 of k - v a r i e t i e s which i s also a group action of G on X . Let • u : G x x > X be a k-morphic action and l e t X(k) £ x be the set of k - r a t i o n a l points of X . I t i s then well known (see [ 1 2 ] p. 2 1 8 ) that i f Y £ X(k) and Z £ X i s k-closed, then Tran (Y,Z) = {x e G | U . ( X , Y ) £ Z} i s k-closed i n G (in the Z a r i s k i k-topology). Thus, i f • u : G x G > G i s defined as u(g,x) = g-x-g ^ , and T, T' are connected k-closed conjugate subgroups of G , then Tran(T(k), T) i s k-closed i n G . Since T(k) £ T i s dense i n the Z a r i s k i k-topology Tran(T(k), T 1) - Tran(T, T') . If also Tran(T, T 1) rVG(k) ? 0 then Tran(T, T 1) i s a G(k) coset of N (T) the normalizer of T i n G . Hence Tran(T, T') i s G • closed i n the W -topology of G . Theorem 2 . 3 . 1 : Let G = lim G. be a connected pro-k-group ( i . e . : G = G^) +- 1 where TT . . : G. — — » G. i s onto for i > j . Let 8. ^ 0 be a cl a s s of ID i : • - D connected k-closed subgroups of G_. such that, i ) I f i' > j and B. e 8. then n. . (B. ) e 8. . - i l ID i D i i ) B., B! £ 8. implies there e x i s t s g £ G. such that i i i I gB^g 1 = B^ . i i i ) 8. i s maximal with respect to condition i i ) . i Then there e x i s t s a non-empty set 8 such that B consists of pro-algebraic k-subgroups of G which s a t i s f y i i ) and i i i ) . Furthermore, each TT . : G > G. induces TT . : 8 >> 8 . l . i l i 28 Proof: By i i ) we can define n. . : B. ——> 8. as TT. . (B. ) = TT. . (B. ) Furthermore TT . i s onto ( e s s e n t i a l l y because each TT-. . i s onto) . iD . iD. . Fix some i and choose B. e 8. . I t i s e a s i l y v e r i f i e d that 8. = G./N (B.) . l l G. l i Since G /N . (B.) i s a homogeneous space for G. we can put a topology on l G . i l i B v i a t h i s i d e n t i f i c a t i o n . (This i s the W -topology on G./N„ (B.) i k l G^^ l discussed at the end of chapter 1). If i > j and B . = TT . . (B.) . Then - D ID i TT . . 8 1 1 8. i D G./N (B.) —> G./N^ , (B.) l G. l ' D G. ] i TT . . D . ID ' commutes and thus TT . . is. a closed, continuous map with our i d e n t i f i c a t i o n . ID' By homogeneity t h i s topology does not depend on our choice of B. . The net r e s u l t i s that {8., 1 S a proper system ( i t s a t i s f i e s 'the assumptions of Theorem 1.2.1). Thus by Theorem 1.2.1 B~ = li m 8. 0 and i n f a c t i TT . : B » 8. i s onto for each i by Theorem 1.2.2. I I By d e f i n i t i o n of F , B e 8 implies that B = {B.|TT (B.) = B.} . 1 i j 1 D Thus l e t 8 = {B = lim B !{B.} e 8} . Then by d e f i n i t i o n 8 consists of •:• x ' 1 2 9 pro-algebraic k-subgroups of G . To complete the proof we must show that B s a t i s f i e s i i ) and i i i ) above. Let B = lim B. , B' = lim B.' e 5 . Then T. = Tran(B.,B|) ? 0 <- X 1 X X X i s W -closed since i t i s a coset of N (B.) . Also T T . . ( T . ) <= T. . k G i 1 • xj x - 3 Thus {T. , TT . .} s a t i s f i e s the assumptions of Theorem 1 . 2 . 1 , so that T = lim T i- 0 . But by d e f i n i t i o n of T , x = (x.) e T means that <- x x x.B x. 1 = B' . Hence x Bx 1 = B' . x x x x To prove that B s a t i s f i e s i i i ) l e t B = lim B. e B , x = (x.) e G . Then xBx 1 e 8 because from Theorem 1 . 2 . 2 xBx 1 = lim x.B.x. and by <- x x x d e f i n i t i o n x.B x. 1 e B. . q.e.d. x x x x Remark: I f { G . , ^ ^ j ^ 1 S a n inverse system of K-groups ( i . e . k = K) then the assumptions of Theorem 2 . 3 . 1 are s a t i s f i e d f o r the following three classes of subgroups. i ) T. ={T. C G . I T . i s a maximal torus} . X X - X X i i ) U. = {u. c G . l u . i s a maximal connected unipotent subgroup} . 1 x - 1 1 I . • • • i i i ) B. = {B. C G .IB. i s a Borel subgroup} . x x i ' x See [ 1 2 ] p.135 for d e t a i l s . For the applications we have i n mind k i s not a l g e b r a i c a l l y closed. Consequently, the r e s u l t s of Theorem 2 . 3 . 1 have to be s u i t a b l y r e f i n e d to y i e l d conjugacy theorems for G(k) = lim G .(k) . • ' ' • - < - 1 Theorem 2 . 3 . 2 : I f G = lim G . i s a connected k-pro-group ( i . e . each <- x TT . . : G . > G . • i s k-proper. See D e f i n i t i o n 2 . 2 . 3 ) , l e t B. ^ 0 be a i j i ' • : 3 class of -connected k-subgroups of G^ such that 30 i ) I f i-> j and B. £ 8. then TT..(B.) e 8; . — J i i 1 . 3 i 3 i i ) I f B , B 1 e 8. then there e x i s t s g £ G. (k) such that i i i ' i gB.g 1 = B! . i i i ) 8 i s maximal with respect to i i ) . l Then there e x i s t s a non-empty set 8 c o n s i s t i n g of pro-k-subgroups of G such that conditions i i ) and i i i ) are s a t i s f i e d f o r 8 . Furthermore each TT : G >> G. induces TT , : 8 >> 8. '. i l l l Proof: Let 8 (k) = {B,(k) B. e 8.} . Then 8.(k) =8. . i l i l - i i i . e . Define r : 8. — > B.(k) ' i i as r (B.) = B. (k) I l and c : 8. (k) — — > 8. I I as c(B. (k)) = B. (k) . ' i i . (where B.(k) denotes closure In the Z a r i s k i k-topology)." I Then r and c are inverses because ' B.(k) £ B. i s dense i n the l l Z a r i s k i k-topology (1.1.3). But j u s t as i n the proof of Theorem 2.3.1 8. (k) = G. (k)/N (B. (k)) l l G.(k) l I Hence 8.(k) and 8. can be given the topology induced by the W -topology i i k on G. (k)/N „-,(B.(k).) . Since by assumption TT . . : G. (k) >. G.(k') i s i G^(k) I 1 3 1 3 closed and continuous TT . . : 8. > 8. i s closed arid continuous. Thus J-3 1 3 the proof proceeds exactly as. i n Theorem 2.3.1. q.e.d. We s p e c i a l i z e these r e s u l t s to maximal k - s p l i t t o r i (see the d e f i n i t i o n s preceding 1.1.5). Lemma 2.3.3: Let <j> : G >•> H be an epimorphism of k-groups. I f T c G i s a maximal k - s p l i t torus then <}>(T,j £ H i s a maximal k - s p l i t d a torus. Proof: Let T, £ T where T i s a maximal torus defined over k . Then — • d by 34.3 of [12], T = T, • T where T i s a k-anisotropic torus defined • d a a over k . Furthermore <j> (T) .= <j> (T,) r<J> (T ) i s a maximal torus of H d a (2.13C of [12]), <j)(T ) i s k - s p l i t and (j) (T ) i s k-anisotropic (p. 219 o a of [4]). But such a decomposition i s unique. q.e.d. Remark: By Lemma 2.3.3 and 1.1.6 the assumptions of Theorem 2.3.2 are s a t i s f i e d for T ( G . ) = {T. C G . I T . i s a maximal k - s p l i t torus} . Thus k l l - i ' I we make the following d e f i n i t i o n . D e f i n i t i o n : Let {G. , TT . .}. be an inverse system of k-groups where . • l 1 3 i e l . . each TT. . i s k-proper (2.2.3). I f T. .<= G . i s a k - s p l i t torus f o r each 1 3 1 ~ 1 i e l such that TT . . (T. ) c T . whenever i > j , then T = lim T. £ G 1 3 1 3 ~ <- 1 = lim G. i s c a l l e d a k - s p l i t pro-torus. If further, each T. i s a maximal k - s p l i t torus, T = lim T. 1 . . . <- 1 i s c a l l e d a maximal k - s p l i t pro-torus. For further reference we state the following s p e c i a l case of Theorem 2.3.2. Theorem 2.3.4: Let G = lim G. be a k-pro-group (2.2.3) . Then G has a ' •<- 1 maximal k - s p l i t pro-torus T = lim T. . If- S = lim S.. i s another 32 maximal k - s p l i t pro-torus then there ex i s t s g e'G-(k) = lim G(k) such that gSg 1 =.T . Proof: This i s a simple a p p l i c a t i o n of Theorem 2.3.2 to the remark following Lemma 2.3.3. q.e.d. 2.4. Imbedded k-pro-groups In the discussion Of pro-algebraic symmetry of minimal algebras (Chapter.3) the t o p o l o g i c a l r e l a t i o n between automorphisms and-endomorphisms becomes an important f o c a l point i n s t r u c t u r a l c l a s s i f i c a t i o n . With the help of Theorem 2.3.4 t h i s can y i e l d i n t e r e s t i n g c a t e g o r i c a l information (3.6.2). Example 2.4.1: Let M ( n , K ) be the set of n x n matrices over the a l g e b r a i c a l l y closed f i e l d K and l e t G£(n,K), Si(n,K) £ M(n,K) be res p e c t i v e l y , the i n v e r t i b l e matrices and the matrices of determinant 1. In the Z a r i s k i K-topology G£(n) = M ( n ) whereas SI (n) £ M ( n ) i s closed. In the f i r s t case conjugacy properties of Gfc'(n,.K) . y i e l d information about H ( n ) whereas conjugacy properties of Si(n) are a p r i o r i i n d i f f e r e n t to M ( n ) . A useful way to get at some of these closure properties i s with 1-parameter subgroups. 1-parameter subgroups D e f i n i t i o n : a) An imbedded pro-k-group i s the inverse l i m i t X = l i m X^ of a f f i n e k - v a r i e t i e s X^  such that I) For each i there e x i s t s G. c x. an open a f f i n e embedding where G. i s a k-group. 1 . . i i ) TT . j i s a k-group morphism. l ] i G. b) If i n addition each X^ has the structure of an algebraic semi-group consistent with the group structure on G^ and each TT^ _. i s a semi-group morphism then X = lim X. . i s c a l l e d a pr o - a f f i n e k-semi-group. <• I ; D e f i n i t i o n : a) Let X be an imbedded pro-k-group. Then a 1-parameter * subgroup i s a morphism X : K > G = lim G. £ X of pro-k-groups. •<- i i . e . : X = lim X where -«- l i) A : K > G. £ X. i s a k-group morphism. i i i K commutes for i • >_ j b) A 1-parameter subgroup X = lim X . i s s a i d to converge i f f o r : a l l i X . : K > G. extends to a morphism i l . X . : K l -> X. such that l i ) K > X. 1 > G-X . l commutes. 34 and i i ) 71 . . I D commutes. Notation: i ) X = lim X. —: *- i i i ) X(0) = lim X(t) t+0 . c) I f X : K > G £ X i s a 1-parameter subgroup l e t -1 * * * X = X ° i : K > G £ X where i : K > K i s the inverse map. I f neither X nor X converge X i s s a i d to diverge. Remark: The notation X(0) = lim X(t) adopted i n part b) makes sense t+0 * — because i f X. : K > G. extends to X. : K —:—> X. then i t extends I I I I uniquely. This i s an elementary consequence of Co r o l l a r y 2.1.2. Lemma 2.4.2: Let H £-X be a commutative sub-semi-group of X , a pr o - a f f i n e k-semi-group. Then H £ X i s commutative. Proof: Let y e I-I and define f : X > x x x as y f (x) = (xy, yx) Y Then H c f ^(Ax) = {x e xlxy = yx} which i s closed (Proposition 2.1.1). Y Thus H e f - 1 ( A x ) for a l l y e H and so H e n f _ 1 ( A x ) = {x e xlxy = yx — y -1 — V . i J J yeH y 35 for y e II} = Z Repeating the above' argument with Z i n place of H implies that H £ (x e x|xz = zx for z e z} . But then H i s commutative. q.e.d. Theorem 2.4.3: a) Let X = lim X. be a pr o - a f f i n e k-semi-group. I f ~ ' -r- 1 * A,^ : K -> G £ X are 1-parameter subgroups which converge to A(0) and UJ(0) re s p e c t i v e l y , then there e x i s t s geG such that g-A(0) -g and w(0) commute. b) If further the induced pro-k-group G = lim G. i s a a k-pro-group ( D e f i n i t i o n 2.2.3) then there e x i s t s g e G(k) = lim G.(k) such that g*A(0)*g ^ and ,u(0) commute. Proof: a) X (K) £ X(K*) £ S and u (K) £ to (K ) £ T where S and T are some maximal k - s p l i t p r o - t o r i . Furthermore S and T are commutative (Lemma 2.4.2) and conjugate (Theorem 2.3.1). Thus a) i s proved. b) I f G = lim G. i s also a k-pro-group Theorem 2.3.2 applies as well and b) i s proved. . q.e.d. 36 Chapter • 3 - Automorphism Groups of Minimal Algebras In t h i s chapter a natural a p p l i c a t i o n of chapter 2 i s achieved by examining the automorphism group of a minimal algebra. 3.1. Preliminaries . Let A be a f i n i t e l y generated connected graded k-algebra, and l e t A = A ® K where K i s an algebraic closure of k . Then i t K k i s w ell known that Aut(Aj,) = {f : A > A^Jf is. an i n v e r t i b l e K-algebra homomorphism} i s ah imbedded a f f i n e k-group. , I.e. G(A ) = A u t ( A ) £ End(A ) = E(A ) is. K K K i s a p r i n c i p a l open subset of E(A ) where E(A ) i s n a t u r a l l y an a f f i n e K K k-variety. If d : A > A i s a degree +1 l i n e a r map and E(A K,d) = {f e E(A K)|f°d = d°f} , G(A K,d) = G(A K) n E(A K,d) the above conclusions hold f o r G(A ,d) £ E ( A ,d) as we l l . , K K .3.2. Minimal Algebras Let M be a 1-connected minimal algebra of f i n i t e type. Recall 37 that t h i s means, i) M i s a d i f f e r e n t i a l graded algebra over Q . i i ) iH 0 = Q , M 1 = 0 i i i ) M i s free as a graded commutative algebra. iv) d(K) £~M+-M+ where M+ = e f l " . n>0 v) dim^M n < °° for a l l n _> 0 . (See [8] or [10] f o r d e t a i l s ) . In many i n t e r e s t i n g cases such an algebra i s not f i n i t e l y generated so one cannot apply the preliminary remarks of t h i s chapter d i r e c t l y . But i n the l i g h t of chapter 2 one can do just,as w e l l . For a minimal algebra M define M £ M as the subalgebra n k generated by ffi M . C l e a r l y M Is a f i n i t e l y generated minimal algebra k<n n and H can be written as . Q = M, £ M _ £ • • • £M £ u M = M . 1 2 n n n>0 where each l e v e l of the f i l t r a t i o n i s canonical and f u n c t o r i a l (see [8] or [10]). Thus 3.1 applies to G •= Aut (M ®K) £ End(M ®K) = E n n n n i . e . G^ i s a p r i n c i p a l open a f f i n e imbedded Q-group i n the Q-affine semi-group E^ . . By f u n c t o r i a l i t y of the above f i l t r a t i o n we have the following commutative diagram of Q-varieties 38 n ,m * -G. m n > E where r i s the r e s t r i c t i o n of. maps. Here, of r m n,m n,m course n > m . Cl e a r l y i) r i s a Q-morphism of v a r i e t i e s n ,m i i ) r (f °g) = r (f) °r (g) n,m n,m n,m i i i ) r = 1 n,n • E iv) r °r = r n,m Jt,n £,m Thus we have an inverse system {E , r ' } of Q-affine algebraic semi-groups n n,m which r e s t r i c t s to an inverse system {G ,r '} of Q-groups. n n,m Before studying the properties' of the i n v e r s e . l i m i t of t h i s system we assemble some of the elementary properties of minimal algebras. 3.3. Q-structures on minimal algebras For the purpose of studying algebraic groups over Q , the r a t i o n a l numbers, i t i s customary (and convenient)to work over K , an algebraic closure of Q and keep track of the r a t i o n a l i t y p r o p e r t i e s . In 3.2 minimal algebras were defined as a s p e c i a l kind of Q-algebra. The obvious modification of t h i s d e f i n i t i o n y i e l d s minimal algebras over any f i e l d of c h a r a c t e r i s t i c 0 . . . . . D e f i n i t i o n : a) Let M be a minimal algebra over K (an algebraic closure of Q) . A 0-structure on M consists of a minimal algebra M 39 over Q , such that i) M £ Jl i i ) K ® M > M- i s a d.g.a. isomorphism (where d(a®x) = a®dx) k " • b) A Q-morphism f : M ——> N of minimal algebras with Q-structure i s a d.g.a. morphism. f such that f(M) £ N . The set of Q-endomorphisms of M i s denoted End(M)(Q) and the i n v e r t i b l e ones Aut(M)(Q) . C l e a r l y , the diagram • l®o _ End(M) > End(M) Aut (M) Aut(M) l®o i d e n t i f i e s Aut.(M) and End (M) with Aut(M)(Q) and End (M) (Q) respectively. For n otational convenience we l e t E = End(M )/ E (Q) = End (M ) (Q) n n n n G = Aut(M '), G (Q) = Aut (M~ ) (Q) . n n n n Then G i s a Q-group and G (Q) i t s subset of Q-rational points, n n In what follows we s h a l l always adhere to the following notation. M i s a minimal algebra over K (the algebraic closure of Q) with Q-structure M £ M /T1 = {.x e M | degree x = n} tt = the subalgebra of M generated by © . n K<n 40 M + = & 'K> 1 D(M) = y (Fl + ® B +) where u : Fi. ® M. > M i s the algebra structure . Q(H) = M+/D (M) • ' ' • 2(M) = {x e M | dx = 0} B(M) = {x e M | x = dy f o r some y e M} H(K) = Z(M)/B(M) S(M) = Z(M)/(Z(M) n D(M)) Each of the above i s n a t u r a l l y graded, Z(M) c i s a subalgebra and B(M) £ Z(M) i s an i d e a l . 1 . I f V i s any vector space over K a Q-structure on V i s a Q-vector space V £ V such that K ® V > V i s an isomorphism. Q Proposition 3.3.1: Let M £ IT be minimal algebra with Q-structure. .Then M £ H induces Q-structures on M n , ft , D(M), Q(M) , Z(M), B(M) , H(M) n and S(M) . Thus for example M c i s a Q-structure on M i n the above n ~ n n . sense. Proof: The proof i s e n t i r e l y elementary and does not depend on the properties of minimal algebras. As a sample l e t M = © N n . Then K ® (®M") z > H = © j ^ ' and K ® (© M n) = © (K ® M n ) . Q Q Thus K ® Hn — — — > M11 . q.e.d. Q = / 41 We now prove a simple but c r i t i c a l lemma that allows us to apply the r e s u l t s of chapter 2 to a s u f f i c i e n t l y large c l a s s of minimal algebras. R e c a l l that for a minimal algebra M , G = Aut(M ) , and n n . r : G > G , i s the r e s t r i c t i o n of maps-. n,n-l n n-1 Lemma 3 . 3 . 2 : Let M be a minimal algebra with Q-structure M c M . Suppose that s" (M) = Z°(M)/(z°(M) n D(M)) = 0 . Then r , : G > G , n,n-l n n-1 1 S Q-proper (see d e f i n i t i o n 2 . 2 . 3 ) . Proof: Let K = ker r . . Then K = {f e Aut (M ) I f I r r = l r r } . n n,n-l n n 1 M 1 n_ n-1 n-1 Since s n(M) =0 , f e K implies that n f = 1 + n where — — n » r\(F^) £ Dn(M) Thus f i s unipotent and consequently. K i s a unip'otent n subgroup of G . By Theorem 1.1.10 r , i s Q-proper. q.e.d. n n,n-l Thus we can now properly examine the inverse l i m i t of {G , r } . n n,m introduced at the beginning of t h i s chapter. R e c a l l i n g notation we have G = Aut(M ) and r : G > G i s the r e s t r i c t i o n of-maps. Let n n n,m n m G = lim G and G(Q) = lim G (Q) . n -(- n Theorem 3 . 3 . 3 : Let M- be a minimal algebra with Q-structure M c M . I f dim S(M) < co then G = lim G i s a Q-pro-group ( D e f i n i t i o n 2 v 2 . 3 ) . Thus K n any two maximal Q^.split p r o - t o r i S and T are conjugate under G(Q). . Proof: By lemma 3.3.2 a l l maps r : G > G are 0-proper f o r n r ra. n m I — n,m > N , where N i s chosen so that S (M) = 0 for I > N . Thus Theorem 2.3.2 applies and the theorem i s proved. q.e.d. Remark: There are other classes of pro-algebraic subgroups of G which s a t i s f y the conclusion of Theorem 3.3.3. Our i n t e r e s t i n t o r o i d a l symmetry i s prompted by i t s usefulness, where applicable, i n s t r u c t u r a l a n a l y s i s . 3.4. Weight S p l i t t i n g s of Minimal Algebras D e f i n i t i o n : Let W be a minimal algebra with Q-structure M £ M . An n-weight s p l i t t i n g of M i s a d i r e c t sum decomposition M = © M (over a e Z Q) such that i) j M m = © ( Mn M m) n a a e Z i i ) d( M) £ M a a i i i ) ..M A M £ M a 6 a + g (where M A - M = p ( M ®.H) ) a 3 a B C l e a r l y such a decomposition M = 9 M induces M= 9 M • 1 • e n a n a a e Z a e Z where M = M « K . A A Q •• In general a minimal algebra M with Q-structure M £ M may possess many n-weight s p l i t t i n g s (for various n) . The following theorem implies that weight s p l i t t i n g s , and ( 0 - s p l i t ) t o r o i d a l symmetry are e s s e n t i a l l y the same. Recall that K i s the set of non-zero algebraic 43 numbers and G = Aut(M) where i s a minimal algebra with Q-structure M c M .. Theorem 3.4.1: Each n-weight s p l i t t i n g M = © • aM determines a aeZ * * _ ' • Q-group morphism <j> : K x ... x K > Aut(M) and conversely. n factors . Proof: I f M = © aM i s an n-weight s p l i t t i n g define ae Z * * _ * i <j> : K x ... x K ——> Aut (H) as follows. Let t = (t , . . . , t ) £ K x . . . x K I n n a i a n and x e M where a = (a, , .. . ,a ) £ Z . Then <J> Ct) (x) = t, • . . . • t • x . a 1 n I n <}> (t) i s uniquely determined by these conditions since M = ® n • <Ht) ° d = d ° <j> (t) • since d( aM) c aM and oi£Z <j)(t) (x-y) = <j)(t) (x)-<Kt) (y) since • A M A g« £- a +p' M * Conversely, i f o>' : K x ... x K > Aut CM) i s a Q-group * * * morphism l e t T = <j> (K x ... x K ) and consider X(T) = Horn (T, K ) I t Q-group i s well known that X(T) = z n f o r some n (see [4] p. 205). For a £ X(T) l e t JA = {x £ M | $ (t) (x) = a(<|)(t)) • x f o r a l l o> (t) £ T} . C l e a r l y M = © aM since the aM are j u s t the various eigenspaces of a£X(T) . T which i s diagona'lizable. q.e.d. Proposition 3.4.2: Let M = ^ aM be an n-weight s p l i t t i n g and l e t a£Z •* * — <J> : K x ... x K > Aut (M) be the corresponding Q-group morphism (as i n the proof of 3.4.1). Then t h i s induces the following d i r e c t sum 44 decompositions. M = © „K , B.(M) = © a B ( M ) , Z ( M ) = © „Z(M). , n • a n a H(M) =9 a H ^ ' Q.(") = 9 ' = 9 a S ( " } • * * Furthermore, a l l eigenvalues of 9 ( K x . . . x K ) are generated m u l t i p l i c a t i v e l y on S ( M ) . • * * — — Proof: Let T = <j> ( K x . . . x K ) . Each of the above spaces (Mn, B(M), etc.) are f t i n c t o r i a l and the induced maps on automorphism groups are maps of Q-groups. For example B : A u t ( M ) > A u t ( B (M ) ) i s a Q-group morphism. But then B(T) i s a Q - s p l i t torus (see [4] p.219) and thus diagonizable over Q . Hence we can write B ( M ) = © a B ( M ) . To prove the second p a r t we observe that Q(M) = S (M) © h(M) (unnaturally). Inductively x e h n (M) implies that 0 f dx e ^ n _ - j _ (where x represents x e h n(M)) so that the condition d( aM) £ aM implies that the eigenvalues corresponding to h n(M) are determined by ^ . q.e.d. 3.5. Types of Weight S p l i t t i n g s We now si n g l e out some of the most important types of weight s p l i t t i n g s . D e f i n i t i o n : Let M = © aM be an n-weight s p l i t t i n g of M , a e Z ' . a = (a. , . .. ,a ) . 1 n n i) I f Hm(M) = 0 whenever V a. / m then © „rl i s c a l l e d a a . , 1 „n u i = l aeZ 45 homology diagonal weight s p l i t t i n g . ' i i ) I f 'M = 0 for a. < 0 and o^ = FP then © „M i s ct i n « cteZ c a l l e d a positive, weight s p l i t t i n g . , i i i ) I f M = 0 f o r a. < 0 then © aM i s c a l l e d a non-negative weight s p l i t t i n g . iv) . I f aM = 0 for a ^ 0 then 9 a M i s c a l l e d the, t r i v i a l cteZ weight s p l i t t i n g . ' • ' Remark: i) In [17] i t i s shown that a minimal algebra M has a homology diagonal weight s p l i t t i n g i f and only i f there ex i s t s a d.g.a. map <j> : M -> H(M) such that : commutes. A t o p o l o g i c a l space X i s c a l l e d formal i f i t s minimal algebra M(X) s a t i s f i e s • t h e above property (since i n such cases H(X) i s a formal * consequence of H (X;Q)) . E a s i l y v e r i f i e d examples of formal spaces include spheres and Eilenberg-Maclane spaces. Using the c l a s s i c a l Hodge theory and other techniques i t i s shown i n [7] that KghIer manifolds are formal. i i ) Morgan [15] proves that open smooth complex v a r i e t i e s have p o s i t i v e weight r a t i o n a l homotopy type ( i . e . : The associated minimal 46 algebra has a p o s i t i v e weight decomposition). Ac t u a l l y he proves more than t h i s so a b r i e f summary i s i n order. By the work of Hironaka [11] any non-singular open complex v a r i e t y X can be written X = V - u D where V i s smooth and p r o j e c t i v e x and a r e smooth d i v i s o r s of V with normal crossings. Generalizing Hodge theory Deligne [5], [6] obtains mixed Hodge structures f o r the de Rham cohomology of such v a r i e t i e s . By carrying t h i s construction through, f i r s t to the d i f f e r e n t i a l forms and then to the minimal algebra Morgan obtains h i s r e s u l t . He also observes that the r a t i o n a l homotopy (minimal algebra) of an open smooth v a r i e t y i s a "formal consequence" of the cohomological properties of i t s Hironaka r e s o l u t i o n . Proposition 3.5.1: Let M = © aM be a non-negative n-weight s p l i t t i n g . o t e Z • Then QM i s a minimal algebra with Q-structure 0 M £ QM . Proof: The conditions QM A Q M C Q M ' and d( QM) £ QM imply that Q M i s a d.g.a. and c l e a r l y K ® M ——> M i s an isomorphism. Q ° ° Since the weight s p l i t t i n g i s non-negative +M = © a H i s a d i f f e r e n t i a l graded i d e a l and c l e a r l y QM = M/+M . Thus M > M >> M / M = M , and so _M i s a r e t r a c t of M . From t h i s o , + o o i t i s easy to deduce that Q(QM) = QQ(M) and D(QM) = D(M) n 0M . Thus M i s minimal. q.e.d.. 47 3.6. 1-parameter subgroups and Weight S p l i t t i n g s Notation: Let M be a minimal algebra with Q-structure M- c M such that dim S (M) < °° . i f M has a p o s i t i v e 1-weight s p l i t t i n g M = © aM K aeZ we say M has p o s i t i v e weights. The category of a l l such algebras w i l l be denoted P . By standard abuse of notation we write M e P i f M has p o s i t i v e weights. At the other extreme we l e t W be the subcategory of a l l minimal algebras M (with the above f i n i t e n e s s condition) which have no n o n - t r i v i a l nbn-negative weight decomposition. Again we write Me N i f M s a t i s f i e s the above property. Proposition 3.6.1: i) M e P i f and only i f there ex i s t s a one parameter subgroup A : K > Aut(M) such that l i m A(t) = e_ . (Recall e_ i s t->0 . M M the composite M -> K ——> M) . i i ) M e W i f and only i f for any one parameter subgroup * — * — A : K > Aut(M),A(K ) i s closed i n End(M) . i i i ) M possesses a n o n - t r i v i a l non-negative 1-weight s p l i t t i n g * — i f and only i f there e x i s t s a one-parameter subgroup A : K ——> Aut(M) e(M) 1e(PT) such that lira A(t) = e some idempotent e ¥• 1_ and A (t) t+0 M Proof: For the "only i f " parts of i) and i i i ) l e t M = © aM be a aeZ non-negative 1-weight s p l i t t i n g . Define A : K > End(M) as follows, X(t) (x) = t a - x i f x e aM , a f 0 0 i f x € aM , t = 0 , a ± 0 i f x. e QM, 48 Then A = A • i s a one-parameter subgroup such that K lira X(t) = e i s an idempotent € ^ 1_ . t->0 M In case the given weight s p l i t t i n g i s p o s i t i v e .( M = M° = Q) A(0) = e . The " i f " parts of a) and c) follow from Theorem 3.4.1 and the f a c t that i f a one-parameter subgroup converges as t > 0 then a l l non zero eigenspaces have non-negative weights. i i ) follows from i i i ) since a convergent 1-parameter subgroup always converges to an idempotent. q.e.d. Remark: In the proof of Proposition 3.6.1 we made no reference to the pro-algebraic character of the r e s u l t s . We can get away with t h i s here because our assumptions include the existence of a c e r t a i n l i m i t i n g object. Our r e s u l t s on inverse l i m i t s are only needed for coherence and existence information... Theorem 3.6.2: Let • M be a minimal algebra with Q-structure M £ M such that dim S(n) < » . Then there e x i s t s a minimal algebra Mil) K unique up to d.g.a. isomorphism such that i ) There e x i s t s d.g.a. maps j : M(l) > M' and p : M > M (1) with p ° j = 1_ . • Mil) * i i ) There e x i s t s a one-parameter subgroup A : K. >'Aut(M) such that lim A(t) = J,°P, • t-K) i i i ) I f N —:—> M > N s a t i s f i e s i ) and i i ) above then there 3 q . • e x i s t s • r' : N. > M(l) and I : M(l) -—•> N 49 such' that r ° I = 1_ K ( D Remark: For M(l) and N as above i t i s also easy to deduce from the * — following proof that there e x i s t s a. one parameter subgroup oi : K ——> Aut(N) such that lim w(t) = £ ° r . Proof: C l e a r l y any two d.g.a.'s s a t i s f y i n g , i ) , i i ) and i i i ) are isomorphic. Let T c Aut(M) be a maximal. Q - s p l i t pro-torus (2.3.4). Since dim S(M) < «••, dim T < °> (see 3.4.2). K Let F c T (closure i n the p r o j e c t i v e l i m i t Z a r i s k i topology) be defined as follows. * e. e r i f there e x i s t s a one parameter subgroup A : K > T i . i such that lim A.(t) = e, . Then F i s a f i n i t e commutative semi-group because T i s f i n i t e dimensional and commutative (2.4.2), and the product of one-parameter subgroups is' yet another. Thus, l e t e = n e . e r e.o 1 . I Of course, Q = lim A(t) where t->0 * * * A : K —7—> K x . . . x K ——-> T A 11A . .• •' .1 A (t) = (t, ... . ,t)' . , and (IIA.)(t t ) = IIA.(t.) . i ' 1 n l x 50 Thus l e t M(l) = e (M) £ M . C l e a r l y M(l) s a t i s f i e s conditions i) and i i ) . Suppose (N, j , q) s a t i s f i e s i ) and i i ) above. Then g ° j = 1^ * and there e x i s t s w : K > Aut(M) such that lim u)'.(t) = j ° q . — — • " — - _t-o But then j ° q e w (K ) and w (K ) c s where S i s some, maximal Q - s p l i t pro-torus of Aut(M) . By Theorem 2.3.4 there e x i s t s u e Aut(M) (Q) = AutCM) such that u S u 1 = T .. Thus e ' = u(j°q)u 1 e T i s an idempotent.so that Q' e T . By d e f i n i t i o n of Q above . ^ ' - o e = e 0 e' = e . C l e a r l y t h i s implies that Q (M) = M (1) i s a r e t r a c t of e' (W) = N . q.e.d. 3.6.3: Notice that t h i s process can be i t e r a t e d as follows. For each M s a t i s f y i n g the conditions of Theorem 3.6.2 we can form M(l) c M -> M(l) , p o j = i _ * — and • ''X : K > Aut (M) such that lim X (t)•= j ° p -t-'O But then Theorem 3.6.2 applies to M(l) . Thus we can f i n d (M(2), j 2 , p 2) such that M(2) £ M(D " > M (2) , • p ° j = 1 j 2 P2. 2 2 H(2) * — and . ' X 2 : K > Aut(M(l)) such that-lim X (t) = j o p 2 , (where J 2 ° p 2 i s "smallest" i n the sense of c>0 2 " -51 condition i i i ) of Theorem 3.6.2). Continuing i n t h i s way we obtain a f i n i t e c o l l e c t i o n j £ , p £ , VC=1 such that i ) M(£) £ M(A-l) > M(£) and L p ° j = 1_ * _ i i ) X . : K > Aut(M(£-l)) i s a one parameter subgroup such that lim X <t> = j £ ° p -t-H) i i i ) If N c.rt(£-l) -—> N . s a t i s f i e s i ) and i i ) then M(2.) i s a i r e t r a c t of N . By proposition 3.6.1 i i i ) the one-parameter subgroup * X ^ : K > Aut(M(£-l)) induces a non-negative 1-weight s p l i t t i n g M(il-l) = © aM(S,-l) , where J l U - 1 ) = M(2.) a e Z Let I(rtU) +) be.the i d e a l i n H(£-l) generated by M(£) + = o M ( £ - l ) + Then //(M(£)+) i s stable under the d i f f e r e n t i a l so M(S,-l)/I(M(t) +) makes sense as a d.g.a.. . For s i m p l i c i t y we write M(£-l)//M(£) for rT(£-l)/f(M(£)+) . Then M //rt (£) i s a minimal algebra. The d i f f e r e n t i a l i s decomposable because the d i f f e r e n t i a l of M(fc-l) i s decomposable. M (1-1).//M (I) i s free as a graded algebra because MU) i s a r e t r a c t of MU-1) Along with S u l l i v a n [17] we c a l l M(£) —T->-H(A-1) J > H(fc-1)//M(£) T „ T T „ 52 an algebraic f i b r a t i o n with se c t i o n . 3.6.3. We can rephrase our r e s u l t s as follows. For any minimal algebra M s a t i s f y i n g the conditions of Theorem 3.6.1 there e x i s t s a tower of algebraic f i b r a t i o n s with sections. (*) M//M(l) << M j H(l)//M(2) <<- M(l) = M(l) rt(2)//W{3) « M(2) = M(2) m m M(m) =• W(lti) and one parameter subgroups X : K > Aut(M(£-l) ) which induce maps of algebraic f i b r a t i o n s . 53 M(£) V U -n(i-i) ——-> M(£-l)//M(£-l) -> M -U -U / /MU) Note: i ) The above data i s e s s e n t i a l l y unique i f we require each (M(£-l), M(£)) to be chosen i n accordance with Theorem 3.6.2. i i ) Each M(£-1)//M(£) has p o s i t i v e weights induced from A^ : K > Aut(M(£-l)//M(£)) because M(£) i s the eigenspace of •. . * A^(K ) with eigenvalue 1 . i i i ) Since the process has to terminate the minimal algebra M (m) * has the property that i f A : K > Aut(M ( T i i ) ) i s a one parameter subgroup then A(K ) i s closed i n End(M(m)) . Thus (*) can be viewed as a composition s e r i e s f o r the minimal algebra JM . The successive quotients have p o s i t i v e weights and the serie s terminates with a minimal algebra M-(m) that has no n o n - t r i v i a l non-negative weights. Equivalently, Aut(M(m)) i s closed i n End(M(?n)) . 3.7. Products, Coproducts and Toroi d a l Symmetry The coproduct i n the category of minimal algebras i s the graded tensor product ([8] p. 3 ). I t i s u n l i k e l y that t h i s category has products.. However/ the homotopy category of minimal algebras and d.g.a. homotopy classes of maps ([8] and [17]) has products. We can construct the product as follows. Let M , N be minimal algebras (with Q-structure). Define M x N = rt+ © N + © K . Then H x N i s a 1-connected d.g.a. with coordinate wise d i f f e r e n t i a l , addition and m u l t i p l i c a t i o n . Here, of course (M x N ) " = © N 1 1 i f n > 0 . I f i s easy to v e r i f y that that M x N i s the product of M and N i n the category of connected d i f f e r e n t i a l graded algebras. But M x N i s not a minimal algebra i n general. To remedy t h i s s i t u a t i o n we proceed as follows. By Theorem 2.5 of [8] there e x i s t s a minimal algebra rt ® N and a morphism p : M O N ——> Jl x N of d i f f e r e n t i a l graded algebras, * * such that p i s an isomorphism on H Suppose f : A > M , g : A > N are morphisms of d.g.a.'s (preserving the t a c i t l y assumed Q-structures). Then there e x i s t s a map 9 : A > M x N such that P N commutes. By Theorem 5.19 of [10] there e x i s t s f ® g : A -unique up to d.g.a. homotopy such that . f © g ^ - - 7 w ° N M X N commutes up to homotopy. -> M O N Thus homotopy commutes. To prove uniqueness suppose f - f : A > M and g - g' Then there e x i s t s F : A 1 > M G : A > N such that F ° X = f , F ° A = f o 1 G ° X = g , G ° X = g 1 o 1 Thus define H : A -—> rM x N as H = ( F x G) ° A (where A(x) = (x,x)) . Then H ° X = f-Iig and H o \ = f<J[g' o 1 Retracing our steps we have proved that whenever and g : A > N are d.g.a. maps there e x i s t s a map f <S> < unique up to homotopy such that M — " " i s> g — A — M O N % ° P N 5 6 homotopy commutes. Before r e l a t i n g these ideas to t o r o i d a l symmetry we observe that H + ( M o N ) - Z ^ - > H + ( M x N) —^ — > H + ( M ) e H + ( N ) and that M , N are P P * P N r e t r a c t s of M o N . The subcategory-of minimal algebras which exhibits the most accessible r e l a t i o n s h i p between t o r o i d a l symmetry and coproducts and products i s P . R e c a l l that P i s the category of minimal algebras which have p o s i t i v e weights. Let Z n = Z x ... * z and Z. ——> Z° be the i n c l u s i o n on the i n factors i - t h f a c t o r . Define VZ. = u z. £ Z n . 1 i=L 1 Theorem 3.7.1: Let M = © aM be a p o s i t i v e n-weight s p l i t t i n g and l e t a e Z M. = • © M . Then i „ a . a e Z . i i) Each M_ i s a minimal algebra. n n i i ) There are maps ® M . — > M > •' o M . . 1=1 J i i 1=1 i i i ) l i j . i s an isomorphism i f and only i f Q(M) = © •• Q(M)' . 1 aevz. i iv) lip. i s an isomorphism i f and only i f H(M) = © H(M) . 1 aevz. l Proof: i) follows from proposition 3.5.1. i i ) For each M. we have maps. M. — > M — — > M. such- that i . 1 3 . P. i 1 1 57 Thus "there e x i s t unique (up to homotopy) maps. n n commute (up to homotopy). i i i ) If -U-j : ® FT •—-> M then QCHjJ : Q (® M J —^-> Q (M) . But © Q('j ) : © Q (M\ ) — Q ( ® -Thus . © Q(j) : © Q(M ).-r~^> Q(M) and © Q(M ) Q(M) • e aQW) a e v z . 1 Thus i f I i i . i s an isomorphism © „QCM) =Q(M) . Conversely, i f I © QW) = Q(M) then • a e v z . i 58 _ . n _ © Q (M) = © Q(M . ) vz. 1=1 l » Q(j/) Q(® M ) Q(M) © Q(M) aevz a l Thus Q.(JLj.) i s an isomorphism. Hence J i j ^ i s an isomorphism because i t induces an isomorphism on (free) generating sets. iv) The proof of iv) i s exactly dual to i i i ) using the f a c t that any map between minimal algebras which induces an isomorphism on cohomology i s an isomorphism'. q.e.d. Remark: In the above theorem i f we l e t e. = j . ° P. '• M ~~~ > M . Then i i i . ^ o = p . and e 0 e = e ° ©. . Condition i i i ) i s then equivalent to i l l l 3 j l n i i i ) ' e. '" e . = £._ i f i / j , and I Q (e.) .= 1 _ • 1 3 M j = l 3 : QW Condition iv) i s equivalent to n i v ) ' e. o e. = e_ i f i ^ j , and £ H(e.) = 1 _ • H(M) Here of course e _ i s the composite M >-K > M 59 Before using t h i s theorem to prove uniqueness of decomposition r e s u l t s for P we must digress s l i g h t l y and prove a stronger version of Proposition 3.6.1. i ) . In order to do t h i s we need a few elementary lemmas about imbedded algebraic groups. Lemma 3.7.2: Let G be an algebraic group and l e t p : G — > G£(n,K) £ M(n,K) be a representation of G , where M(n,K) i s the set of a l l n x n matrices over K . I f 0 ve p(G) ( Z a r i s k i closure) then 0 ye p(T) where T i s any maximal torus of G(Here V = K " ) . Proof: (R. Body ). Let D(n) £ M ( n , K ) be the ( Z a r i s k i closed) subset of a l l matrices with zeros o f f the diagonal. Let x : D(n) > K N be the composite D(n) £ M(n,K) ——> K [ t ] / ( t n ) = where 9 (fi) = det(tl- -M) •. Then x 1 S a f i n i t e dominant map ([12] p. 31 ). Thus i t i s also closed. Without loss of generality there e x i s t s a maximal torus T £ G such that p (T) £ D(n) . Notice that p (T) £ D(n) .. Thus 0 .e x(P (G)) £ x(P (G)) = X (P (T) ) = X(P (T) ) The f i r s t i n c l u s i o n i s true by c o n t i n u i t y . The f i r s t e q u a l i ty i s true since i) X ( D = X(af1a~1) a e G«,(n,K) and i i ) x ( a ) = X ( a ) ( a 1 S the semi-simple part of a (1.1.4) ). s s The second equality follows from the f a c t that x i s a closed map. But since X _ 1 ( 0 ) = {0 v},0 v e p(T) . q.e.d. 60 Lemma 3.7.3: Suppose p : G —> G2,(n,K) £ M(n,K) i s a representation of the algebraic group G such that 0 y e p(G) . Then there e x i s t s a * one-parameter subgroup ' A : K. > p(G) such that A extends to A : K — — > p.(G) with A(0) = 0 v Proof: By lemma 3.7.2 0^ e p(T) where T £ G i s some torus. But p(T) £ p (T). i s a t o r o i d a l imbedding i n the sense of [13]. Furthermore 0 y i s a f i x e d point under the action p (T) x p (T) > p (T) * Thus by Theorem 2 of [13] there e x i s t s a one-parameter subgroup A : K > p(T) such that A extends to A : K > p (T) with A(0). = 0 • .' q.e.d. R a t i o n a l i t y properties are also n i c e l y preserved under t h i s construction i n the following sense. Lemma 3.7.4: Let G be a k-group and p : G ——> G£(n,K) £M(n,K) be a k - r a t i o n a l representation. I f 0 y e p(G) then there e x i s t s a one parameter subgroup A : K* > p(G) defined over k such that A extends to A : K —-> p(G) with A(0) = 0 v Proof: By lemma 3.7.3 there e x i s t s a one parameter subgroup A :' K > p.(G) extending to A : K — : — > p(G) with A(0) = 0 y . Now A(K ) £ S some maximal torus S defined over k (see p. 4 of [9]). By p. 219 of [4] we can write S = S ' x s where S, i s k - s p l i t and S i s k-anisotropic. d a d a Thus Horn (K ,S) = Horn (K ,S,) © Horn (K ,S ) . Thus we K-group K-group d K-group a 0 61 * * can write A = (A . A ) . But Hoin; (K ,sj = Horn, (K , S j d a K-group d k-group d * - * because S, and K are k-split. Thus A : K > p (G) i s a d d k-group morphism. Furthermore by p. 218 of [4]- s A = ( ' ^ e r S ) acHorn, (S ,K ) k-group Thus det "'"(l) 2 s • Hence A extends to 0 and a d • \ (0) = 0 q.e.d. d v Lemma 3.7.4 applies at once to y i e l d the following version of Proposition 3.6.1 i ) . Proposition 3.7.5: Let M be a f i n i t e l y generated minimal algebra with Q-structure M c M . Then the following are equivalent i ) M has p o s i t i v e weights. i i ) E _ e Aut(M) c End(M) . M * i i i ) There e x i s t s a one parameter subgroup A : K ——> Aut(M) extending to A : K > End(M) with A(0) = e_ . M Proof: By proposition 3.6.1 we need only prove that i i ) implies i i i ) . Since M i s f i n i t e l y generated there e x i s t s a f i n i t e dimensional subspace V £ M such that p : Aut (M) > Gil (V) i s a representation s a t i s f y i n g the assumptions-of Lemma 3.7.4. q.e.d. Proposition 3.7.6: Let M be a minimal algebra with; Q-structure M £ M such that dim S(M) < 0 3 . Then M has p o s i t i v e weights i f and only M K n has p o s i t i v e weights for a l l n >_ 0 . ' Proof: I f M has p o s i t i v e weights then so does each M , because each . n 6 2 weight space aM i s an eigenspace of some diagonalizable map under which M i s i n v a r i a n t , n Conversely, i f dim S (M) < oo then there e x i s t s N > .0 such that • K the r e s t r i c t i o n r : Aut(rt ) > Aut(M ,) has a unipotent kernel n n n-1 (Lemma 3.3.2), f o r n > N . Since each M has p o s i t i v e weights there n e x i s t s an integer L > 0 such that dim T = dim T , f o r n > L , n n+1 where T £ Aut(M_) . i s a maximal Q - s p l i t torus (The • Q - s p l i t rank cannot keep decreasing) Hence i f T c Aut(M ) and T c Aut(M ) are respective n — n n-1 n-1 maximal Q - s p l i t t o r i with r (T ) c T . then r :. T —-> T * ^ n n ~~ n-1 n n n-1 for n > max{N,L} . . By Lemma 3.3.2 and Theorem 2.3.2 there e x i s t s {T • | T £ Aut(M ) 1 n n n. i s a maximal Q - s p l i t torus and r (T ) £ T ) n n n-.l-Now l e t i i = ®- M be a p o s i t i v e wieght s p l i t t i n g n _ .a n acZ (n > max{N,L}) and l e t X K -> Aut(M ) be the associated one-parameter n subgroup (Theorem 3.4.1). By the above remarks we have the following diagram 'n+2 rn+2 n+2 "n+1 Aut(M l .) n+1 "n+1 / T / ^ n+1 • / i / Aut (M n K ~ - > T A . n Aut (fl ) n-1 63 Thus we can f i l l i n the dotted arrows with Q-group morphisms A : K > T „ such that the whole diagram commutes. This ri+£ . n+i determines a morphism of Q-pro-groups * A = lim A : K > lim T = T <• n «- n By Theorem 3.4.1 t h i s determines a 1-weight s p l i t t i n g H = ® 'M . cxcZ Since {a e Z I M ^ 0} i s a d d i t i v e l y generated by {a e Z I S(M) ^ 0} (Proposition 3.4.2) the above weight s p l i t t i n g i s p o s i t i v e . q.e.d. Theorem ,3.7.7: Let M be a minimal algebra with Q-structure M £ M such that dim S(M) <. » . Then the following conditions are equivalent i ) e_ e Aut(M) M i i ) M has-positive wieghts * — i i i ) There e x i s t s a one parameter subgroup A : K -> Aut(M) extending to A : > End(M) with A(0) = e M Proof: i i ) and i i i ) are equivalent by Proposition 3.6.1 and c l e a r l y i i i ) implies i ) . Thus i t s u f f i c e s to show that i ) implies i i ) . • I f e_ e Aut(M) then = e_ e Aut(M ) . Thus by proposition J l n 3.7.5 M n has p o s i t i v e weights (M^ i s f i n i t e l y generated). Since this, holds for a l l n >_ 0 M has p o s i t i v e weights by Theorem 3.7.6. q.e.d. Remark: Condition i) of Theorem 3.7.7 i s a purely t o p o l o g i c a l condition on the imbedding Aut(rt) £ End(M) . The proof of the following proposition exhibits the usefulness of such a theorem. . 64-Proposition 3.7.8: Suppose M i s a f i n i t e l y generated minimal algebra and has p o s i t i v e weights. I f N i s a r e t r a c t of M then N has p o s i t i v e weights. Proof: [2] Suppose N —:—> M > N where r ° j = 1 . C l e a r l y N i s ~ 3 r N f i n i t e l y generated. Define ty : End(M) > End(W) as T|J (f) = r ° f ° j . Then ij; i s a morphism of- Q - v a r i e t i e s . Since 'M has p o s i t i v e weights there ex i s t s a one-parameter subgroup A : K -—> Aut(M) which extends to A : k > End(M) with A(0) = e_ . Thus e_-e A(K) = L £ L (in f a c t equal) ( Z a r i s k i closure i n M M End(M)j . L i s i r r e d u c i b l e (not the union of two proper closed subsets) since K i s i r r e d u c i b l e and A i s continuous. Thus L i s irreducible.. But then ty (L) i s i r r e d u c i b l e because i s continuous. Also ty(L)£ IJJ(L) i s open because ip (L) i s c o n s t r u c t i b l e . Let D = Aut (N* )"n ip(L) . Then 1_ e D 0 . Thus D £ ^ (L) N i s open and non-empty. Hence D £ ^(L) i s open. Since (L) i s i r r e d u c i b l e D = ty (L). . Thus ty (L) £ D £ Aut (N) . Finally., e_ = ijj(e_) e tyCL) £ Aut (N) . Therefore, by Theorem 3.7.7 N has p o s i t i v e . N M weights. q.e.d. Co r o l l a r y 3.7.9: Let N be a minimal algebra with p o s i t i v e weights such that dim S(M) < » . If N i s a r e t r a c t of M then N has p o s i t i v e K weights. Proof: I f N.—^> M > N s a t i s f i e s r°j = 1 then N ——> M ——> N : r 77 n : r n N n n s a t i s f i e s r ° j = 1 for a l l n > 0 . Thus Proposition 3 . 7 . 8 applies n N n and N has p o s i t i v e weights f o r a l l n > 0 . By Proposition 3 . 7 . 6 N n • • ~ has p o s i t i v e weights;. q.e. Proposition 3 . 7 . 1 0 : Let M , N be minimal algebras such that dim S ( M ) < ° ° K and dim S ( N ) < <» . Then the following are equivalent. K - • i ) M and N have p o s i t i v e weights. i i ) M ® N has p o s i t i v e weights, i i i ) M O N has p o s i t i v e weights. Proof: C l e a r l y i i i ) or i i ) implies i ) because and N are r e t r a c t s of M ® N and M © N . Thus we need only show that i ) implies i i ) and that i ) implies i i i ) . i ) implies i i ) . Let M = . © M and N = © N be p o s i t i v e weight s p l i t t i n g s aeZ . aeZ of M • and N r e s p e c t i v e l y . Then fi ® N = © ( © ^ M ® N ) = © ( M ® N ) i s a.positive aeZ weight s p l i t t i n g of K ® N where ( M ® N ) = © M . ® N . a 6+y=a . . Y i) implies i i i ) . Recall that M o N i s a minimal algebra f o r M X N so there e x i s t s a weak equivalence p : M © N > M x N . Suppose M = © M and N = © N are p o s i t i v e weight s p l i t t i n g s aeZ 3eZ 66 f o r M arid N r e s p e c t i v e l y and define A ^ : M * N ——> M x N as a 8 — — * A (x,y) .= (p -x, p -y) for x e M and y e QN and p f 1 e Q p a p C l e a r l y , A i s a d.g.a. automorphism. Thus by Theorem 2.13 of [8] P there e x i s t s to : M ® N > M © N such that © N -^-> Ji x N A p M'© N ~ > M x N • commutes. P to i s an automorphism because the above diagram forces i t to be a weak equivalence. io* : H ( M s> N) > H(M. © N) i s diagonalizable because A i s . * * Thus to = to where oo i s the semisimple part of u> (see Theorem 2.2.5). s s A straightforward induction argument on {(M © N } „ shows that i f n n^ .0 — — . ' • ' • * to € Aut(M o N) ' i s semisimple and to i s diagonalizable then to i s s s s i n f a c t diagonalizable. Thus to i s diagonalizable and induces a p o s i t i v e weight s s p l i t t i n g on H (rt © N) (because A does). By Proposition 3.4.2 to P s induces a p o s i t i v e weight s p l i t t i n g on M © N because the eigenvalues of to are already m u l t i p l i c a t i v e l y generated on S (Ji © N) . q.e.d. s In the language of [ l ] Proposition 3.7.10 says that the category . P of minimal algebras M with p o s i t i v e weights s a t i s f y i n g dim S (M) < oo i s productive with respect to ® and © . K D e f i n i t i o n : Let M e P , M f Q = K . 67 i) I f M = N ® L implies that M = N o r M = L then M i s c a l l e d ®-irreducible. i i ) I f M = N O L implies that M = N or rt = L then r/1 i s c a l l e d O - i r r e d u c i b l e . C l e a r l y any M e P can be written rt = ® M where each i = l M. e P i s ®-irreducible. One s t a r t s with any n o n - t r i v i a l such s p l i t t i n g and r e i t e r a t e s the process on each f a c t o r . The process terminates because dim S(M) < oo . Each 11. e P because of Proposition 3.7.10. K i _ tn S i m i l a r i l y any N e P can be written N = ® N. where each , i = l 1 N'. e P i s ©-irreducible. . " ' i As i n the proof of Proposition 3.7.10 there are weight s p l i t t i n g s such that M = © „tt and N = © N aeZ BeZ .M. = © M and N. = © - ' N 1 a • 1 o P-; where Z = Z' > z" i s the i n c l u s i o n on the i - t h f a c t o r . By Theorem 3.7.1 _ n — Q(M) = © Q(M) = © Q'(M.) a . , l aevz. , i = l I and _ • — • . TI\ — H(N) •=• © DH(N) = © H(N.) . f3evzi D=l The above weight s p l i t t i n g s determine Q-group morphisms * * — : K x ... x K :—> Aut.(M) n-factors and 68 * * \p : K x ... x K -> Aut (N) -m- factors a a _ where <j> (t, , . . . ,'t. ) (x) = t , 1 . . . ' t n • x , x e M , a = (a , .. . ,a ) 1 n 1 n a 1 n . \ A _ and (s ,•. . . ,s ) (x) = s 1 •. . . - s ^ " x , x e , B = (6^, .. . ,8^) . * * * Let A . : K > K x ... x K s a t i s f y X. (t) = (t, t, .. . , t, 1, t, .. . , t) i l n-factors ( i - t h factor) . and a). •:. K -> K x ... x K s a t i s f y u_. (s) = (s, ... , s , 1 ,s, . .. ,s) fti -factors ( j - t h factor) . Then ° A . : K -> Aut(M) and 4*. = ° 0 0 . : K > Aut(tf) are one-parameter subgroups such that lim <J>. (t) •= e . where e. (M) t->-o = M. and lim \\> As) = f. where f.(N) = N. t-K) 1 3 3 3 * 4-Thus {e ,...,e } £ 9 ( K X . . . X K ) c Aut(M) I n . . * * . — and {f . . . . , f } c ijj (K x . . . X K ) c Aut (N) 1 m (1) (2) By Theorem 3.7.1 2 e i = e i (S . ° e = e i f ' i ^ £ 1 M 59 n C3) '. l Q(e ) = 1 _ i = l Q(M) and dually PI) f 2"= f . . 3 .. 3 P2) f o f • = .£ i f j jt K P3) I H( f-) = 1 j = l 3 H(N) Conversely, (again by Theorem 3.7.1) any c o l l e c t i o n {e ,...,e } <= End(M) such that CI) - C3) hold determines a ®-splitting 1 n . -_ n • rt = ' ® e. (M) . i = l 1 • Dually, any c o l l e c t i o n {f . . . . , f } £ End(N) such that PI) - P3) 1 m _ . m _ hold determines a o - s p l i t t i n g N = o f. (N) . • 3 = 1 3 A b r i e f summary i s i n order here. I f M , N e P then a _ n _ ®-splitting M = ® M, determines i) {<?...., e } £ End(M) s a t i s f y i n g CI) - C3) such that 1 • n e . ( M ) = M. . x 1 • • i i ) A maximal Q - s p l i t torus T £ Aut(M) such that {e, e } £ T the Z a r i s k i closure of T . A ©-splitting I n — w _ N = ON determines 3=1 1 . i ) ' {f , . . . , f } c End(r7) s a t i s f y i n g PI) - P3) such that 1 ?a ~ i i ) ' A maximal Q - s p l i t torus . S £ Aut(N) such that {f ,...ff^}.£ S the Z a r i s k i closure of S . Notation: I f {e ...., e } • c End(M) s a t i s f y CI) - C3) and e. (M) i s • . 1 n ~ 1 ®-irreducible f or a l l i , {e ,...,e } i s c a l l e d a ©-splitting.' I n Dually, i f {f , . . . r f ^ } £ EndtN) s a t i s f i e s PI) - P3) and f.(N) i s ©-irreducible for each j , {f„>..., f } i s c a l l e d a ©-splitting. . If {e ,...,c 1 c T and { e e ' • } £ T r are ©-splittings 1 n 1 n then by Theorem 2.3.4 there e x i s t s g e Aut(M)(Q) such that {ge'g \ . . . ,ge' ,g 1} c T . Since T i s commutative (Lemma 2.'4.2) 1 n / - l g g commutes with e_. for i = l , . . . , n ' and j = l , . . . , n . Dually, i f {f^} , {f^} £ S' are ©-splittings then by the same reasoning there e x i s t s h e Aut(N)(Q) such that {hf^,.../hf^.h" 1} £ S . In the language of [1] P i s a f l e x i b l e , productive category of minimal algebras with respect to both ^ - s p l i t t i n g s and ©-splittings Thus Theorem 2 of [I] implies that each n o n - t r i v i a l M e P s a t i s f i e s n i) M = ® M. i= l 1 i i ) Each M. i s ®-irreducible _ n' _ i i i ) I f M = ® M! s a t i s f i e s i) and i i ) then n = n' and i= l 1 M. = M' where cr" is a permutation on n - l e t t e r s . I a ( i ) 71 The dual argument y i e l d s the following. Each n o n - t r i v i a l N e P s a t i s f i e s — m — i) . N = © N . . j = i 3 i i ) Each N . i s ©-irreducible. 3 m' i i i ) I f • N = © N ! s a t i s f i e s i) and i i ) then m=rn' and j = l 3 N. = N' ,. where x i s a permutation of TC\ . 3 T (D) 72 References [I] R. Body, R. Douglas, "Rational Homotopy and Unique F a c t o r i z a t i o n " , P a c i f i c J . Math. 68(1977). [2] R. Body, R. Douglas, "Unique F a c t o r i z a t i o n of Rational Homotopy Types Having P o s i t i v e Weights"., Pre p r i n t . [3] R. Body, D. S u l l i v a n , "Homotopy Types Having P o s i t i v e Weights", Preprint (USCD - La J o l l a ) . [4] A. Borel, "Linear Algebraic Groups", Benjamin, New York (1969). [5] P. Deligne, "Theorie de Hodge I", Actes de Congres International des Mathematiciens, Nice (1970). [6] P. Deligne, "Theorie de Hodge I I " , Publ. IHES 40(1971). [7] P. Deligne, P. G r i f f i t h s , J . Morgan, D. S u l l i v a n , "Real Homotopy Theory of Kahler Manifolds", Inven. Math 29(1975). [8] A. Deschner, "Sullivan's Theory of Minimal Models", U.B.C. Master's Thesis (1976). [9] R. Godement, "Groupes Lineaieres Algebrique sur un Corps P a r f a i t " , S£minaire Bourbaki, (1960/61). [10] S. Halperin, "Lectures on Minimal Models", Universite Des Sciences et Techniques de L i l l e (1977). [II] H. Hironaka, "Resolution of S i n g u l a r i t i e s of an Algebraic Variety over a f i e l d of C h a r a c t e r i s t i c 0", Ann. of Math 79(1964). [12] J . Humphreys, "Linear Algebraic Groups", GTM, Springer Verlag, New York, (1975). [13] G.' Kempf, F. Knuds.en, D. Mumford, B. Saint-Donat, "Toroidal Embeddings I", Lec. Notes i n Math., 339, Springer Verlag, New York,' (1973).. [14] M. Mimura, H. Toda, "On p-equivalences and p-universal Spaces", Comment. Math. Helv., 4(1971). [15] J . Morgan, "The Algebraic Topology of Smooth Algebraic V a r i e t i e s " , . Preprint (IHES) . [16] I. Stewart, "Lie Algebras Generated by Finite-Dimensional Ideals", Pitman Publishing, London, (1975). [17] D. S u l l i v a n , " I n f i n i t e s i m a l Computations i n Topology", Preprint (IHES). 

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