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Algebraic homotopy theory, groups, and K-theory Jardine, J. F. 1981

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ALGEBRAIC HOMOTOPY THEORY, GROUPS, AND K-THEORY by JOHN FREDERICK JARDINE B.Math., Uni v e r s i t y of Waterloo, 1973 M.Math., Univ e r s i t y of Waterloo, 1974 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS We accept t h i s t hesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1981 @ John Frederick Jardine, 1981 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library 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 reference and study. I further agree that permission for extensive copying of t h i s thesis for s c h o l a r l y purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Mathematics::  The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date April 27, 1981 DF-fi ( 2 / 7 9 } i i Thesis Supervisor: Dr. Roy R. Douglas Abstract Let M be the category of algebras over a unique f a c t o r i z a t i o n k domain k, and l e t ih d - A f f ^ denote the category of pro-representable functors from to the category E of sets. It i s shown that ind-Aff i s a closed model category i n such a way that i t s associated homotopy category Ho(ind-Aff^) i s equivalent to the homotopy category Ho(S) which comes from the category S^  of s i m p l i c i a l sets. The equivalence i s induced by functors S^: ind-Aff• • S^  and R^ S^  • ind-Aff^. In an e f f o r t to determine what i s measured by the homotopy groups ir.(X) := ir. (S X) of X i n ind-Aff i n the case where k i s 1 I K . K. an a l g e b r a i c a l l y closed f i e l d , some homotopy groups of a f f i n e reduced algebraic groups G over k are computed. I t i s shown that, i f G i s connected, then 7 TQ(G) = * i f and only i f the group G(k) of k - r a t i o n a l points of G i s generated by unipotents. A f i b r a t i o n theory i s : developed f or homomorphisms of algebraic groups which are s u r j e c t i v e on r a t i o n a l points which allows the computation of the homotopy groups of any connected algebraic group G i n terms of the homotopy groups of the uni v e r s a l covering groups of the simple algebraic subgroups of the associated semi-simple group G/R(G), where R(G) i s the solvable r a d i c a l of G. The homotopy groups of simple Chevalley groups over almost a l l f i e l d s k are studied. I t i s shown that the homotopy groups of the s p e c i a l l i n e a r groups SI and of the symplectic groups Sp 0 converge, n zm i i i r e s p e c t i v e l y , to the K-theory and ^L-theory of the underlying f i e l d k. I t i s shown that there are isomorphisms ir (SI ) = H_(S1 (k);Z) = K 0(k) f o r n > 3 and almost a l l f i e l d s k, 1 - n 2 n _ — and TT (Sp_ ) = H 0(Sp„ (k);Z) = n L 0 ( k ) for m > 1 and almost a l l 1 zm z zm -± z ' — f i e l d s k of c h a r a c t e r i s t i c ^ 2, where Z denotes the r i n g of integers." I t ,is also shown that i r i ^ s P 2 m ^ ~ H 2 ^ S p 2 m ^ ' Z ^ ^ i f k i s a l g e b r a i c a l l y closed of a r b i t r a r y c h a r a c t e r i s t i c . A s p e c t r a l sequence for the homology of the c l a s s i f y i n g space of a s i m p l i c i a l group i s used f o r a l l of these c a l c u l a t i o n s . i v Table of Contents Introduction 1 Chapter I. Algebraic homotopy theory 9 § 1. Preliminary Lemmas 9 §2. The main r e s u l t s 22 I I . Algebraic groups 28 Chapter Introduction . . . . , 28 §1. Preliminaries 30 §2. Path-components of algebraic groups . . . . 34 I I I . F i b r a t i o n s of algebraic groups 40 Chapter Introduction 40 §1. Sheaves and torseurs 43 §2. Devissage . . . . 48 §3. Chevalley groups 56 IV. Algebraic K-theory 63 Chapter Introduction 63 §1. Q u i l l e n K-theory 66 §2. The fundamental group 76 § 3. A vanishing r e s u l t 83 Bibliography 87 V Acknowledgements I would, f i r s t of a l l , l i k e to thank my supervisor Dr. Roy Douglas f o r h i s patience, encouragement, and provocative questions during the preparation and w r i t i n g of t h i s t h e s i s . I would also l i k e to thank Dr. Larry Roberts f o r reading various d r a f t s and suggesting several t e c h n i c a l , grammatical and aesthetic improvements. Dr. B i l l Casselman was h e l p f u l on c e r t a i n t e c h nical points. Dr. John MacDonald i s to be thanked f o r h i s i n t e r e s t i n some of the consequences of Chapter I. F i n a l l y , I would l i k e to thank my f r i e n d and colleague Lex Renner; a series of stimulating conversations with him was the source of much of the material of Chapter I I . 1 Introduction In an a r t i c l e [24] which appeared i n Topology i n 1977, Kan and M i l l e r showed that, i f k i s a unique f a c t o r i z a t i o n domain, then the homotopy type of a f i n i t e s i m p l i c i a l set K can be recovered from i t s k-algebra A°K of Sullivan-de Rham 0-forms. A°K i s the evaluation at K of a contravariant functor A : S[ > M^ from the s i m p l i c i a l set category to the category M^ of algebras over k. For a s i m p l i c i a l set X, A°X i s defined by A°X = _(X,k^), the s i m p l i c i a l set homomorphisms from X to the s i m p l i c i a l k-algebra k^, whose algebra of n-simplices i s given by n k n = k t x 0 ' • • • » x n l / ( I x _ ~ 1)» i=0 with faces and degeneracies induced by d.x. = i J x. ^ - 1 3 < i j = i j > i , and s .x. i 3 x. 1 x+1 x 3+1 3 < i 3 = i 3 > i , respectively. There i s a contravariant functor F : M^ y S_, which i s adjoint to A° i n the sense that there i s an isomorphism S_(X,F°A) = 1^ (A,A°X) which i s na t u r a l i n both v a r i a b l e s . For a k-algebra A, F°A i s defined by s p e c i f y i n g i t s set °f n-simplices to be the set of k-algebra homomorphisms M^CAjk^). What Kan and M i l l e r proved i n [24] i s that the canonical map ri : K — > F^A^K i s a weak homotopy equivalence i f K i s K. a f i n i t e s i m p l i c i a l set. Perhaps the most i n t e r e s t i n g aspect of t h i s r e s u l t i s that i t suggests that there i s a homotopy theory for algebras over a unique f a c t o r i z a t i o n domain k. By t h i s I mean that one would hope to f i n d a closed model structure i n the sense of Q u i l l e n [37, 39] on the category M^ l n such a way that the adjoint functors F^ and A^ would induce an equivalence of i t s associated homotopy category with the homotopy category Ho(S) which comes from s i m p l i c i a l sets. I t turns out that set theoretic d i f f i c u l t i e s a r i s i n g from the construction of products of algebras preclude the existence of such an equivalence of categories. The adjunction homomorphism n :~. X —»• F^A^X i s not a weak equivalence i n A general, even i f k i s a f i n i t e f i e l d . A proof of t h i s f a c t appears i n [23]. The problem referred to above can be avoided by passing to the pro-category pro-M^, whose objects consist of a l l small l e f t f i l t e r e d diagrams i n M^. In [23] i t was shown that, i f k i s an a r b i t r a r y unique f a c t o r i z a t i o n domain, then pro-M^ i s a closed model category. Moreover, there are contravariant functors A: S_ —>• pro-M^ and F: pro-M^ *" §_> which are adjoint and induce an equivalence of the associated homotopy categories Ho(S) and Ho(pro-M^). A and F are c l o s e l y r e l a t e d to A^ and F^; there are canonical isomorphisms 0 " 0 AK - A K and FA - F A f o r f i n i t e s i m p l i c i a l sets K and k-algebras A r e s p e c t i v e l y , and the natural map n : X — > FAX i s a weak equivalence A. 3 f o r a l l s i m p l i c i a l sets X i n a way which generalizes the r e s u l t of Kan and M i l l e r . The proofs of these r e s u l t s are reproduced here; they are the subject of Chapter I. The category of pro-representable functors from M^ to the set category IS w i l l be denoted by ind-Aff . The r e s u l t s of Chapter I imply that there are covariant functors R^: —*• ind-Aff ^.v • S^: ind-Affk • _S and a closed model structure on ind-Aff^_ i n such a way that R^ and induce an equivalence of Ho(S) with Ho(ind-Aff^), and t h i s f or any unique f a c t o r i z a t i o n domain k. For X e S^ , ^ s the functor which i s represented by AX. In view of the f a c t that the i n c l u s i o n k[x^,... .x^] c "Mxo> • • • J x n l induces an isomorphism kfx.. ,... ,x ] = k for each n ^ 0, the n-simplices of S (T) f o r JL XT XT _C T € i n d - A f f ^ are s p e c i f i e d by n S k ( T ) _ = l n d - A f f k ( A i > T ) where A^ r i s the representable functor M^(k [x^ , . . . ,x^ ], ). We s h a l l also write A^ = Spec^(k [ x ^ , . . . , x I')5, i n keeping with the usual d e f i n i t i o n of the contravariant functor Spec^: M^ —*• ind-Aff^, which assigns to each k-algebra A the representable functor Spec^(A) = M^(A, ). Such functors are c a l l e d a f f i n e k-schemes. Applying the functor Spec^ to the s i m p l i c i a l k-algebra k^ gives a c o s i m p l i c i a l k-scheme, which i s denoted by A^. As noted previously, each i s isomorphic to the n hyperplane Spec^(k[xQ,. . . ,x ]/( I x ^ ~ 1)) i n Spec^(k[xQ,. . . , x n H , and 1=0 one can check that the cofaces and codegeneracies of A^ are defined i n the same way as those of the usual c o s i m p l i c i a l space A which i s gotten from the standard n-simplices A n i n baryceritric co-ordinates. For t h i s 4 reason, i s c a l l e d the algebraic singular functor, and i s c a l l e d the algebraic r e a l i z a t i o n functor, with! .respect to k. These constructions appear to y i e l d a natural way of defining homotopy theory i n the a f f i n e algebraic geometric s e t t i n g of ind-Aff over a large c l a s s of bases which includes the integers. (There are other constructions of homotopy theories for schemes; see [2], for example.) How natural i s i t ? In p a r t i c u l a r , what i s measured by the homotopy groups of the spaces S. T a r i s i n g from T e ind-Aff, ? The k ° k rest of t h i s t hesis consists of an attempt to answer t h i s question i n the case where T i s a reduced a f f i n e scheme of f i n i t e type over an a l g e b r a i c a l l y closed f i e l d k. Let me hasten to remark, however, that I do hot yet know any geometric c r i t e r i a f o r an a r b i t r a r y a f f i n e v a r i e t y X to be even path-connected i n the sense that TTQ(S^X) = *. Perversely, path-connectedness i s not a l o c a l property, so that the problem of f i n d i n g such c r i t e r i a seems to be very d i f f i c u l t i n general. But answers begin to emerge i f one assumes that X has more structure. In f a c t , one can use c l a s s i c a l algebraic group theory to show that an algebraic group G over k of a r b i t r a r y c h a r a c t e r i s t i c , which i s connected i n the usual sense, i s path-connected i f and only i f i t s group G(k) of k - r a t i o n a l points i s generated by unipotent elements. This i s the subject of Chapter I I . Because S^G i s a s i m p l i c i a l group and hence a Kan complex when G i s an algebraic group, G i s path-connected i f f o r any r a t i o n a l point x e G(k) there i s a k-scheme morphism co: A^ — > G with UJ(0) = e and u)'(l) = x, where 0 and 1 l i e i n A.1 (k) = k. I f the c h a r a c t e r i s t i c 5 of the underlying f i e l d k i s 0 and x i s a unipotent element of the group Gl^Ck) of r a t i o n a l points of the general l i n e a r group G l ^ , then a "path" OK — • G l ^ from e to x i s defined f o r s e ^ k ^ ) ^y o>(k)(s) = exp'(sr. l o g ( x ) ) . This observation was the s t a r t i n g point f o r the r e s u l t s of Chapter I I . No such r e s u l t may be used, however, i n the p o s i t i v e c h a r a c t e r i s t i c case. There, one must use a well-known r e s u l t which says that the underlying k-scheme of a connected unipotent group U over k i s isomorphic to an a f f i n e space A^, where n i s the dimension of U. The most e f f i c i e n t proof of t h i s r e s u l t uses the theory of fpqc torseurs i n the sense of Demazure and Gabriel [13]. These are defined f o r the fpqc topology analogously to the notion of p r i n c i p a l f i b r a t i o n . For example, any homomorphism iv:G *• H of algebraic groups over k which i s s u r j e c t i v e on r a t i o n a l points i s a K-torseur over H, where K i s the group-scheme kernel of IT. The pullback p: Xx G —*• X H of ir over any k-scheme morphism f: X —*• H i s a K-torseur over X, and f l i f t s to G i f and only i f the induced K-torseur p i s t r i v i a l i n the sense that Xx G i s equivariantly isomorphic to XxK over X. In t h i s way H one sees that the obstruction to l i f t i n g a map f: X —>- H over an algebraic group homomorphism ir: G —*• H which i s s u r j e c t i v e on r a t i o n a l points i s contained i n the pointed set of K-torseurs over X, where K i s the group-scheme kernel of ir. The u t i l i t y of t h i s idea l i e s i n the fa c t that such an algebraic group homomorphism IT i s a f i b r a t i o n i n the sense of Chapter I i f and only i f every k-scheme morphism A^ —>• H l i f t s to G, and t h i s for a l l n > 0. Chapter I I I i s devoted to i d e n t i f y i n g several d i f f e r e n t f i b r a t i o n s 6 of algebraic groups. The long exact sequences which r e s u l t allow one to represent the homotopy group of a path-connected group G over k of a r b i t r a r y c h a r a c t e r i s t i c i n terms of the homotopy groups of Chevalley groups over k which are of u n i v e r s a l type. The s p e c i a l l i n e a r group SI i s univ e r s a l of type A f o r a l l n+± n n ^ 1. I t i s easy to see, using the determinant map, that SI , i s the • n+l * path component of the i d e n t i t y i n Gl ,,, and that TT„(G1 ..,) = k , the n+l 0 n+l group of non-zero elements of k. I t follows that the inductive algebraic group SI i s the path component of the i d e n t i t y e e Gl(k) i n the inductive group Gl, and T\Q(G1) = k . But the homotopy groups of Gl are defined to be the homotopy groups of the s i m p l i c i a l group Gl(k^) which i s gotten by applying the general l i n e a r group functor to the s i m p l i c i a l k-algebra k A, and i t has been known for some time [17] that there i s an isomorphism (1) Tr j.(Gl(k^)) = K ± + 1 ( k ) f o r i > 0, where the groups K i + 1 ( k ) , i > 0, are Quillen's algebraic K-groups of the underlying f i e l d k. A new proof of the isomorphism (1) i s given i n Chapter IV. The main tools which are used f o r t h i s proof consist of Quillen's homotopy property f o r the K-theory of a regular Noetherian r i n g , and a s p e c t r a l sequence f o r the homology of the c l a s s i f y i n g space of a s i m p l i c i a l group. This s p e c t r a l sequence i s also used, together with c e r t a i n r e s u l t s of Matsumoto, to show that there are isomorphisms (2) V ^ n + i ) = H 2 ( S 1 n + l ( k ) ; Z ) " K 2 ( k ) f ° r n " 1 » a n d 7 (3) 7 r i ( S P 2 m ) " H 2 ( S p 2 m ( k ) ; Z ) " K 2 ( k ) f ° r ~ 1 ' where Sp 2 m(k) x S t' i e § r o u P °^ k - r a t i o n a l points of the symplectic group a n C * Z denotes the integers. Such isomorphisms also hold i n some cases where k i s not a l g e b r a i c a l l y closed. In p a r t i c u l a r , one has (4) ^ i ( S 1 n + l ) ~ V S 1 n + l ( k ) ; Z ) = K 2 ( k ) f o r n - 2 ' for SI ., and TT, (SI ,.,) defined over almost a l l f i e l d s k, and n+1 1 n+i (5) V S p 2 m } ~ H 2(Sp 2 m(k);Z) a . ^ ( k ) for m > 1, for Sp- and ir, (Sp„ ) defined over almost a l l f i e l d s k of 2m 1 Zm c h a r a c t e r i s t i c not equal to 2. _^L^(k) i s one of the Karoubi L-groups [25] of the f i e l d k; i t coincides with K 2 ( k ) k i s a l g e b r a i c a l l y closed. Jan Strooker has recently informed me that the isomorphism (4) can also be obtained from a r e s u l t of Krusemeyer [27] by d e s t a b i l i z i n g a r e s u l t of Rector [43] which r e l a t e s the Karoubi-Villamayor K-groups of the f i e l d k to the groups T r ^ ( G l ( k ^ ) ) . I t i s also shown i n Chapter IV that c e r t a i n r e s u l t s of Bass and Tate [4] on of a number f i e l d may be used to show that of an a l g e b r a i c a l l y closed f i e l d k vanishes when the Kroenecker 6 ( k ) of k s a t i s f i e s <S(k) < 1, and i s a n o n - t r i v i a l uniquely d i v i s i b l e abelian group otherwise. Some consequences f o r the homotopy groups of a r b i t r a r y Chevalley groups of type and C^ over k are also discussed. I t seems that much more can be said i n the way of computing homotopy groups of Chevalley groups. For example, I conjecture that, i f i s a Chevalley group over an a l g e b r a i c a l l y closed f i e l d k which i s 8 u n i v e r s a l f o r an indecomposable root system $, then there i s an isomorphism ir^CG^) = H^CG^Ck) ;Z) which generalizes the isomorphisms (2) and (3) above. Beyond t h i s , one would l i k e to know to what extent the homotopy groups of an a r b i t r a r y Chevalley group G^ of un i v e r s a l type over k coincide with the K-theory of k when k i s assumed to be a l g e b r a i c a l l y closed. One also hopes for p e r i o d i c i t y r e s u l t s f o r the groups 7 1 which would r e f l e c t the apparent p e r i o d i c i t y i n Quillen's c a l c u l a t i o n of the K-theory of the algebraic closure of a f i n i t e f i e l d . 9 I. Algebraic homotopy theory §1. Preliminary Lemmas As noted i n the introduction, t h i s chapter contains a reproduction of those r e s u l t s of [23] which are relevant to the present work. We begin by e s t a b l i s h i n g some terminology. This exposition i s not self-contained however; i f necessary, the reader should consult the Appendix of [2] for more d e t a i l s . Let k be a unique f a c t o r i z a t i o n domain, and l e t denote the category of k-algebras. R e c a l l that a pro-object i n ( l e* a n object i n the category pro-M^) i s a contravariant functor T:I_—»- M^ ., where I_ i s a small f i l t e r e d category (see [2, p. 154]),, I f S: J _ — * • i s another pro-object then a pro-map §: T *• S i s defined to be an element of the set l i m l i m M^T^S ), «-J where the l i m i t s are taken over j e and i e I, and M^T^S^.) i s the set of k-algebra morphisms from T^ to S_. . i s a f u l l subcategory of pro-M^ i n such a way that, f o r A e M^, pro-M k(T,A) = l i m M^T ,A) . — * - i This i s a set of equivalence classes of homomorphisms 8: T^ —*• A; the class that 9 represents i s denoted by [0], Thus, a pro-map <|>: T -—~*• S can be thought of as a c o l l e c t i o n of simpler kinds of pro-maps 10 <f>j : T • S , j e J , such that f o r each $: j ' —>• j i n J , the diagram commutes i n the sense that <[>..,; - [S^ ° T ] , where T represents <j>^., This information i s summarized by saying that a c o l l e c t i o n of.maps 3 represents the pro-map <j> i f (ambiguously) [<}>_. ] = <}>.. for every j e J_. With t h i s terminology, i t i s convenient to say, given another pro-object U: K *• and a c o l l e c t i o n of maps representing the pro-map \p: S *• U, that the c o l l e c t i o n L l ( j (k)) j (k) -»- Ufc, k e K, represents the composite IJJ 0 <f>: T —>• U i n pro-M^. Now we describe a contravariant functor A: S pro-M^., where denotes the category of s i m p l i c i a l sets. R e c a l l that the (contravariant) 0-forms functor A°: j[:-.—+ i s defined, f o r X e S, by 11 A°X = S/X,k^), where i s the s i m p l i c i a l k-algebra of the Introduction. Now for X e S^  l e t Fin(X) be the small f i l t e r e d category which has a l l f i n i t e subcomplexes K of X as objects and a l l i n c l u s i o n s between them as morphisms. Then a contravariant functor AX: Fin(X) —*• i s defined on morphisms i : K >• L by 0 0 0 AX(i) = A ( i ) : A L — • A K. I f f: X — * Y i s a s i m p l i c i a l map and K i s a f i n i t e s i m p l i c i a l complex of X then f (K) i s a f i n i t e subcomplex of Y. Let f t : K —>• f (K) be K the r e s t r i c t i o n of f to K. Then i t i s . easy to see that the c o l l e c t i o n A°(f| •): A°(f(K)) • A°K, K e Fin(X) , represents a pro-map Af: AY — • AX, and that the assignment f i—>• Af determines a contravariant functor A: — > pro-M^. I t i s worth noting that A i s the r i g h t Kan extension of the r e s t r i c t i o n of the composition A° ——y c pro-M^ to the f u l l subcategory F i n of f i n i t e s i m p l i c i a l sets, along the i n c l u s i o n of F i n i n S_. Re c a l l that the contravariant functor F°: — • S_ i s defined for A e by r e q u i r i n g the n-simplices ^ A of F°A to be the set of homomorphisms M^CAjk ). S i m i l a r l y , a contravariant functor F: pro-M^ >• S_ i s defined f o r T e pro-M^ by s p e c i f y i n g that FT_ = p r o - M ^ T , ^ ) . 0 Obviously FA = F A for A e M^, and i t i s straightforward to show 12 0 Lemma 1.1: There i s a natural map ?:A X —»- AX f o r X e with c = A ^ ( i ) : A^X —>• A^K for K e Fi n (X) , where i i s the i n c l u s i o n K -1 of K i n X. t, i s an isomorphism i f X i s f i n i t e ; i n t h i s case r. i s represented by 1.0. A X There i s a natural map i | i : pro-M^T.AX) —»-_S(X,FT), such that, f o r g: T — • AX and x e X , n > 0, ^g(x) i s the n composition T - S - AX - ^ U AA n -^L A°A n - U k , n where i : A n — • X i s the c l a s s i f y i n g map for x and j i s the isomorphism A^A n = S(A n,k.) — • k defined by j ( f ) = f ( i ), where — * n n i € A n i s the standard n-simplex. In f a c t , we have n n Proposition 1.2: f i s a na t u r a l b i s e c t i o n , so that A and F are adjoint on the r i g h t . Proof.: By d e f i n i t i o n , i f X e S_ and T: I_—»• i s a pro-object, then pro-ML (T,AX) = l i m l i m K ( T ,A°K), -«—K — * - i 1 where the l i m i t s are taken over K € Fin(X) and i e _I. But there i s a natural isomorphism 0 •* 0 l i m l i m M, (T.,A K) = l i m l i m S(K,F T.), «-K ->i ~* 1 <-K -*1 1 13 i n view of the f a c t that A° and F° are adjoint on the r i g h t , and a nat u r a l map lim l i m S_(K,F°T.) _(X,FT) •<—K - * - i 1 which i s gotten by taking c o l i m i t s . The map c i s an isomorphism, since lim_(K,F°T.) = _(K,FT) —y± for each f i n i t e K e S_. I t i s an exercise to show that ip i s the composition of the isomorphisms c and <j>. QED The key point i n a l l that follows i s Lemma 1.3: Consider the s i t u a t i o n A u I — >• B where A and B are k-algebras, B i s a unique f a c t o r i z a t i o n domain, I i s a non-zero i d e a l of A, and h i s a non-zero m u l t i p l i c a t i v e k-module homomorphism. Then there i s a unique k-algebra homomorphism h^: A > B extending h. Proof: Choose u e I such that h(u) * 0. Then f o r a l l x e A there i s a 8 e B such that x h(xu) = B h(u). In e f f e c t , i f h(xu) = 0 then obviously 3 =0, and i f h(xu) * 0 then 14 the equations h ( x u ) k = h ( x k u ) h ( u ) k - 1 , k > 2, give the r e s u l t , f o r i f p i s a prime and p m|h(xu), p m + x / h ( x u ) and p |h(u), then ( k - l ) r < km for a l l k S 2. Moreover, since B i s an i n t e g r a l domain, 3^ i s unique. Observe that i f there i s another v e I such that h(v) * 0, with corresponding i d e n t i t y h(xv) = yh(v), X then 3 h(u)h(v) = h(xuv) = y h(v)h(u), and so 3 x = y • Now define h^: A • B by h^(x) = 3 . QED Some tec h n i c a l lemmas for pro-M^ w i l l now be l i s t e d . Lemma 1.4: Let I be a small f i l t e r e d category and consider the pullback diagram a X • Y w y z Y o i n the category M^— of contravariant functors I —*• M^, where 6\ : Y^ — > Z^ i s s u r j e c t i v e f o r every i e I_. Let B e M^ be a unique f a c t o r i z a t i o n domain. Then the diagram 15 pro-I^(Z,B) • pro-M^W.B) pro-I^(Y,B) *• pro-M^CX.B) i s a pushout of sets. Proof: Suppose that there i s a commutative diagram of sets pro-M^Z.B) pro-M^Y.B) Y pro-M^W.B) E Take a pro-map $: X —»- B and l e t K. = ker{6.: Y. Z.} = ker{p.:X. —>-> W.} l i l l i l l Then: (i) I f there i s a representative <J> : X^ —*• B of $ such that <t>(K^ ) = 0, then there i s a unique pro-map ty: W —»- B such that = (j>. ( i i ) I f there i s no representative <(>^  of if) which k i l l s K^, then there i s a unique pro-map n: Y —»- B such that na = <|>. The uniqueness of if) i n ( i ) follows from the fact that 3 i s an epi of pro-M^; t h i s w i l l be proven i n Lemma 1.7. To see ( i i ) , consider the diagram X. ¥ Y. a. I 16, By Lemma 1.3, there i s a unique map n : —*• B such that n £ a ^ = •f^ * Lemma 1.3 also guarantees that n = [n ] i s independent of the choice of representative <j> of <|>, and that n i s the unique pro-map such that na = cj>. Now, i n order to get a commutative diagram * Y pro-M^Z,.) > pro-M^W.B) pro-M^Y.B) • pro-^.(X,B) we are obliged to define >(*) = { g(if>) i f <J> s a t i s f i e s ( i ) , and [ f(n) i f <j> s a t i s f i e s ( i i ) . * * A straightforward case check shows that 0a = f and 98 = g, and the Lemma i s proved. QED Corollary 1.5: Let X ——*• Y W be a pullback diagram of k-algebras i n which <5 k-algebra B be a unique f a c t o r i z a t i o n domain, diagram i s a pushout of sets: i s s u r j e c t i v e and l e t the Then the following 17 W^(Z,B) >• M^W.B) M^Y.B) • M^X.B). Proposition 1.6: (Kan, M i l l e r [24]) The natural map n^: K —>• F°A°K i s a weak equivalence i f K i s a f i n i t e s i m p l i c i a l set. Proof: In e f f e c t , each f i n i t e complex K i s b u i l t up v i a a f i n i t e sequence of pushouts . n 3A L A • L \ where L and L' are f i n i t e , and 9 A n i s the subcomplex of A n which i s generated by the (n-l)-simplices d i i n > 0 < i < n. By Corollary 1.5, the induced diagram _0,0 ,n N F A (3A ) V F°A°i F A (A ) - F°A°L Y F°A°L' i s also a pushout of s i m p l i c i a l sets. Moreover, F°A°i i s an i n c l u s i o n of s i m p l i c i a l sets, hence a c o f i b r a t i o n . Now, n .:'A n F°A°A n i s a A n weak equivalence by the Extension Lemma of [5, p. 3]. Thus, i f we as sume that n and n are weak homotopy equivalences, then the Brown 3 A n L Glueing Lemma of [19, p.80] implies that n , i s a weak homotopy equivalence J_J as w e l l . An induction on dimension f i n i s h e s the proof. QED The missing l i n k i n the proof of Lemma 1.4 was 18 Lemma 1.7: (i) Take X, Y: I_ • i n pro-M^. and l e t ir: X > Y be a natural transformation such that TT.: X. *• Y. i s su r j e c t i v e 1 1 I J for every i e _I. Then ir represents an epimorphism of pro-M^. ( i i ) In every pullback diagram a S y X -y Y i n pro-M^ with TT as above, p i s epi. Proof: (i) Take f.g^Y -—>• A, A e M^, with fir = gir, and l e t f^: Y^ *• A and g^: Y^, >• A represent f and g respectively. Then there i s a commutative diagram TT . Y v X - ^ Y -a / 1 y 1 / \ „ / \ \ f i X.„ ^ Y.„ \ l l " A x \ Y<*2 \ / g., a2 X., — Y . t 1 1 TT . . x' so that f.Y = g.,Y and f = g. The general case follows e a s i l y , ( i i ) Suppose that T i s the pro-object T: Z_ > M^. By Section 3 of the Appendix of [2], there i s a small f i l t e r e d category M , c o f i n a l p functors <|>: frL y J and : M_ »• I, a natural transformation — P — ~P — 3 : Ttj) • and a commutative diagram 19 -> Y can can T<f> i n pro-M^, where the v e r t i c a l arrows are canonical isomorphisms which are represented by i d e n t i t y maps. Moreover, the diagram X -»• Y can can X\p »• Y1J1 commutes i n pro-M^, where in}i i s represented by the obvious natural transformation. Thus, by [2, A .4.1], our pullback i s isomorphic to the pullback S' -> Xip Yip of •nib : Xiii m m -»• i s s u r j e c t i v e f or every m e M^  , so that p' i s an epi of pro-M^ by ( i ) QED It i s necessary at t h i s point to r e c a l l b r i e f l y the construction of f i l t e r e d inverse l i m i t s i n pro-M^ from [2, A.4.4]. Let J be a small f i l t e r e d category, and consider a contravariant functor T: J > pro-M^. In p a r t i c u l a r , we have T ( j ) : I_(j) *• M^ for every 20 j £ J . Let K be a category having as objects a l l pa i r s ( j , i ) with j e £ and i e I_(j), and such that a morphism (a,<J>): ( j , i ) y (j ' , i ' ) consists of an arrow a: j y j ' of J, together with a k-algebra map <j>: - ( j 1 ) ^ y T(j") i representing TCa)^. K i s small and f i l t e r e d . Let L: K y be the pro-object which i s defined on morphisms by L(ot,cf>) = <J>. The pro-maps ir_. : L y T(j) with ^ ^ ± represented by ^T(J).; ^ o r m a l i m i t i n g cone i n pro-M^. As an example, take A^ e M^, where the i ranges over some index set X. The product of the A. i n pro-M, , denoted by II A., 1 ic ., ^„ i leX i s the functor P: Fin(X) y M^, where Fin(X) denotes the f i n i t e subsets of X considered as a small f i l t e r e d category, and with P(K) = II A., for K e Fin(X) , ieK 1 t h i s f i n i t e product being taken i n M^. Lemma 1.8: For T: J y pro-M^ as above, i f T(a): T(j') y T(j) i s an epi of pro-M^ for every a: j *- j ' i n J , then the maps TTj = L »- T(j) , j € J, are also epimorphisms of pro-M^. Proof: Take f, g: T(j) y M^, with A e M^, such that fir = gir^ . , and l e t f : T ( j ) i y A and g^,: T C j ) ^ y A represent f and g resp e c t i v e l y . By using the f i l t e r e d condition on i f necessary we can assume that there i s a commutative diagram i n pro-M^ of the form 21 U j , i ) = T ( j ) . L(a,<j>) L ( j " , i " ' T ^ T ( J " ) , ; L(a,i|j) L ( j , i ' ) = T ( J ) . - ; But $ and ^ both represent Ta, so fTa = gTa and f = g. The general case again follows e a s i l y . QED Corollary 1.9: For T with L as i n Lemma 1.8 and A e M^, pro-M^CL.A) = pro-M^lim T.,A) = l i m pro-M^T. ,A) . Proof: Every map L > A i s represented by a map T(j)_^ >• A of 4 - QED Corollary 1.10: Let B e be an i n t e g r a l domain. Then pro-M^C II A i,B) = [_| ^ ( A ^ B ) . This isomorphism i s natural i n B. i e l i e l •3 22 § 2. The main r e s u l t s The reader w i l l r e c a l l [37, 39] that s p e c i f y i n g a closed model structure f o r a category C_ requires the d e f i n i t i o n of three classes of maps, c a l l e d f i b r a t i o n s , c o f i b r a t i o n s , and weak equivalences r e s p e c t i v e l y , such that the following axioms are s a t i s f i e d : CM1: i s closed under f i n i t e d i r e c t and inverse l i m i t s , f g CM2: Given U > T • S i n C, i f any two of g, f and g ° "f are weak equivalences, then so i s the t h i r d . CM3: If f i s a r e t r a c t of g i n the category of arrows of C^  and g i s a c o f i b r a t i o n , f i b r a t i o n , or weak equivalence, then so i s f. CM4: Given any s o l i d arrow diagram i n of the form U > T i / P / V y S, where i i s a c o f i b r a t i o n and p i s a f i b r a t i o n , then a dotted arrow e x i s t s making the diagram commutative i f e i t h e r i or p i s a weak equivalence. CM5: Any map f may be factored as (i) f = p i where i i s a c o f i b r a t i o n and a weak equivalence and p i s a f i b r a t i o n , and ( i i ) f = qj where j i s a c o f i b r a t i o n and q i s a f i b r a t i o n and a weak equivalence. I t should be pointed out that the notions of t r i v i a l f i b r a t i o n and 23 c o f i b r a t i o n , and r i g h t and l e f t l i f t i n g property have t h e i r customary meanings here (see [37]). Now say that, i n pro-M^: (1) f: T *• S i s a c o f i b r a t i o n i f f has the l e f t l i f t i n g property with respect to a l l maps of the form A°(i): A°(A n) • A°(A°), n > 1, 0 < r < n, where i s the subcomplex of A n which i s generated by a l l of the faces of i except d i , and i : A n <=• A n i s the n r n r in c l u s i o n . (2) f: T S i s a weak equivalence provided that Ff i s a weak equivalence of and (3) f: T > S i s a f i b r a t i o n i f f has the r i g h t l i f t i n g property with respect to a l l c o f i b r a t i o n s which are also weak equivalences. .Then we have Theorem 2.1: With these d e f i n i t i o n s , pro-M^ i s a closed model category i f k i s a unique f a c t o r i z a t i o n domain. Proof: CM1 comes from [2, A.4.2]. CM2 and CM3 are t r i v i a l . One shows f i r s t of a l l that the f a c t o r i z a t i o n axiom CM5(ii) holds. Take f: T • S and consider the set of a l l diagrams D of the form D r , n„ T • A°(A ) A°i s > A o ( A V " 24 Form the pullback pr S0 = s 0 nn -> n A U ( A U ) D n A 0 -n A i ( V ; A o , A V y n A (A ) , D "D 0 where II A i i s the obvious natural transformation. A l l of the maps occurring i n II A ^ i are s u r j e c t i v e and so i s epi by Lemma 1.7. " 'A 0 Lemma 1.8, together with Proposition 1.6, ensures that F ( n A i ) i s t r i v i a l c o f i b r a t i o n of S^. Thus, by Lemma 1.4, so i s Fa . Let f • T -> be the unique map such that * n A°(A n D) D * .0 n- A x • n A°(A n D) (3 D) D r D commutes, and now i t e r a t e the construction to produce a tower of maps °3 CT2 °1 - s 3 - — . s 2 — y S l — > s Q = s , together with a cone 25 Let S„ = l i m S., with l i m i t i n g cone <—i TT. : S -* S., i > 0. 1 0 0 l Then f has a f a c t o r i z a t i o n f = IT f , where f i s the unique map U CO 00 induced by the f . , i ^  0. f i s a c o f i b r a t i o n , since any 1 oo 0 n 3: S > A (A ) factors through some S , according to the construction oo r m of f i l t e r e d inverse l i m i t s i n pro-M^. TCQ has the r i g h t l i f t i n g property with respect to a l l c o f i b r a t i o n s , so i n p a r t i c u l a r TT^  i s a f i b r a t i o n . F i n a l l y , one uses Lemma 1.8 to show that Fir^ i s a t r i v i a l c o f i b r a t i o n of jS, and so TT^  i s a weak equivalence as well as a f i b r a t i o n . The f a c t o r i z a t i o n CM5(i) i s obtained s i m i l a r l y , by using the fac t that f: T > S i s a t r i v i a l c o f i b r a t i o n of pro-M^ i f and only i f f has the l e f t l i f t i n g property with respect to a l l maps A°i: A°(A n) — • A ° ( 3 A n ) , n > 0, induced by the i n c l u s i o n i : 3 A n c A n, where "3A° =0 by convention. The n o n - t r i v i a l part of CM4 i s a standard consequence of the construction which i s used f o r the proof of CM5(ii). QED Theorem 2.2: F and A induce an equivalence of Ho(S) with HoCpro-M^) Proof: We begin by showing that the unit of the adjunction, n x: X • FAX, i s a weak equivalence f o r a r b i t r a r y X e S^. F i r s t of a l l , one uses the n a t u r a l i t y of the map £ of Lemma 1.1 together with the fac t that n (x) A. for x e X , i s the composition 26 A i -1 s. ' x ^ £ ~ j ... A A n ,U,n AX > AA \ • A A >• k , n to show that there i s , f o r every s i m p l i c i a l set X, a commutative diagram n x _ X >• FAX n x I Ff F°A°X = FA°X Thus, by Lemma 1.1 and Proposition 1.6, n i s a weak equivalence f o r A f i n i t e X e S_. For the general case, X can be regarded as a f i l t e r e d c o l i m i t X = l i m K >• KeFin(X) i n JS, while by Corollary 1.9, FAX = l i m FAR . KeFin(X) Thus, TI i s a weak equivalence, since i t i s a f i l t e r e d c o l i m i t of X weak equivalences n : K > FAK, K e Fin(X). Now, the counit of the K adjunction, e T : T > AFT, i s a weak equivalence of pro-M^ by the t r i a n g l e i d e n t i t y T?P o = I " T 'FT FT A preserves weak equivalences since n- i s a natural weak equivalence, and F preserves weak equivalences by d e f i n i t i o n . The theorem follows e a s i l y . QED This section closes with the following observation: Proposition 2.3: Suppose that C i s a closed model category and that the functor F: Cj >- D i s a (resp. contravariant) equivalence of categories with quasi-inverse G: I) >- C_. Say that an arrow f of I) i s a f i b r a t i o n (resp. c o f i b r a t i o n ) , c o f i b r a t i o n (resp. f i b r a t i o n ) or weak equivalence i f G(f) i s , res p e c t i v e l y , a f i b r a t i o n , c o f i b r a t i o n , or weak equivalence of C. Then, with these d e f i n i t i o n s , D i s a closed model category i n such a way that the functors F and G induce an equivalence of Ho(C) with Ho(D). What i s meant by saying that F: C_ > D_ i s an equivalence of categories with quasi-inverse G: I) • C_ i s that there are natural isomorphisms 1 = GF and 1 = FG. 28 I I . Algebraic groups  Introduction The category ind-Aff, of inductive a f f i n e schemes over a it unique f a c t o r i z a t i o n domain k i s contravariantly equivalent to the category pro-M^ of pro-objects i n the category of k-algebras. In view of the r e s u l t s of Chapter I, ind-Aff^_ has a closed model structure i n such a way that Ho(ind-Aff ) i s equivalent to the ordinary homotopy K. category Ho(_S) which i s associated to the category of s i m p l i c i a l sets. In other words, what we have here i s an a f f i n e algebraic geometric s e t t i n g f o r homotopy theory. In p a r t i c u l a r , one may associate to each a f f i n e k-scheme X i t s homotopy groups TT^(X), i S 0, i n a seemingly very natural way, j u s t by passage to the s i m p l i c i a l category. One could reasonably ask what these invariants measure i n the c l a s s i c a l case where k i s an a l g e b r a i c a l l y closed f i e l d . The f i r s t problem that one encounters i n t h i s programme i s that of f i n d i n g n a t u r a l l y occurring algebraic geometric objects whose homotopy groups can be computed, at l e a s t p a r t i a l l y . It seems that the category of a f f i n e reduced schemes of f i n i t e type over k i s not quite adequate for t h i s i n v e s t i g a t i o n . Even the notion of path-connectedness i s d i f f i c u l t to understand there. It i s a d i f f i c u l t problem to see how to "draw l i n e s " on an a r b i t r a r y a f f i n e (even smooth) v a r i e t y X i n the sense of f i n d i n g enough homomorphisms A^ >• X. I t can be seen, for example, that, while the a f f i n e l i n e A^ i s c o n t r a c t i b l e i n the sense that i t s associated s i m p l i c i a l set i s c o n t r a c t i b l e , the s i m p l i c i a l set which i s associated to the open subset A?" - {0} = Spec(k[x,x x ] ) i s a d i s c r e t e s i m p l i c i a l group 29 * on the group of units k of the underlying f i e l d k. There i s , however, at least one c l a s s i c a l geometric s e t t i n g i n which drawing l i n e s i s very n a t u r a l . Let G± n(k) be the group of r a t i o n a l points of the general l i n e a r group and suppose that the c h a r a c t e r i s t i c of the a l g e b r a i c a l l y closed f i e l d k i s 0. I t i s well-known (see [22, p. 96]) that, i f an element x of Gl (k) i s unipotent, then there i s a n homomorphism •- <f>: a • Gl of algebraic groups from the addit i v e group OL^ , which i s defined on _^.(k) = k by <Ka) = exp(a« log(x)) f o r a e k. Moreover, <J> defines an isomorphism of onto the unique smallest closed algebraic subgroup of Gl which contains x i n i t s group of n r a t i o n a l points. The underlying v a r i e t y of i s the a f f i n e l i n e A^, so that t h i s r e s u l t implies that, i f G i s an algebraic group over k and x e G(k) i s unipotent, then there i s a scheme homomorphism _: • G with u(0) = e and w(l) = x i n G(k), where e denotes the identity-.of G(k) . .We c a l l such an to a path from e' to x. An•algebraic group 6 i s said to be path-connected i f there i s a path from e to any x of G(k). This observation i n c h a r a c t e r i s t i c 0 was the s t a r t i n g point f o r the c l a s s i f i c a t i o n of path-connected algebraic groups over k of a r b i t r a r y c h a r a c t e r i s t i c which i s given i n the second section of t h i s Chapter. It turns out that the theory of algebraic groups i s r i c h enough that one can begin to calculate t h e i r homotopy groups. In succeeding Chapters we w i l l set up a f i b r a t i o n theory f o r homomorphisms of algebraic groups and start to compute the fundamental group. For t h i s , i t w i l l be necessary to work i n several categories which are described i n the f i r s t section. 30 §1. Preliminaries Let k be an a l g e b r a i c a l l y closed f i e l d and l e t Sch^ _ be the category of schemes over k. F, w i l l denote the f u l l subcategory K. of Sch^ _ whose objects are reduced and of f i n i t e type over k. A group-scheme i s a group-object of Sch^_, while an algebraic group over k w i l l be a group-scheme which i s a f f i n e and l i e s i n F_ . Where i t i s convenient, we s h a l l follow the standard pr a c t i c e of i d e n t i f y i n g objects of F^ and algebraic groups i n p a r t i c u l a r with t h e i r sets of k - r a t i o n a l points, together with the induced topology and sheaf of r i n g s . The language and basic r e s u l t s of c l a s s i c a l algebraic group theory w i l l be assumed; the reference i s [22], As before, l e t be the category of algebras (or modeles; see [13]) over k. Associated to each k-scheme X i s a covariant functor, which i s also denoted by X, from to the category E_ of sets. X i s defined by X(A) = Sch k(Spec(A),X) for A e M^. We c a l l X(A) the set of A-points of X. As i s well known, thi s a s s o c i a t i o n determines a f u l l y f a i t h f u l functor from Scli^ to the category M^E of functors from to E, so that Sch^ _ i s equivalent to i t s image i n M^F- Thus, a k-scheme X can be i d e n t i f i e d with the functor that i t "represents"; a f f i n e k-schemes are associated with honestly representable functors i n t h i s way. M^E also contains the f u l l subcategory of pro-representable functors. These are a l l of the functors of the form 31 Sp k(T)(A) = l i m M ^ T ^ A ) , for A e ^ , i e l where T: 1 > i s a pro-algebra as i n Chapter I. The contravariant functor which i s gotten from T 1—>• Sp^CT) for T e pro-M^ i s f u l l y f a i t h f u l , so that pro-M^ i s contravariantly equivalent with the category of pro-representable functors i n M^Es which w i l l be denoted by ind-Aff i n view of the equivalence with the usual category of inductive a f f i n e k-schemes. I f we take for each X e ind-Aff a fixed representing object TX e pro-M^, then a functor T: ind-Aff > pro-M^ i s defined, which i s the inverse of Sp up to natural isomorphism. If K i s a f i n i t e s i m p l i c i a l set, we s h a l l often write K for the s i m p l i c i a l set i t s e l f , f o r the a f f i n e k-scheme Spec(A^K), and for the representable functor M^CA^K, ) i n M^E- l a t h i s notation, A n i s the a f f i n e n-space A^ for n > 0 i n either Mj^ jS o r the scheme category, since A^A n = k[x^,...,x n] i n M^. The c o s i m p l i c i a l k-scheme that one gets by applying Spec to the s i m p l i c i a l algebra k}V w i l l be denoted by A k. I f f: X >- Y i s a morphism of ind-Aff^, we may now say the following: (i) f i s a f i b r a t i o n i f i n every s o l i d arrow diagram i n i n d - A f f k of the form A n • X r <f / / f *i - 4 >x • 32 there i s a dotted arrow making the diagram commute, ( i i ) f i s a weak equivalence i f Yf i s a weak equivalence of pro-M^ (equivalently, i f FTf i s a weak equivalence of s i m p l i c i a l s e t s ) , and ( i i i ) f i s a c o f i b r a t i o n i f f has the l e f t l i f t i n g property with respect to every map which i s both a f i b r a t i o n and a weak equivalence. In view of the r e s u l t s of §2 of Chapter I, we have Theorem 1.1: With these d e f i n i t i o n s , the category ind-Aff i s a closed K. model category i n such a way that the functors r and Sp, induce a K. contravariant equivalence of Ho(pro-M^) with Ho(ihd-Aff ). Corollary 1.2: The covariant functors F°T: ind-Aff *• S^  and Sp^oA: _S_ »• ind-Af f induce an equivalence of Ho (ind-Af f ) with Ho(S). One may think of F.of as an analogue of the ordinary singular functor. We s h a l l write! 1 = Fop and c a l l i t the algebraic singular functor. S i m i l a r l y , = Sp^oA i s c a l l e d the algebraic r e a l i z a t i o n functor. Observe that,'for X e ind-Aff , S (X) i s isomorphic to the s i m p l i c i a l set X(k^), which comes from applying the functor X to the s i m p l i c i a l algebra k. . The homotopy group TT.(X,X) of X based at " I th x e X(k) i s defined to be the i homotopy group TT (| X(k^) | ,x) of the ordinary geometric r e a l i z a t i o n i n the to p o l o g i c a l category of the s i m p l i c i a l set x ( k A ) . Observe that a morphism X »- Y of ind-Af f, i s a weak equivalence i f and only i f i t induces an isomorphism of the homotopy groups so defined. The rest of t h i s thesis consists of computations of some of the 33 homotopy groups of a f f i n e group-schemes G of f i n i t e type over k. In f a c t , i t s u f f i c e s to consider only algebraic groups i n the sense described above, since the reduced part G , of G i s a closed red subgroup-scheme of G 'and we have the elementary Proposition 1.3: The closed immersion G r e (j c G induces an isomorphism " G ( k * } 34 §2. Path-components of algebraic groups One of the p r i n c i p a l reasons that something may be said about the set •"'QCG) °f path-components of an algebraic group' G has to do with the fac t that the s i m p l i c i a l group G(k^) i s a Kan complex. I t follows from the combinatorial d e f i n i t i o n of the homotopy groups of a Kan complex (see [33]) that G(k A) i s connected as a s i m p l i c i a l set i f and only i f G s a t i s f i e s the property that f or every x e G(k) there i s a k-scheme morphism co: > G with w(0) = e and u ( l ) = x i n G(k), where e denotes the i d e n t i t y of G(k). In e f f e c t , one i d e n t i f i e s G(k[x^]) with the k-scheme homomorphisms A^ >• G, and i d e n t i f i e s the face maps d^ and d^ of G(k A) with the functions from S c h ^ A ^ G ) to ___j S.(A^,G) = G(k) which are induced by the choice of r a t i o n a l points 1 and 0 r e s p e c t i v e l y i n A^ - k. An algebraic group which s a t i s f i e s t h i s property i s said to be path-connected. The morphism CJ i s c a l l e d a path from e to x. When one says that G i s connected, on the other hand, one means, as usual, that G i s i r r e d u c i b l e as a k-scheme. I t i s not the case that a l l connected algebraic groups are path-connected. Let y_ denote the m u l t i p l i c a t i v e group Spec(k[t,t "*"]). The i n c l u s i o n k c k[x l 5...,x ] 1 n induces an isomorphism u (k) = k = k[x,,...,x ] = u (k[x.,...,x ]) for n > 1. -ic 1 n — k ± n It follows that the s i m p l i c i a l group ^ ( ^ A ^ ^ s d i s c r e t e on the group k of units of k, so that , while i t i s connected, i s not path-connected. Similar remarks hold f or any torus. I t i s the case, however, that we have 35 Proposition 2.1: A l l path-connected algebraic groups over k are connected. Proof: Use the fa c t that the continuous image of an i r r e d u c i b l e t o p o l o g i c a l space i s i r r e d u c i b l e . QED In f a c t , we s h a l l prove Theorem 2.2: A connected algebraic group G over k i s path-connected i f and only i f the group G(k) of r a t i o n a l points of G i s generated by unipotent elements. We begin the proof of t h i s theorem with a construction. Let e i 1 G (k) = {x e G(k)| there i s an w: *• G with w(0) = e and u ( l ) = x i n G(k)}. Then we have Lemma 2.3: G (k) i s a closed, connected, and normal subgroup of G(k). Proof: G (k) i s a normal subgroup of G(k) by m u l t i p l i c a t i o n and conjugation of paths. Let be the image i n G(k) of A^(k) under a path to. Then Y^ c G 6(k). The subgroup H of G(k) generated by a l l such Y^ i s closed and connected [22, p. 55]. C l e a r l y H = G 6(k). QED e It follows that G (k) i s the group of r a t i o n a l points of a closed algebraic subgroup G e of G. Using the fa c t that a l l of the a f f i n e n e spaces A^ are reduced one can show that the s i m p l i c i a l group G (k^) i s the connected component of the vertex e i n G ( k A ) . For t h i s reason, we e c a l l G the path-component of the i d e n t i t y i n G. 36 Lemma 2.4: Let G be a connected algebraic group and l e t <Gu>(k) be the subgroup of G(k) which i s generated by the unipotent elements of G. Then <G >(k) i s a closed, connected, normal subgroup of G(k). Proof: Let B be a Borel subgroup of G with unipotent subgroup B^. By the Density Theorem [22, p. 139], G(k) = u x B ( k ) x _ 1 . xeG(k) Thus, every unipotent element of G(k) i s contained i n the unipotent subgroup B (k) of some Borel subgroup B of G. A l l such B are u o r u connected closed algebraic subgroups of G [22, p. 122], so that the subgroup H of G(k) generated by a l l of the ^ ( k ) i s closed and connected as w e l l . But H = <G u>(k), and <G^>(k) i s normal i n G(k). QED Again, <G^>(k) i s the group of r a t i o n a l points of a connected algebraic subgroup < ^ u > °f G. The connectedness assumption on G was used i n Lemma 2.4 i n that the Density Theorem was invoked. Assume that G i s connected from now on. Now we have Lemma 2.5: G i s a closed algebraic subgroup of < ^ u > • s e Proof: I t s u f f i c e s to show that G (k) <= <G u>(k). Consider the exact sequence of groups P e • <G >(k) • G(k) »• (G/<G >)(k) > e. u u G/<G^ > i s a connected algebraic group, and every element of (G/<Gu>)(k) i s semi-simple by the n a t u r a l i t y of the Jordan-Chevalley decomposition. Thus, G/<G> i s a torus (see [22, p.'il37]). Every torus i s discre t e i n 37 i n a hotnocopical . sense, so that, i f x e G (k) , then p(x) = e, whence x e <Gu>.(k) . QED It i s w e l l known that any connected unipotent algebraic group U over k i s isomorphic as a k-scheme with A^, where n i s the dimension of U. The most e f f i c i e n t proof of t h i s f a c t uses the theory of fpqc-torseurs; t h i s w i l l be sketched i n the next Chapter. But i t i s easy to see that there i s an isomorphism of s i m p l i c i a l sets A^(k A) = k Ax...xk A (n ^copies), so that A^(k^) i s co n t r a c t i b l e by the Extension Lemma of [5]. I t follows, i n p a r t i c u l a r , that any connected unipotent algebraic group over k i s path-connected. This proves Lemma 2.6: < ( ^ u > 1 S a closed subgroup of G . The proof of Theorem 2.2 i s complete as w e l l . Examples of path-connected algebraic groups abound i n nature, i n view of Theorem 2.2. Any semi-simple group, for example, i s generated by unipotent elements. Thus, a l l of the c l a s s i c a l simple algebraic groups are path-connected. I t i s also easy to see that the s p e c i a l l i n e a r group S l ^ i s the path-component of the i d e n t i t y for the general l i n e a r group Gl , f o r a l l n > 1. n It follows from the proof of Lemma 2.5 that the group '"'g^) of path-components of a connected algebraic group G i s the group of r a t i o n a l points of a torus T(G). By appealing to a fundamental r e s u l t of the next Chapter, we may e s t a b l i s h the rank of T(G) as follows: 38 Proposition 2.8: Let G be a connected algebraic group. Then the rank of T(G) i s the rank of a maximal torus of the solvable r a d i c a l R(G). Proof: Let T be a maximal torus of R(G). Let R (G) be the unipotent u r a d i c a l of G and write H = G/R (G). Then R(H) i s a torus, and u rk(R(H)) = r k ( T ) . Moreover, a r e s u l t of the next Chapter implies that, since ^ u ( ^ ) ^ s a connected unipotent algebraic group, the sequence P e • R u(G)(k^) • G(k^) • H(k^) • e i s an exact sequence of s i m p l i c i a l groups, where p i s induced by the canonical p r o j e c t i o n . I t follows that p induces an isomorphism p^: TQ(G) = TTQ(H), and that the t o r i T(G) and T(H) have the s.ame rank. Since H/[H,H] i s a torus and the commutator subgroup [H,H] of H i s semi-simple, we have < H u > = [H,H], and there i s an exact sequence of groups e > F • R(H)(k) • Tr Q(H) = (H/[H,H])(k) • e, where F i s a f i n i t e abelian group. The conclusion follows. QED Remark: In the proof above, the canonical p r o j e c t i o n G *• H a c t u a l l y induces an isomorphism T(G) = T(H) of t o r i , by a smoothness argument. An easy consequence of Proposition 2.8 i s the following c r i t e r i o n for path-connectedness: Corollary 2.9: A connected algebraic group G i s path-connected i f and only i f i t s solvable r a d i c a l R(G) i s unipotent. Some consequences of t h i s C o r o l l a r y are: 39 Corollary 2.10: Let H be a closed, connected, normal algebraic subgroup of a path-connected algebraic group G. Then H . i s path-connected. Proof: Use R(H) c R(G). QED Corollary 2.11: Let L be an a l g e b r a i c a l l y closed f i e l d containing k, and suppose that G i s a connected algebraic group which i s defined over k. Let G 0, L denote the base extension of G over L. Then G ®, L i s k k path-connected over L i n the obvious sense i f and only i f G i s path-connected over k. Proof: G ® L i s connected over L, and R(G) ®. L = R(G ®, L ) . Then k k k R(G) i s unipotent i f and only i f R(G ® L) i s unipotent. QED 40 I I I . F i bratibns of algebraic groups  Introduction It was shown i n Chapter II that a connected algebraic group G, which i s defined over an a l g e b r a i c a l l y closed f i e l d k, i s path-connected i f and only i f i t s group G(k) of r a t i o n a l points i s generated by unipotent elements. In t h i s Chapter, we begin to answer the question of what i s measured by the higher homotopy groups of an algebraic group; the main purpose here i s to give a d e s c r i p t i o n of the higher homotopy groups of a path-connected algebraic group i n terms of the homotopy groups of Chevalley groups of un i v e r s a l type. This i s done by developing a theory of f i b r a t i o n s f o r algebraic groups. A homomorphism f: G >- H of algebraic groups i s said to be a f i b r a t i o n i f i t induces a f i b r a t i o n S, f: S, G • S, H i n the k k k s i m p l i c i a l set category. This means p r e c i s e l y that f i s a f i b r a t i o n of ind-Aff i n the sense described i n the l a s t Chapter. Now, S f i s a ic ic homomorphism of s i m p l i c i a l groups, and i t i s w e l l known that every s u r j e c t i v e s i m p l i c i a l group homomorphism i s a f i b r a t i o n of s i m p l i c i a l sets. Thus, for S f to be a f i b r a t i o n , i t s u f f i c e s that every k-scheme homomorphism A^ H can be l i f t e d to G, "for every n ^ 0. Part of t h i s condition (the 0-simplices part) says that f i s s u r j e c t i v e on r a t i o n a l points. Such a homomorphism of algebraic groups w i l l be said to be s u r j e c t i v e henceforth. One could ask i f every s u r j e c t i v e homomorphism of algebraic groups i s a - f i b r a t i o n . I t w i l l be seen i n §2 that t h i s i s the case i f the c h a r a c t e r i s t i c of k i s 0; i t i s not true i n p o s i t i v e c h a r a c t e r i s t i c s . The s i t u a t i o n i n a r b i t r a r y c h a r a c t e r i s t i c i s f a r from unmanageable, 41 however. S t a r t i n g with a path-connected algebraic group G, one divides o f f by i t s solvable r a d i c a l R(G) to get the associated semi-simple group S = G/R(G), together with a s u r j e c t i v e homomorphism (1) G • G/R(G) = S, which i s j u s t the canonical map. Consider now the f i n i t e c o l l e c t i o n of minimal closed normal, connected subgroups of p o s i t i v e dimension S, ,...,S of S. These subgroups c e n t r a l i z e each other and generate S, 1 n so that a s u r j e c t i v e algebraic group homomorphism (2) m: S.x. . .xS • S I n i s defined using the m u l t i p l i c a t i o n map of S. I t w i l l be shown, also i n §2, that the homomorphisms (1) and (2) are f i b r a t i o n s . I t w i l l follow from an a p p l i c a t i o n of the long exact sequence for a f i b r a t i o n n that TT.(G) = © TT.(S.) f or i > 1, and that there i s a short exact 1 • -i 1 3 3 = 1 sequence of abelian groups 0 > 9 v (S.) • Tr (G) y r y 0, j = l J where T i s a f i n i t e abelian group. By the C l a s s i f i c a t i o n Theorem [10] for simple algebraic groups, each of the groups S. i s isomorphic to a Chevalley group G over k 2 P j which comes from a f a i t h f u l representation of a simple L i e algebra L. over the complex numbers C. I f $. denotes the root system of L. 3 3 3 and G^ i s the Chevalley group over k which i s universal of type $., j 2 then :there i s a s u r j e c t i v e comparison homomorphism 42 (3) . • % — * G $ , j j j for j - l , . . . , n . These homomorphisms are shown to be f i b r a t i o n s i n §3, a f t e r a review of d e f i n i t i o n s and notation. The main tools are the "big c e l l " and a r e s u l t §2, The technical point that i s exploited throughout t h i s Chapter i s that the obstruction to l i f t i n g a map A^ > H over a s u r j e c t i v e algebraic group homomorphism f: G • H l i e s i n the pointed set fi^(A^;K) of fpqc-torseurs over A^ with c o e f f i c i e n t s i n the group-scheme kernel K of f. The f i r s t section of t h i s Chapter contains a quick introduction to t h i s theory, together with proofs of r e s u l t s about unipotent algebraic groups that were used i n Chapter I I . The approach to torseur theory that i s taken here follows that of the book of Demazure and Gabriel [13]. 43 §1. Sheaves and torseurs We s h a l l work i n the Demazure-Gabriel version [13] of the Or category M^ E_ of fpqc-sheaves over the a l g e b r a i c a l l y c l o s e d . f i e l d k. The category of schema's over k i s a f u l l and f a i t h f u l subcategory of M^E. Let G be a sheaf of groups over k and take S e M^F. Then we r e c a l l that a r i g h t G-torseur over S consists of: .(i) a map p: X • S i n a n d ( i i ) a r i g h t G-action m: XxG • X on X, such that p(xg) = p(x) for a l l x e X(A), g e G(A), and a l l A e M^, and such that there i s a sheaf epi T »• S for which the pullback Tx X >• T i s equivariantly isomorphic to the pro j e c t i o n pr : TxG > T. R e c a l l further that any G-torseur which i s e q u i v a r i a n t l y isomorphic to the torseur P r g : SxG *• S i s said to be t r i v i a l , and that a G-torseur p: X *• S i s t r i v i a l i f and only i f p has a section. The class of ^1 isomorphism classes of G-torseurs over S i s denoted by H (S;G). The o-l class of t r i v i a l torseurs provides H (S;G) with a base point. Every s u r j e c t i v e homomorphism IT : G *• G' of algebraic groups over k i s a K-torseur over G', where K i s the group-scheme kernel of IT, and every quotient map - G >- G/H, for H a closed normal algebraic subgroup of G, i s an H-torseur over G/H. A key point (see [13, p. 361]) i s : Proposition 1.1: Let p: X >• S be a G-torseur over S, and l e t g: T > S be a k-sheaf morphism. Then g l i f t s to X i f and only i f the G-torseur Tx X -> T, which i s induced by pullback over g, i s t r i v i a l . 44 Suppose that p: X * S i s a G-torseur over S, and that f: G H i s a homomorphism of sheaves of groups over k. Define a l e f t G-action on X XH v i a g*(x,h) = (x«g ,f(g)*h), and l e t the k-sheaf G ^ XV H be the coequalizer i n M^E of the p a i r of maps m •y G x ( X x H ) X X H , P^XxH where m i s the G-action and pr i s the obvious p r o j e c t i o n . H acts A X n md there through the composition ^ G on XV H on the r i g h t and i s a unique map XV H > S f a c t o r i n g pr P XxH X y S, Q which makes Xv H into an H-torseur over S (see [13, p. 368]). This ^1 Oil operation defines a natural pointed set map f ^ : H (S;G) y H (S;H) ^1 and makes H (S; ) a functor on the category of sheaves of groups over k. Now consider an exact sequence (1) e y K -^-y G H y e of .sheaves of groups over k. Re c a l l that, i f G and H are algebraic groups, then exactness of (1) means p r e c i s e l y that TT i s s u r j e c t i v e (on r a t i o n a l points) and that K i s the group-scheme kernel of TT . For S e M^E and G a sheaf of groups over k, we define H°(S;G) = M^E(S,G). Then i t f i s w e l l known that H^S; ) i s the f i r s t r i g h t derived functor of H (S; ) i n the sense that we have Proposition 1.2: For any S e M^ JS a n ^ a n y e x a c t sequence (1) as above, there i s a 6-term sequence of pointed sets '45 * • H (S;K) • H U(S;G) • H U(S;H) • H A(S;K) • H (S;G) • H A(S;H), which i s exact i n the sense that kernel = image everywhere. The eoboundary map <5 i s defined by pullback. The basic t e c h n i c a l device, i n the sections that follow w i l l be to show, f o r group-scheme kernels K of c e r t a i n s u r j e c t i v e algebraic group % l m homomorphisms, that H (A^jK) = * for a l l m > 1, and then invoke Proposition 1.1. This w i l l involve Proposition 1.2 and proceed from basic f a c t s . Proposition 1.3: [13, p. 367] Suppose that X i s a scheme which i s of ^1 f i n i t e type over k. Then H (X;G1 ) i s i n one to one correspondence n with the set of quasi-coherent X-modules which are l o c a l l y free of rank n. If X i s a f f i n e , then the class of X-modules of t h i s l a s t Proposition coincides with the class of p r o j e c t i v e A-modules of rank n, where X = Spec(A). Thus, i f X i s one of the a f f i n e spaces A^, then the Serre "conjecture" for k [42] can be restated as ^1 m Proposition 1.4: H ( A ^ G l ^ ) = * for a l l m,n > 1. In p a r t i c u l a r , since G l ^ i s the m u l t i p l i c a t i v e group P^, and any torus i s a f i n i t e product of copies of j ^ , we have Corollary: 1..5: H (A^T) = * for any torus T and a l l m > 1. Corollary 1.5 may also be i n f e r r e d from the vanishing of the Picard 46 group Pic(A^) of A™ (see [13, p. 371]). Another basic f a c t i s Proposition 1.6: [13, p. 383] I f X i s an a f f i n e k-scheme, then HV;^) = *. We may now prove Proposition 1.7: Let U be a connected unipotent algebraic group and suppose that X i s an a f f i n e k-scheme. Then H (X;U) = *. Proof: In e f f e c t , U has a connected closed normal algebraic subgroup which i s isomorphic to oc_ [22, p. 115, 131], and U/a, i s a connected K. it unipotent algebraic group of lower dimension. Proceed by induction on the dimension of U, using the exact sequence of Proposition 1.2. QED It follows from Proposition 1.7 that taking the quotient of a connected algebraic group G by i t s unipotent r a d i c a l R U ( G ) gives a f i b r a t i o n G > G/R (G). This f a c t was used i n the proof of Proposition 2.8 of u Chapter I I . We also have the following well-known r e s u l t : Proposition 1.8: The underlying k-scheme of a connected unipotent algebraic group U i s isomorphic to A^, where n i s the dimension of U. Proof: As i n the proof of Proposition 1.7, U has a closed connected normal algebraic subgroup isomorphic to a,. Then the it quotient map U > U/a, i s a r i g h t O L -torseur over the a f f i n e k-scheme K. i t U/a^, so that U = o^^CU/a^) by Proposition 1.6. The proof i s f i n i s h e d by induction on the dimension of U, as before. QED 47 It follows that every connected unipotent group U i s weakly equivalent to a point Spec(k) i n the sense of Chapter I. This completes the proofs of Proposition 2.8, Lemma 2.6, and Theorem 2.2 of Chapter I I . 48 §2. D6yissage Let G be a connected algebraic group over k, with solvable r a d i c a l R(G). The f i r s t task w i l l be to show that the canonical homomorphism G > G/R(G) i s a f i b r a t i o n . Proposition 2.1: Let S be a connected solvable algebraic group. Then H (A ;S) = * for a l l m > 1. k Proof: Let U be the set of unipotent elements of S and l e t T be a maximal torus. Then U i s a closed connected normal subgroup of S and there i s a s p l i t exact sequence of algebraic groups of the form e >• U >• S > T >• e . But H (Aj^U) = * by Proposition 1.7 and H (A^.;T) = * by Proposition 1.5. An a p p l i c a t i o n of the exact sequence of Proposition 1.2 f i n i s h e s the proof. QED Corollary 2.2: Let G be a connected algebraic group with solvable r a d i c a l R(G). Then the canonical map G > G/R(G) i s a f i b r a t i o n of algebraic groups. Now l e t T be a f i n i t e l y generated abelian group, and consider the functor D(D: >• _E, which i s defined at A e by D(r)(A) = Ab(r,A*), where Ab denotes the category of abelian groups. D(r) i s an a f f i n e groups-scheme over k which i s represented by the group algebra k[P] of T over k, where the c o m u l t i p l i c a t i o n A:k[r] > k[P] ®^ k[T], 49 the counit e; k[T] ^ k, and the in v o l u t i o n <y: k [ r ] k [ r ] are s p e c i f i e d r e s p e c t i v e l y by A;(x) = x®x, e (x) = e, and o(x) = x \ for x c F, and-together-give-the Hopf algebra structure f o r k[T]. Here, r i s written m u l t i p l i c a t i v e l y with i d e n t i t y e. Such an a f f i n e group-scheme D(r) over k i s said to be m u l t i p l i c a t i v e (see [13, p. 144]). Any subgroup-scheme of a torus i s m u l t i p l i c a t i v e ; i n f a c t , the m u l t i p l i c a t i v e group-schemes over k as defined above are p r e c i s e l y the subgroup-schemes of t o r i . Perhaps the most useful general r e s u l t of t h i s Chapter i s Theorem 2.3: Let T be a f i n i t e l y generated abelian group and l e t D(r) be i t s associated m u l t i p l i c a t i v e group-scheme over k. Then, for a l l m > 1, „ 1(A£;D ( r ) ) = * . Proof: One may show that applying D to an exact sequence of abelian groups i P r s e >• Z >- Z • T >- e gives an exact sequence P i e • D(r) »• D(Z S) f D(Z r) y e of group-schemes over k [13, p.473]. Take m ^ 1 and consider the 6-term exact sequence of Proposition 1.2 * • H°(A^;D(r)) • H°(A^;D(Z S)) H°(A^;D(Z r)) 6 — * H V ^ D ) — H^(A™;D(Z S) ) — . ^ ( A ^ D ( Z r ) ) . D(Z S) i s a torus of rank s, so that E1 {A^;T){ZS) ) = * by Corollary 1.5. On the other hand, there i s a commutative diagram 50 H ° ( A £ ; D ( Z S ) ) — • ^ °CAjjD(ZE)) D(Z S) ( k [ X ; L , .. . , x j ) D(Z r) ( k [ X l , . . . , x j ) J * i (k) D(Z S)(k) y D ( Z r ) ( k ) , where i ' denotes the i n c l u s i o n k c k[x,,...,x ]. But the v e r t i c a l 1 m * * * maps j * are isomorphisms since k = k[x^,...,x m] , and i (k) i s su r j e c t i v e because i i s f a i t h f u l l y f l a t . The theorem follows. QED Corollary 2.4: Any s u r j e c t i v e algebraic group homomorphism with m u l t i p l i c a t i v e group-scheme kernel i s a f i b r a t i o n . The argument of Theorem 2.3 breaks down i f the f i e l d k i s not assumed to be a l g e b r a i c a l l y closed. I f A™ i s the a f f i n e m-space over the r a t i o n a l numbers Q and D^(Z/2Z) i s the group-scheme over Q corresponding to D(Z/2Z) above, then one may show that there i s an ^1 m . * * 2 isomorphism H (A^;D Q(Z/2Z)) = Q /(Q ) , and t h i s f o r a l l m > 0. We need some more notation at t h i s point. Let X: ^ E. be a functor. Take A e and l e t M denote the category of A-algebras. Then a functor X : M > E i s defined at B e M by X (B) = X(B), A A A A since B e M. as w e l l . I f Y i s another element of M, E, then a He-functor hom(X,Y) : > _E i s defined at A e by hom(X,Y)(A) = MJ2(X ,Y ) . hom(X,Y) i s the canonical function complex A A A for M^ E^  i n that there i s a natural isomorphism 51 ^.E(XxS,Y) = M kE(S >homCX,Y)) > which i s natural i n X, Y and S. The functors End(X) and Aut(X) which are associated to X e M^E have corresponding obvious d e f i n i t i o n s (see [13, p. 1 6 0 ] ) . Now suppose that T i s a maximal torus of a semi-simple algebraic group G, and i d e n t i f y these algebraic groups with t h e i r associated group-valued functors on M^. T acts on G by conjugation, and the above natural isomorphism associates to t h i s action a natural transformation p: T • Aut(G), which i s defined f o r t e T(A), g e G(B), and B e M by P ^ ^ B ^ S ) = T B § T B Here, following [13], t g denotes the image i n T(B) of t under the map T(A) >• T(B) which i s induced by the T structure map of the A-algebra B. A functor G : > E i s defined at A e by T —1 G (A) = {x e G(A)| tx„t = x_ for a l l B e M and t e T(B)}. a a —A T T G i s a subgroup-scheme of G [13, p. 165]. I t s group G (k) of k- r a t i o n a l points can be i d e n t i f i e d with the c e n t r a l i z e r C G ^ ( T ( k ) ) of T(k) i n G(k). But T(k) = C G ^ ( T ( k ) ) since G i s reductive T [22, p. 159]. Moreover, G i s smooth over k, hence reduced, by the "Smoothness of C e n t r a l i z e r s Theorem" of [13, p. 240]. I t follows that T the subgroup-schemes G and T of G coincide. This i s the e s s e n t i a l point i n the proof of the following Theorem: Theorem 2.5: Let S l S...,S be the f i n i t e c o l l e c t i o n of minimal closed 1 n connected normal subgroups of a semi-simple algebraic group G. Then the algebraic group homomorphism m: S-x...xS > G which i s defined 1 n ' 52 by (s., . ..,s ) I • s * ...•s > i s a f i b r a t i o n . J 1 n 1 n Proof: That m i s a homomorphism of algebraic groups comes from the fact that the subgroups S^(k) of G(k) c e n t r a l i z e each other [22, p. 167] I t follows that the subgroups S^(A) of G(A) c e n t r a l i z e each other, and hence that the group K(A) of A-points of the group-scheme kernel K of m i s c e n t r a l i n (S,x...*S )(A), for a l l A e M . Let T be a I n ~k maximal torus of the semi-siirroie algebraic group S x... xs . Then K(A) \ •" 1 n i s c e n t r a l i z e d by T(A) f o r every A e M^, hence K i s a closed subgroup-scheme of T by the above remarks. Thus, K i s m u l t i p l i c a t i v e and the Theorem i s proved by applying Co r o l l a r y 2.4. QED Since the kernel K i n the proof of Theorem 2.5 i s m u l t i p l i c a t i v e and K(k) i s f i n i t e [22, p. 167], i t follows that K has the form D ( r ) , where r i s a f i n i t e abelian group. Write r = r i f char(k) = 0, and F / r ^ p ) i f char(k) = p, where Vdenotes the p-primary part of T. Then the s i m p l i c i a l group D ( r)(k A) i s di s c r e t e with group of v e r t i c e s T, so that the long exact sequence for a f i b r a t i o n gives Corollary 2.6: If G, S^,...,S are as i n Theorem 2.5, then there i s a short exact sequence of abelian groups 0 • 9 TT (S.) y TT (G) • r • 0, i = l and isomorphisms 9 TT.(S.) - TT . (G) which are induced by m for a l l j > 1. i = 1 J i J 53 I f the c h a r a c t e r i s t i c of the underlying f i e l d k i s 0, then every s u r j e c t i v e homomorphism of algebraic groups over k i s smooth, and hence has reduced group-scheme kernel. This i s what allows us to prove Theorem 2.7: Suppose that char(k) = 0. Then every s u r j e c t i v e Proof: Let K be the group-scheme kernel of f. As noted above, K i s reduced and normal i n G. Moreover, H can be i d e n t i f i e d with the quotient G/K as a sheaf of groups over k. Assume for the present that the Theorem has been proven i n the case where both G and K are connected. Now suppose only that G i s connected, and consider the diagram i n which the morphisms are canonical projections and hence s u r j e c t i v e . The kernel of TT ' i s K/YP , where denotes the connected component of the i d e n t i t y i n K. K/K^ i s a closed normal f i n i t e subgroup of G/K^ ..' The connectedness of G/YP implies that Y/YP i s c e n t r a l i n G/K, hence i s abelian. Every f i n i t e abelian algebraic group over k i s m u l t i p l i c a t i v e i f char(k) = 0 [13, p. 517], so that TT' i s a f i b r a t i o n by Corollary 2.4. It follows from the assumption that TT i s a f i b r a t i o n as w e l l . homomorphism f: G •> H of algebraic groups over k i s a f i b r a t i o n . G (1) For the case i n which G and K are a r b i t r a r y , r e c a l l that the canonical map TT : G homomorphism G^*K -—> G/K i s a f i n i t e union of copies of a s u r j e c t i v e * (G •KJ/K, and that there i s an isomorphism 54 cj>: (G°'K)/K G°/CG°nK) of sheaves of groups over k such that the following diagram commutes: G°«K -> (G°-K)/K -y G°/(G°nK) By a t r a n s l a t i o n argument, we may assume that any map y G/K factors through (G^-K)/K and hence l i f t s to G^ c G^'K <= G, whence TT: G y G/K i s a f i b r a t i o n . I t remains to prove the Theorem i n the case where G and K are connected. Observe, f i r s t of a l l , that, i f R(G)' i s the solvable r a d i c a l of G, then there i s an isomorphism of algebraic groups (G/K)/(R(G)/(KnR(G))) = (G/R(G))/(K/(KnR(G))). These two quotients can be i d e n t i f i e d with a connected algebraic group H i n such a way that there i s a commutative diagram KnR(G) c K n R(G) ->• K/ (KnR(G) ) «-3 m c G -y G/R(G) m S x . . . x S 1 r S x...xS 1 n R ( G ) / ( K n R ' ( G ) ) 4= G/K -> H m pr S ,_x...xS • r+1 n Here, S^,...,S are the simple normal algebraic subgroups of the semi-simple algebraic group G/R(G), and S ,...,S are those simple normal algebraic subgroups which are contained i n the closed 55 connected normal subgroup K/(K0R(G)) of G/R(G). pr i s the obvious proj e c t i o n map; i t s p l i t s , so i t i s a f i b r a t i o n . The maps m and m' are s u r j e c t i v e with f i n i t e c e n t r a l kernel [22, p. 167], so that m" i s as w e l l . I t follows from Cor o l l a r y 2.4 that m" i s a f i b r a t i o n . p' i s a f i b r a t i o n by Proposition 2.1. KnR(G) i s a solvable algebraic group, so that an argument s i m i l a r to that which was used for the diagram (1) together with Proposition 2.1 shows, that p i s a f i b r a t i o n as w e l l . To put i t another way, we have seen that the homomorphisms pr, m", p', and p are s u r j e c t i v e on the l e v e l of k[x^,...,x n]-points f or a l l n ^ 1. A diagram chase using the exactness of the sequences i n sight now shows that TT as well i s s u r j e c t i v e on k [ x ^ , . . . j X ^ J - p o i n t s f or a l l n ^ 1. QED Theorem 2.7 does not hold i f the c h a r a c t e r i s t i c of the underlying f i e l d k i s non-zero. The homomorphism F : a, • a, , which i s p k • —k defined over k of c h a r a c t e r i s t i c p by F (x) = x P for x e a (A) = A, P k A e M^, i s not even s u r j e c t i v e on the l e v e l of k[x^]-points, since x^ has no p*"*1 root i n k[x^] . 56 §3. Chevalley groups In previous sections, we have seen that the problem of computing TT^(G) for a connected algebraic group G over k can be effectively reduced to the case where G is a simple algebraic group, via the fibration theory. A l l simple algebraic groups may be constructed from representations of simple Lie algebras over the complex numbers C, in a systematic way which is due to Chevalley [9] and Steinberg [45]. Groups which are produced in this way are called Chevalley groups over k. There is a nice class of naturally occurring surjective homomorphisms between Chevalley groups of a fixed type, which I c a l l comparison maps. The purpose of this section i s to show that these comparison maps are fibrations. Among other things, this w i l l allow us to compute n of a Chevalley group in terms of i t s "geometric fundamental group" and TT^ of the Chevalley group which is universal of the same type. We begin with a quick review of notation. For more detail the reader should consult [21] and especially the notes of Steinberg [45]. The data for a Chevalley group consists of the following: (1) a simple Lie algebra L over C, together with a choice of maximal toral subalgebra H, (2) the root system $ . of L relative to H, with a choice of a set of simple roots A = { _} ' c $ t where the rank of $ is r = dim (H), u (3) a Chevalley basis B = {h., i = l , . . . , r ; x , a e $} for L — i a (see [21, p. 146], [45]), (4) a faithful representation p : L »• gl(V), and 57 (5) an admissible l a t t i c e M for p. The subgroup G (k) of G1(M ® k) which i s generated by a l l of the x ( t ) , p - _ • a a e $, t e k, where x (t) i s the s p e c i a l i z a t i o n of exp(Tp(x )) £ G1(M ® Z[T]) to G1(M ® k) at t , i s what i s known as a Chevalley group over k r e l a t i v e to the above data. Since x (t+u) = x (t)*x (u) for t,u e k, a e $, G (k) i s a a a P the group of k - r a t i o n a l points of an algebraic group G^  which i s defined oyer the prime s u b f i e l d of k, and moreover the assignment t l > x (t) determines a homomorphism x : OL • G of algebraic Ot Ot rC P groups over k. We w i l l say that the x ^ ( t ) , a e $, t e k, are elementary matrices of G^(k). Within Gp(k), one has the following subgroups: (1) U(k), which i s generated by the x ^ C t ) , t e k, where a i s p o s i t i v e , (2) U (k), generated by the x ^ C t ) , t e k, where a i s negative, (3) N(k), generated by the w ^ C t ) , where -1 * w (t) = x (t)x (-t )x ( t ) , f o r t e k , a e $, and a a -a a (4) H.(k), which i s generated by the h ( t ) , where a h (t) = w (t)w (-1), for t e k , a e $. The groups U(k), U (k), and H(k) are groups of k - r a t i o n a l points of algebraic groups U, U , and H re s p e c t i v e l y . H i s a maximal torus of Gp, and U and U are maximal unipotent algb r a i c subgroups of G^  • which are normalized by H. B = U*H and B = U *H are Borel subgroups of G . Also, N(k) i s the normalizer of H(k) i n G (k), and P p 58 an isomorphism from the Weyl group W of $ to the quotient N(k)/H(k) i s defined on a simple r e f l e c t i o n a of W by a i >• w' (1). a • ' a a The weight l a t t i c e V of p can be i d e n t i f i e d with the character P group X(H) of H. G i s said to be univer s a l of type $ i f T i s the P P e n t i r e abstract weight l a t t i c e r . If TT i s another f a i t h f u l representation of L and V^ <= y then there i s a unique homomorphism X : G > G , which i s s u r i e c t i v e on r a t i o n a l points, such that p , TT p TT J R X (x (t)) = x ( t ) . X i s c a l l e d a comparison map, at le a s t for our P , T T a a P , T T — purposes. One sees that every G^  has a covering by a Chevalley group ' of univ e r s a l type, which i s unique up to isomorphism and w i l l be denoted by G^. Other comparison-type maps i ? ^ : GP0^ *" G^  a r i s e from the r e s t r i c t i o n s of the representation p to the Lie sublagebras L' of L which are generated by connected subdiagrams A_' of A_. Again i . ( x (t)) = x ( t ) , G . i s of univer s a l type i f G i s , and i . i s a closed immersion i n general. The s p e c i a l " l i n e a r groups S l n + ^ ( k ) , n > 1, and the symplectic groups sP2m^' m S 1, are the groups of r a t i o n a l points of Chevalley groups over k • which are un i v e r s a l of type A^, n > 1, and of type C^, m > 1, re s p e c t i v e l y . The elementary matrices of S l n + ^ ( k ) are the matrices X. .(t) = I + t e. ., t e k , 1< i * i < n+1, where e. . i s the matrix which i s 1 i n the ( i , j ) - p o s i t i o n and 0 elsewhere. The elementary matrices of Sp„ (k) are matrices of the types: I. I + t«(e. i»j - e m+j,m+i 1 < i?ej < m, I I . I + f e . i,m+i 1 < i < m, I I I . 1 + f (e. l,m+j + e. j ,m+i 4 ) . 1 < l * j < m, 59 IV. I + t.e , 1 < i < m, and xarx, l V ' I + t* C em+i,j + e m + j . i } ' 1 S ^ • The canonical i n c l u s i o n s SI (k) c s i ,, (k) and Sp„ (k) <= Sp„ , 0(k) n n+l _m _m+z ar.e instances of the maps i . In any given i r r e d u c i b l e root system there are at most two root lengths. The root systems C are m characterized by the f a c t that any two long roots are orthogonal. For t h i s reason, a Chevalley group G i s said to be symplectic i f _A i s one of the root systems up to isomorphism, and non-symplectic otherwise. Observe that Sl„ = Sp„ i s symplectic, and the SI are 2 2 n non-symplectic f o r n > 3. The geometric fundamental group II (G ) of G^  i s defined to be the quotient r,/r . II (G ) i s a f i n i t e abelian group. If G i s of 1 P P P type A n, n > 1, then n(G ) i s a quotient of Z/(n+l)Z. If Gp i s of type C m, m > 1, then n(G ) i s a quotient of Z/2Z. I t follows that the groups of type which can occur are S p 2 m and the adjoint group P S p2m' Suppose that p and TT are. representations of \ L. with ^TT C ^p * ^ e a r e n O W "*"n a P o s^- t l o n t o show that the r e s u l t i n g comparison map _: G^  >• G^ i s a f i b r a t i o n of algebraic groups. We have the subgroups U(k), U (k), ffi(k), N ( k ) , a n d B(k) of - G p ( k ) T h e corresponding subgroups of G (k) are denoted by U (k), U~(k), H (k), N (k), and B (k) . IT T r . T f . . T r u TT Lemma 3.1: A p U~(k) •.B._(k)) '= U _ ( k ) B ( k ) i n G (k) . Proof: We use the Bruhat decomposition. E x p l i c i t l y , G (k) = u B(k)wB(k), p weW 60 with B(k)w 1B(k) = B(k)w 2B(k) i f and only i f w- = w2 (the w i n B(k)wB(k) i s r e a l l y a coset representative i n N(k) f o r w e W v i a the isomorphism W • N(k)/H(k)). Observe that A preserves a l l of the • p ,TT subgroups mentioned above, and induces an isomorphism N(k)/H(k) -> N7r(k)/Hii,(k) which commutes with the respective isomorphisms with W, so that A preserves the Bruhat decomposition as w e l l . Now choose WQ e W such that wQ(°0 < 0 i f a > 0 i n $. Since WQ*G(k) = G(k) and wQ*W = W, we see that G (k) = u U (k)wB(k), p weW with U (k)w.jB(k) = U (k)w 2B(k) i f and only i f w.^  = w2, and that X preserves t h i s decomposition. Thus, i f x e U (k)wB(k) and P ,TT X (x) 6 U (k)B (k), then w = e and x e U (k)B(k), and the Lemma P , TT TT TT i s proved. QED U B = U HU i s the big c e l l of G . I t i s an a f f i n e open subvariety of G^, and the m u l t i p l i c a t i o n map m: U XHXTJ -> U HU i s an isomorphism of k-schemes.' I t follows from Lemma 3.1 that the commutative square U XHXU m -> G P »TT U xH xu — •> G , TT TT Tf TT i n which the map on the l e f t i s induced by the r e s t r i c t i o n of X to P,TT each f a c t o r , is. a pullback i n the category of schemes over k. 61 For the structure of U (respectively U ), choose an ordering TT of the p o s i t i v e roots of $ which i s consistent with addition, and suppose that n i s the number of such roots. Then the map A^ >• U defined by (t„,...,t ) i -> x (t,)«...«x (t ), where the a. are i n a, 1 a n 1 1 n arranged i n the stated order, i s an isomorphism of k-schemes [45, p. 63]. A s i m i l a r statement holds for U (respectively U ). I t follows that TT the maps X - and X I„ are isomorphisms of group-schemes over k. P , T T'U P , 7 T'U Theorem 3.2: X i s a f i b r a t i o n of algebraic groups, with group-scheme kernel K given at A e M, by K (A) = Ab(r / r ,A ). P ,TT ° -4c J P , T T — p T r Proof: From the discussion above, we see that K (A) c H(A) for every P ,TT A e M ^ , and hence that ^ i s a closed subgroup-scheme of H. But then K i s m u l t i p l i c a t i v e , and so X i s a f i b r a t i o n by C o r o l l a r y 2.4. Steinberg shows [45, p. 43] that every h e H(k) can be written h = h ( t 1 ) * . . . , h (t ), where A = {a.,...,a }. I t i s an exercise to a, 1 a n — 1 ' n I n show that, for y e r , the assignment u(h ) y(h ) y(h^ ( t , ) - . . . - ^ (t„)) = t, 1 - ...'t_ n , a 1 a n 1 1 n n where the h are co-roots corresponding to the a. i n the maximal a. l l t o r a l subalgebra of L, defines a character y i n X(H) and an isomorphism n : T • X(H). I t follows that the diagram P P n r — ^ X(H ) TT TT n X p,TT r — ^ > x\nj 62 commutes. But then % H(A) * H (A) i s the map 1 p jTT 1 tt Tr Ab(r ,A ) ^ Ah(r ,A ) for every A e M, , whence the Theorem. QED P TT —K Let n(G ) = T./r he the geometric fundamental group of G p i p P as before and define n(c- ) = i p JI(G p) i f char(k) =0, and n ( G p ) / n ( G p ) ( p ) i f char(k) = p, where II (G ) ^  i s the p-primary component of the f i n i t e group II (G ) P P Then we have Corollary 3.3: There i s a short exact sequence of abelian groups X — • » • Tr, ( ( P' v p ' 0 • T T (G ) T T G ) • n(G ) — * 0, 1 <p ± p p ' where X A i s the homomorphism which i s induced by the u n i v e r s a l covering map X: G, > G . . $ p a. Proof: n(G p) 1 S the group of r a t i o n a l points of the group-scheme kernel of X. ggn 63 IV. Algebraic K-theory  Introduction Perhaps a h i s t o r i c a l remark i s i n order at t h i s point. We have been using a s i m p l i c i a l algebra k A which i s defined over an a l g e b r a i c a l l y closed f i e l d k i n order to define homotopical invariants of a f f i n e k-schemes. Obviously the construction of k A generalizes to produce complexes A A for a r b i t r a r y rings A. My i n t e r e s t i n such complexes o r i g i n a l l y came from the fact that Q A may be used to define the Q-algebra A^(X) of Sullivan-de Rham 0-forms of a s i m p l i c a l set X, ju s t by s e t t i n g A^(X) = S^X,Q A). This construction f i r s t appeared e x p l i c i t l y i n a paper of Swan [46], and l a t e r i n an AMS Memoir of Bousfield and Guggenheim [5]. But the complexes A^ have a h i s t o r y i n algebraic K-theory as wel l . Rector showed [43] i n 1971 that the homotopy groups of the s i m p l i c i a l group G1(A A) coincide with the Karoubi-Villamayor K-theory of the r i n g A, up to a dimension s h i f t . Using t h i s r e s u l t , Gersten showed [17, p.28] that there i s an isomorphism (1) K i + i ( A ) ~ ^ ( G K A ^ ) ) , f o r i > 0, i f A i s a regular Noetherian r i n g , where the K-theory i s that of Q u i l l e n . Anderson gives a proof of t h i s r e s u l t i n [1] which uses some r e s u l t s of Quillen's on the group: completion of a s i m p l i c i a l monoid. For us, the i n t e r e s t i n t h i s isomorphism l i e s i n the fact that the homotopy groups T r.(Gl(k )) are the homotopy groups of the inductive a f f i n e algebraic group G l , as defined i n Chapter I I . I t seems worthwhile, therefore, to include a very e x p l i c i t proof of the 64 isomorphism (1) here. This i s done i n the f i r s t section of t h i s Chapter a f t e r a few pre l i m i n a r i e s about Q u i l l e n K-theory are mentioned. The proof that i s given here d i f f e r s from, that of Gersten or Anderson i n that the bar construction BGl(A^) of the s i m p l i c i a l group Gl(A^) i s e x p l i c i t l y i d e n t i f i e d as BG1(A) + when A i s assumed to be Noetherian and regular. These assumptions on the r i n g A are made i n order that we may use the theorem of Q u i l l e n which asserts that the canonical i n c l u s i o n A c A[x^] induces an isomorphism K_^(A) = K_^(A[x^]) for a l l i ^ 1 i f A i s Noetherian and regular. This isomorphism may be thought of as good stable behaviour, i n a way that w i l l become clear i n the second section, where t h i s r e s u l t i s used together with c e r t a i n theorems of Matsumoto to show that there are isomorphisms (2) T T ^ S I ^ = K 2(k) for n > 2, and (3) 7 T i ( S P 2 m ) " K 2 ( k ) f o r m - 1 ' where SI and Sp„ .'are the obvious group-schemes defined over an n 2m a l g e b r a i c a l l y closed f i e l d k which correspond to s p e c i a l l i n e a r and symplectic groups re s p e c t i v e l y . No assumption i s made on the c h a r a c t e r i s t i c of k here. I t should be pointed out, however, that the lower bound for the f i r s t isomorphism and the lack of a condition on the c h a r a c t e r i s t i c of k f o r the second depends very much on the fact that k i s a l g e b r a i c a l l y closed. Something can T be said about the case where k i s not a l g e b r a i c a l l y closed; t h i s w i l l be sketched at the end of the section. The t h i r d and f i n a l section of t h i s Chapter contains a vanishing theorem for K^(k) of an a l g e b r a i c a l l y closed f i e l d k, which depends 65 on an analysis of of a global f i e l d that was given by Bass and Tate [4]. E x p l i c i t l y , K^ C k ) = 0 i f the Kroenecker dimension of k i s s t r i c t l y l e s s than 2, but i s a n o n - t r i v i a l uniquely d i v i s i b l e abelian group otherwise. A c a l c u l a t i o n of the fundamental group of an a r b i t r a r y Chevalley group of type A or C i n terms of K„(k) and the n m 2 geometric fundamental group i s also given. In c l o s i n g , l e t me say that Jan Strooker has kind l y informed me that the isomorphism (2) could have been i n f e r r e d from a r e s u l t of Krusemeyer [27, p. 23], by showing that the r e s u l t of Rector which was c i t e d above can be d e s t a b i l i z e d . Krusemeyer has a notion of path-connectedness of an algebraic group which coincides with that given i n Chapter I I . He also c i t e s r e s u l t s which are c r u c i a l f o r the proof of the isomorphism (3). 66 § 1. Q u i l l e n K-theory As mentioned i n the Introduction, t h i s section begins with a b r i e f review of what, f o r us, are the basics of Quillen's K-theory of a unitary r i n g . D e t a i l s of these constructions can best be found i n Loday's thesis [28]. One s t a r t s with a unitary r i n g A and considers the c l a s s i f y i n g space BG1(A) of the d i s c r e t e group G1(A) as the r e a l i z a t i o n of the nerve of the appropriate one-object groupoid. The subgroup of elementary matrices E(A) i s a perfect normal subgroup of G1(A) = T T ^ B G I C A ) ) , by the Whitehead Lemma. By adding 2 - c e l l s and 3 - c e l l s to BG1(A) one forms a complex BG1(A) + i n such a way that the i n c l u s i o n i : BG1(A) c BG1(A) + induces an isomorphism i n homology with any (twisted) c o e f f i c i e n t s , and induces a map i ^ on the l e v e l of the fundamental group which f i t s i nto a commutative diagram TT 1(BG1(A)) > T T 1(BG1(A) +) G1(A) • G1(A)/E(A), where the bottom arrow i s the canonical s u r j e c t i o n . An obstruction-theoretic argument shows that i : BG1(A) • BG1(A) i s u n i v e r s a l i n the pointed homotopy category for maps f: BG1(A) *• X s a t i s f y i n g f ^ ( E ( A ) ) = e i n ^ ( X ) , i n the sense that there i s a map f + , which i s unique up to pointed homotopy, such that the diagram 67 BG1(A) • BG1(A) + f X homotopy commutes. An easy consequence i s that the assignment A i > TI\(BG1(A) +) determines a functor from unitary rings to abelian groups for a l l i > 1. We write K^A) = - (BG1 (A)"*"), f o r i ^ 1; these are the Q u i l l e n K-groups of the r i n g A. Observe that K^(A) - G1(A)/E(A). One may also show that there i s an isomorphism K 2(A) = H 2(E(A);Z). It i s also important to know that the group homomorphism ©: Gl(A)xGl(A) >• G1(A), which i s defined by (MSN). . = < ^ i f ( i , j ) = ( 2 k - l , 2 r - l ) , Nfc i f ( i , j ) = (2k, 2r), and otherwise, for matrices M and N, induces the structure of an H-space on BG1(A) +. I t follows from t h i s and the un i v e r s a l property of the + construction that, i f a CW complex X i s an H-space and there i s a map BG1(A) >- X which induces an isomorphism on i n t e g r a l homology, then X has the homotopy type of BG1(A) +., We conclude t h i s survey of p r e l i m i a r i e s by noting the following r e s u l t of Q u i l l e n : Theorem 1.1: The i n c l u s i o n A <= A[x^] induces isomorphisms R\(A) = K^(A[x^]) f o r a l l i £ 1 i f A i s a Noetherian regular r i n g . 68 The proof of Theorem 1.1 i s started i n [41] and f i n i s h e d i n [18]. Now we prove Theorem 1.2: There are isomorphisms n\ (GICA^)) = K_ +^(A) for a l l i > 0 i f A i s a regular Noetherian r i n g . The proof of t h i s Theorem requires a c e r t a i n amount of s i m p l i c i a l group theory. Let G be a s i m p l i c i a l group. Applying the c l a s s i f y i n g space f i b r a t i o n functor E ( ) > B( ) to each of the groups G p of p-simplices of G y i e l d s a homomorphism E(G) > B(G) of b i s i m p l i c i a l sets, where E(G) = E(G ) and S(G) = B(G ) . The diagonal p.q P q p,q p q s i m p l i c i a l set dZ?(G) of E(G) has p-simplices dZ?(G)p = { ( g p , . . . , g 0 ) | g. € G p}, and faces and degeneracies defined by d.(g p,...,g 0) = (d 1g p»d ig p_ 1,...,d.g p_..d ig p_._ 1,...,d.g 0), and s i ( g p " " ' 8 0 ) = ( s l 8 p , " , , S i g p - i ' e , S i 8 p - i - l - - - ' s i g 0 ) (see, f o r example, [12, p. 161]). The actions of G on E(G ) y i e l d P P P an action of G on di?(G) i n such a way that the map d£"(G) >• dB(G) i s a p r i n c i p a l Kan f i b r a t i o n . This f i b r a t i o n i s c l a s s i f i e d by a G-equivariant morphism d£"(G) »• E(G), which i s defined on p-simplices by (gp.-'-jgQ) 1 *• ( g p , d Q g p , .. . ,d Pg Q) e E(G) . Since the r e a l i z a t i o n of a Kan f i b r a t i o n i s a Serre f i b r a t i o n [38], we obtain a morphism of Serre f i b r a t i o n s 69 dE'(G) * E(G) d9(G) * B(G), with common f i b r e G. A Lemma of Q u i l l e n [41, p. 86] asserts that the space dff(G) i s the r e a l i z a t i o n of the s i m p l i c i a l space whose object i s the r e a l i z a t i o n of E(G ). F i n a l l y , since the spaces E(G ) are a l l P P co n t r a c t i b l e , a Theorem of May [34, p. 107] implies that the space dE'(G) i s c o n t r a c t i b l e as we l l . I t follows that dB(G) has the homotopy type of B(G); i n p a r t i c u l a r , these two spaces have isomorphic i n t e g r a l homology. The homology s p e c t r a l sequence of a double complex [30, p.340], together with a generalized Eilenberg-Zilber Theorem of Dold and Puppe [15, p. 213], gives Lemma 1.3: Associated to any s i m p l i c i a l group G, there i s a s p e c t r a l sequence, whose E^-term i s given by E^ " = H (G ;Z), and which p,q q P converges to (B(G);Z). This construction i s f u n c t o r i a l i n G. Lemma 1.4: For an a r b i t r a r y r i n g A, the d i r e c t sum homomorphism 9: Gl(A j t)xGl(A i t) »• Gl(A^) induces an H-space structure on BGKA^). Proof: The proof i s achieved by mimicking the proof given by Loday [28, p. 321] for the corresponding r e s u l t about BG1(A) +. Let u: N > N be an i n j e c t i v e function, where N denotes the set of natural numbers, u induces a homomorphism u: Gl(A^) *• GICA^) v i a M^_ i f ( i , j ) = (u(k),u(r)), and u(M) . 1,3 otherwise, 70 for matrices M. u preserves elementary matrices, and hence induces u: E(A A) >• E(A^). Think of str i n g s of the form (e, ,e) as a base point e f or BGl(A^), and observe that the composition BGl(A^) ( i d > £ ) > BGl(A j t)xBGl(A j,) - > BGl(A^) i s the map u: BGl(A^) >• BGl(A^) which i s induced by u ( i ) = 2 i - l . A s i m i l a r statement holds on the r i g h t . The Lemma w i l l be proved, then, once we have shown that a l l such maps u are pointed-homotopic to the i d e n t i t y on BG1(A A). This i s done by showing that u i s a homotopy equivalence, for then a Lemma of Loday [28, p. 322], which asserts that the monoid of i n j e c t i v e self-maps of N has t r i v i a l Grothendieck group, f i n i s h e s the job. But BE(A^) i s simply-connected since E(A A) i s path-connected, so i t s u f f i c e s to show that u: BE(A A) >• BE(A^) i s an isomorphism i n i n t e g r a l homology, for then u induces a homotopy equivalence on E(A^), hence on Gl(A^), and hence on BGl(A^). F i n a l l y , to see that u i s an i n t e g r a l homology isomorphism, we use a re s u l t of Loday [28, p. 321] which says that u: BE (A) >• BE (A) i s an H^(_;Z) isomorphism for a r b i t r a r y ' A, and then compare the s p e c t r a l sequences of Lemma 1.3. QED What remains for the proof of Theorem 1.2 i s Lemma 1.5: The canonical map j : BG1(A) > BGICA^) induces an H A(_;Z) isomorphism i f A i s a regular Noetherian r i n g . Proof: The map j comes from thinking of A as a discre t e s i m p l i c i a l r i n g , and then taking the map which i s induced by j : G1(A) > GICA^), which i s i n turn induced on the l e v e l of n-simplices by the i n c l u s i o n 71 A c A[x^,...,x n]. The Lemma i s proved by comparing the sp e c t r a l sequences of Lemma 1.3 for G1(A) and G1(A;J;) v i a j , using the observation that Theorem 1.1 implies that the induced map BG1(A) »- BG1 (A[x. , . . . ,x ]) 1 n i s an isomorphism on H^(_;Z). QED It follows from Lemma 1.4 and Lemma 1.5 that BGlCA^) i s pointed homotopy equivalent to BG1(A) + i f A i s a regular Noetherian r i n g . This f i n i s h e s the proof of Theorem 1.2, for then K. + 1(A) = T T . + 1 ( B G 1 ( A a ) ) = T r.(Gl(A A)) i f i > 0. In the context of Chapters II and III of inductive a f f i n e k-schemes over an a l g e b r a i c a l l y closed f i e l d k, Theorem 1.2 implies that the homotopy groups of the inductive algebraic group Gl coincide with the K-theory of k, up to a dimension s h i f t . A well known r e s u l t of Qu i l l e n [40, p. 40], when paraphrased, asserts that K 2 i - 1 ^ = ^ and ^ ^ ( k ) = 0 for a l l i ^  1 i f k i s the algebraic closure of a f i n i t e f i e l d . Not much i s known, however, about the higher K-theory of more general a l g e b r a i c a l l y closed f i e l d s k, beyond a c e r t a i n amount of number-theoretic information about ^ ( k ) . The theory of K^ of a f i e l d i s based on a Theorem of Matsumoto [32], that i s now described, which gives a presentation f o r the group 1L, (G^(k);Z), where G^ i s a Chevalley group which i s defined over k and i s of univer s a l type for an indecomposable root system $. The following r e l a t i o n s are s a t i s f i e d i n G. (k): Rl: x (t)»x (u) = x (t+u) for a e $, and t, u e k, — a a a R2: [x (t) ,x (u)] = n x. ,.„(C. . t V ) , i f a+f3 * 0, where the 72 product i s taken over a l l roots ia+jf3, i , j e N, arranged i n some f i x e d order, and where the C. . are integers which depend i> J only on the ordering and on a and 3• Moreover, C 1 = N , ±,l a,p which i s the integer such that [x ,x n] = N „x i n the simple 6 a' 3 a,3 cx+3 L i e algebra L which gives r i s e to G $ (see §3 of Chapter I I I ) . _2 * R3: w (t)'x (u)«w (-t) = x (-t u) for t e k and u e k — a a a -a ( r e c a l l that w (t) = x (t)«x ( - t - 1 ) ' x ( t ) ) , and a a -a a M: h (t-s) = h (t)*h (s) for t, s e k ( r e c a l l that — a a a h (t) = w (t)-w (-1)). a a a Steinberg shows [45, p. 78] that, since k i s large enough ( i e . i n f i n i t e ) , the symbols x ^ ( t ) , a e $, t £ k, with the r e l a t i o n s R l , R2, R3 and M together give a presentation for G^(k). One sees that G^(k) i s perfect by using the r e l a t i o n (see [45, p. 52]) [ h a ( t ) , x a ( u ) ] = x a ( ( t 2 - l ) - u ) , * v a l i d f o r a l l a e $, t e k , and u e k. The Steinberg group St^(k) i s defined to be the group which i s generated by the symbols x^(u), u e k, a e $, subject to the r e l a t i o n s Rl and R2 i f rank($) > 1, or R l and R3 i f rank($) = 1, where the elements w (t) and h (t) a a are defined analogously to the above. If rank($) > 1, then the r e l a t i o n R3 i s a consequence of Rl and R2. Steinberg has shown that the canonical homomorphism TT : St^(k) —>-*• G^(k)' i s a u n i v e r s a l c e n t r a l extension of G^ (k). I t follows that Ker ( T r ) i s isomorphic to the 73 group H 2 (G $ (k) ; Z) . R^CG^(k);Z) measures the f a i l u r e of the r e l a t i o n s R l , R2, and R3 to imply that h^ i s m u l t i p l i c a t i v e i n S t $ ( k ) . More p r e c i s e l y , l e t a be a long root of $ and define elements C a ( t , s ) of Ker(ir) f o r t, s e k by c (t,s) = h (t)«h (s)-h ( f s ) " 1 . a a a ce Matsumoto shows that c = : : c s a t i s f i e s the following r e l a t i o n s [32, p. 26]: SI: c(x,y)*c(xy,z) = c(x,yz)«c(y,z), i _ i _ i S2: c ( l , l ) = 1, c(x,y) = c(x ,y ), and S3: c(x,y) = c(x,(l-x)y) i f x * 1, for a l l x, y, and z i n k . Moreover, i f G $ i s non-symplectic, then the existence of a long root B which i s not orthogonal to a forces c = c to s a t i s f y : a Sl°: c(xy,z) = c( x , z ) • c ( y , z ) , S2°: c(x,yz) = c(x,y)•c(x,z), and S3°: c(x,l-x) = 1 i f x * 1. * * The group S(k ) of Steinberg cycles over k i s defined to be the A abelian group which i s generated by the symbols c(x,y), x, y e k , o * and subject to the r e l a t i o n s SI, S2, and S3. The group S (k ) A of b i l i n e a r Steinberg cycles over k i s the abelian group which i s A generated by c(x,y), x, y e k , but subject to the r e l a t i o n s 74 Sl°, S2°, and S3°. There i s a canonical s u r j e c t i v e homomorphism * o * . ' S(k ) »-»- S (k ) . Matsumoto's Theorem i s the following [32, p. 30]: Theorem 1.6: Let a be a long root of $. Then the homomorphism * * S(k ) »- Ker(TT) , which i s defined by c(t,s) I *• c a ( t , s ) for t, s £ k , i s an isomorphism i f i s symplectic, and factors through an isomorphism o * S (k ) *• Ker (IT) i f G i s non-symplectic. The f i e l d k does not need to be a l g e b r a i c a l l y closed for Theorem 1.6 to hold; the proof given i n [32] requires only that k be i n f i n i t e . For such a f i e l d k, define a Steinberg cocycle over k with c o e f f i c i e n t s i n an abelian group A to be a member of Ab(S(k ) ,A) . B i l i n e a r Steinberg cocycles are defined s i m i l a r l y . The Steinberg cocycles and the b i l i n e a r Steinberg cocycles with c o e f f i c i e n t s i n A form abelian * o * groups, which are denoted by S(k ,A) and S (k ,A) r e s p e c t i v e l y . o & & S (k ,A) i s a subgroup of S(k ,A). Matsumoto has shown [32, p. 28] Proposition 1.7: I f c i s a Steinberg cocycle over k with c o e f f i c i e n t s i n A, then the function c^, which i s defined on the set * * 2 * k xk by C2(x,y) = c(x,y ), i s a b i l i n e a r Steinberg cocycle over k 2 2 with c o e f f i c i e n t s i n A. Moreover, c(x,y ) = c(x ,y) for a l l x and y i n k . Krusemeyer remarks i n [27, p. 23] that i t i s a consequence of t h i s l a s t r e s u l t that, i f an i n f i n i t e f i e l d k i s q u a d r a t i c a l l y closed, then every Steinberg cocycle on k i s b i l i n e a r . C o r o l l a r y 1.8: The canonical map S(k*) —>-*• S°(k*) i s an isomorphism 75 i f k i s a l g e b r a i c a l l y closed. The constructions of Matsumoto's Theorem also respect the comparison maps i ^ of § 3 of Chapter I I I . This y i e l d s the following s t a b i l i t y r e s u l t ( r e c a l l that S l 2 i s symplectic): Co r o l l a r y 1.9: I f k i s an i n f i n i t e f i e l d , then (1) the canonical i n c l u s i o n i : SI (k) c si ,,(k) induces n n+1 isomorphisms H„(S1 (k);Z) = H„(S1 ( 1 ( k ) ; Z ) = S°(k*) f o r n > 3, <L x\ z n+± and (2) the following diagram commutes: * o * S(k ) ~ S (k ) ** H 2 ( S l 2 ( k ) ; Z ) y H 2 ( S l 3 ( k ) ; Z ) , where the v e r t i c a l maps are the isomorphisms of Theorem 1.6. Putting C o r o l l a r y 1.8 together with Co r o l l a r y 1.9 shows that there are isomorphisms H 2 ( S l n ( k ) ; Z ) = H 2 ( S l n + 1 ( k ) ; Z ) = S°(k*) for n > 2, i f k i s an a l g e b r a i c a l l y closed f i e l d . 76 §2. The fundamental group We begin with some g e n e r a l i t i e s about s i m p l i c i a l groups. As i n § 1 , l e t G be a s i m p l i c i a l group, and r e c a l l the s p e c t r a l sequence of Lemma 1 . 3 . Suppose that ( 1 ) the groups G Q and G ^ are perfe c t . 2 Then the E -term of the s p e c t r a l sequence i s as follows: 2 • ( I 2 1 0 0 • 0 0 0 0 0 0 1 2 3 It follows that there i s a well-defined composition H 2 ( G 0 ; Z ) " E 0 , 2 - ~ E 0 , 2 - ~ E 0 , 2 = H 2 ( B G ; Z ) -1 2 Observe that the map 2 — » - > • EQ 2 * S t^e cokernel of the map ( d A . - d-.): H0(G..;Z) >• H_(G -Z). Thus, the above composition i s an isomorphism i f the extra conditions -( 2 ) s Q ^ : H 2(G Q;Z) • H^G^Z) i s an isomorphism, and (3) G 2 i s perfect are s a t i s f i e d by G. F i n a l l y , i f we assume that (4) G i s path-connected, then we may i d e n t i f y H 2(BG;Z) with ir (G), f o r then BG i s simply-77 connected, and there are isomorphisms H2(BG;Z) = Tr (BG) = TT (G). A l l of these i d e n t i f i c a t i o n s and maps are natural f o r the s i m p l i c i a l groups G for which they make sense. Now consider the algebraic group S l ^ , which i s defined over an a l g e b r a i c a l l y closed f i e l d k, and suppose that n ^ 3. TT (SI ) ± n i s defined, as before, to be the fundamental group of the s i m p l i c i a l group SI (k.) . SI (k) and SI (k[x..]) are perfect, since both are n K n n 1 generated by elementary transformations, and the respective elementary transformation groups are perfect by the r e l a t i o n [ h a ( t ) , x o ( u ) ] = x a ( ( t 2 - l ) u ) , A which i s v a l i d f o r a l l a e A u e k[x,], and t e k . Observe also n—1 1 that, f o r a l l n, the s i m p l i c i a l groups SI (k.) are path-connected n * by the r e s u l t s of Chapter I I , hence so i s SlCk^). By the discussion above, there i s a commutative diagram H 2 ( S l n ( k ) ; Z ) — v > H 2 ( B S l n ( k A ) ; Z ) = ^ ( S l ^ ) ) H 2(Sl(k);Z) H 2(BSl(k A);Z) = ^ (SKk^)), for which the v e r t i c a l maps are induced by the canonical i n c l u s i o n Sl^Ck^) <= s l ( k A ) , and the h o r i z o n t a l maps come from the respective s p e c t r a l sequences. Observe that H 0(S1 (k);Z) *• H 0(Sl(k);Z) z n L i s an isomorphism f or n S 2 by the remark following C o r o l l a r y 1.9. 78 The i n c l u s i o n k c k[x.,...,x ] induces an isomorphism 1 n K_(k) = K^(k[x^ x n ^ a ^ n ~ ^' ^ Theorem 1.1. I t i s an easy consequence of the ''K- isomorphism that E(k[x 1,...,x ]) = Sl(k[x.,. 1 1 n 1 i n Gl(k[x^,...,x n]) f or a l l n > 1, so, i n p a r t i c u l a r , S l ( k [ x ^ , x 2 ] ) i s p e r f e c t . Observe also that the homomorphism s Q * : H 2(Sl(k);Z) • H 2(SI(k[x^]);Z) may be i d e n t i f i e d with the homomorphism H 2(E(k);Z) >- H^(E(k[x^.]) ;Z) which i s induced by k c k[x t h i s i s an isomorphism, again by Theorem 1.1. We have shown that the s i m p l i c i a l group Sl(k^) s a t i s f i e s the conditions (1) - (4) which are given above. It follows that the bottom h o r i z o n t a l map i n the above diagram i s an isomorphism, so that the top map i s an isomorphism as w e l l . We have proved Proposition 2.1: There are isomorphisms r r / S l ) = H_(S1 (k);Z) = K_(k) i n _ .n I for a l l n > 2 i f k i s a l g e b r a i c a l l y closed. One of the basic components of the proof of Proposition 2.1 was the fa c t that the groups SI (k[x n]) are perfect f o r n > 2. In a n 1 s i m i l a r way, s t a r t i n g from a r e s u l t of Matsumoto [31, p. 102], which says that the groups G^(k[x^]) of k[x^]-points of a Chevalley group G $ of u n i v e r s a l type over k i s generated by elementary transformations, one can show that a l l such groups G $(k[x^]) are perf e c t . Thus, there i s a s u r j e c t i v e homomorphism (*) H 2(G $(k);Z) — H 2 ( B G $ ( k A ) ; Z ) = ^ ( G ^ k * ) ) , 79 which a r i s e s from the s p e c t r a l sequence f o r H.(BG.(k.);Z) j u s t as before. Now we may prove Theorem 2.2: I f k i s an a l g e b r a i c a l l y closed f i e l d , then there i s an isomorphism H„(Sp. (k):Z) = TT. (Sp„ ), for every m > 1. z zm 1 zm Proof: The fix e d points of the n o n - t r i v i a l diagram automorphism of the Lie algebra s ^ ^ C ) form a simple L i e algebra of type C m i n such a way that the root f o r s ^ ^ C ) corresponding to the (m,m+l) entry of the matrices involved r e s t r i c t s uniquely to a long root a of C^. Moreover, the induced representation of L i s of uni v e r s a l type. Thus, the Chevalley group over k which a r i s e s from t h i s representation can be i d e n t i f i e d with SP2 m' and gives r i s e to an imbedding i : Sp2 m(k) <= S ^ ^ k ) which has the property that the elements of the form x (u) and x (u) may be i d e n t i f i e d with X ,,(u) and a -a J m,m+l X ,, (u) re s p e c t i v e l y , f o r a l l u e k. I t follows that there i s a m+1, m commutative diagram S(k*) S°(k*) H2 ( S p 2 m ( k ) ; Z ) " H2 ( S 1 2 m ( k ) ; Z ) ' i n which the top h o r i z o n t a l map i s the canonical one. But t h i s homomorphism i s an isomorphism since k i s a l g e b r a i c a l l y closed, by Cor o l l a r y 1.8, so that i ^ i s an isomorphism as w e l l . Comparing s p e c t r a l sequences as i n the proof of Proposition 2.1 y i e l d s a commutative diagram 80 H 2(Sp 2 m(k);Z) - ~ H 2(BSp 2 m(k,);Z) ~= ^ ( S p ^ k , ) ) 1 A H 2 ( S l 2 m ( k ) ; Z ) > H 2 ( B S l 2 m ( k A ) ; Z ) = ^ ( S l ^ k * ) ) , i n which the bottom morphism i s an isomorphism by Proposition 2.1. I t follows that the top homomorphism i s i n j e c t i v e as w e l l as s u r j e c t i v e , and the Theorem i s proved. QED An imbedding Sp„ (k) c SI. (k) which i s suitable f o r the proof of _m _m Theorem 2.2 may also be obtained by applying a Lemma of Matsumoto [32, p. 37] One conjectures that there are isomorphisms W a W k ) ; Z ) a V k ) ' i f i s any Chevalley group of u n i v e r s a l type over any a l g e b r a i c a l l y closed f i e l d k. There i s some basis for b e l i e v i n g that the conjecture i s true over more general f i e l d s . Steinberg showed [45, p. 72], for example, that H 2(G^(k);Z) - 0 for a l l f i n i t e f i e l d s k and indecomposable root systems $ such that |k| > 4 and |k| * 9 i f rank($) = 1 (these are the cases where St^(k) may not be a unive r s a l c e n t r a l extension of G $(k) [45, p. 78]). The Matsumoto r e s u l t that was used to construct the homomorphism (*) above i s broad enough so that there i s such a su r j e c t i v e homomorphism for a l l f i e l d s k outside of the small l i s t j u s t r e f e r r e d to (from now on the expression "almost a l l f i e l d s " w i l l mean every f i e l d except those i n the l i s t ) . This proves Proposition 2.3: H 2(G $(k);Z) = ir (G $) = 0 for a l l f i n i t e f i e l d s k 81 and indecomposeable root systems $ such that |k| > 4 and |k| * 9 i f rank($) = 1. Proposition 2.1 i s also true much more generally, i n view of the remark of the l a s t section which says that Theorem 1.6 i s v a l i d over a l l i n f i n i t e f i e l d s . I t follows that the argument given for Proposition 2.1 generalizes to a l l i n f i n i t e f i e l d s , provided that we avoid the symplectic group SI,,. Putting t h i s together with the l a s t Proposition y i e l d s Theorem 2.4: There are isomorphisms i r 1 ( S l n ) = H 2 ( S l n ( k ) ; Z ) = K 2 ( k ) , for a l l n > 3, and for any f i e l d k such that |k| > 4. Another one of the main devices that i s used i n the proofs of both Proposition 2.1 and t h i s l a s t Theorem i s the homotopy property for Q u i l l e n K-theory, which i s Theorem 1.1. A s i m i l a r property holds for the Karoubi _^L-theory of a commutative r i n g A (see [28], [25]), which i s defined, f or n ^ 1, by -L (A) — TV (BSp(A) ), where Sp i s the —I n n i n f i n i t e symplectic group, and the space BSp(A) + i s defined by analogy with BG1(A) +. E x p l i c i t l y , t h i s homotopy property says that, i f A i s a regular Noetherian r i n g which contains 1/2, then the i n c l u s i o n A c A[x^] induces an isomorphism _^L^(A) = _^L^(A[x^]) for every n > 1. Proceeding as i n the proof of Theorem 2.4 gives Theorem 2.5: For almost a l l f i e l d s k such that char(k) * 2, there are isomorphisms 82 H 2(Sp 2 m(k);Z) = T T 1 ( S p 2 m ) = S(k*), for a l l m S 1, where S(k ) i s the group of Steinberg cycles over k . The "char(k) * 2" condition of Theorem 2.5 i s forced upon us by the "containing 1/2" part of the statement of the homotopy property for Karoubi's _^L-theory. I do not know i f t h i s condition may be removed outside of the a l g e b r a i c a l l y closed case. The same phenomenon appears i n the following analogue of Theorem 1.2: Theorem 2.6: Suppose that A i s a commutative Noetherian regular r i n g which contains 1/2. Then BSp(A^) i s an H-space which has the homotopy type of BSp(A) +, and there are isomorphisms T.(Sp(A.)) = .L.^CA) for a l l i > 0. 83 3. A vanishing r e s u l t I t was established i n the previous Section that, i f k i s an a l g e b r a i c a l l y closed f i e l d and S l ^ i s defined over k, then Tr (SI ) = K„(k), and t h i s f o r a l l n > 2. R e c a l l that SI n i s the 1 n 2 n+l univer s a l Chevalley group over k of type A^. In the l a s t Chapter, i t was shown that the covering map X: SI ., >• G , onto a r n+l p Chevalley group G p over k of the same type, i s a f i b r a t i o n , with a group-scheme kernel whose A-points consist of the abelian group homomorphisms Ab(II(G p),A ) for a l l A e M^, where n ( G p ) i s the geometric fundamental group of G . In t h i s case, II (G ) i s a quotient P P of Z/(n+l)Z. R e c a l l that the long exact sequence of a f i b r a t i o n gives isomorphisms TT . (SI ,., ) = TT . (G ), which are induced by X for a l l i > 1, x n+l x p ' J ' and a short exact sequence ° — * — * w — * *<v — * °» where n(Gp) i s II (G p), mod i t s p-primary component i f char(k) = p. A s i m i l a r r e s u l t holds i n the symplectic case. In t h i s Section, well-known Theorems on the structure of K^(k) are applied to the study of 7 T i ( S l n ) a n d 1 Tl^ S p2m^ ''"n t* i e a x S e b r a i c a H y closed s e t t i n g . A r e s u l t of Steinberg that was quoted i n the l a s t Section implies that K 2(k) = H 2(Sl(k);Z) = 0 i f k i s the algebraic closure of a f i n i t e f i e l d . I t can be shown that K 2(k) i s a uniquely d i v i s i b l e group f or any a l g e b r a i c a l l y closed f i e l d k (see [4, p. 357]). This implies that K 2(k) i s i n f e c t i v e , so that we have 84 Proposition 3.1: Let G^  be a Chevalley group over the a l g e b r a i c a l l y closed f i e l d k which i s of type A or C for a l l n, m > 1. Then n m there i s an isomorphism TT. (G ) = K n (k) $ W(G ). 1 p 2 p One may ask when the groups S l n and S p 2 m a r e simply-connected i n the sense that they have t r i v i a l fundamental group; these groups are simply-connected i n a geometric sense, so that one i s asking when ( i e . for what f i e l d s k) these two notions coincide. This i s the case, f o r example, i f k i s the algebraic closure of a f i n i t e f i e l d . Milnor has shown, however, that K 2(C) i s uncountable [35, p. 107], so that K 2(k) does not vanish i n general. In p a r t i c u l a r , ^ i ^ l +-^) does not coincide with the usual fundamental group of the L i e group S l n + ^ ( C ) when S l n + ^ i s defined over C. Milnor's method of proof may be used to say more. The Kroenecker dimension 6(k) of k i s defined by 6(k) = i Tr.deg^(k) + 1 i f char(k) = 0, and [ Tr.deg F (k) i f char(k) = p. P Then, following Milnor, we show Lemma 3.2: K 2(k) i s n o n - t r i v i a l i f 6(k) > 1. Proof: Since <5(k) > 1, k i s eit h e r an algebraic extension of a f i e l d F(x), where F contains Q, or F contains ^ (y) i f char(k) = p. Take the x-adic valuation v: F(x) »• Z and extend i t to a valuation of k. The r e s t r i c t i o n of t h i s valuation to a f i n i t e algebraic extension L of F(x) determines a d i s c r e t e valuation v: L • Z i n such a way that, i f L' i s a f i n i t e algebraic extension of L, then there i s a 85 commutative diagram A A A F(x) c L c L ' v V V * • z • z, xn x m where the p o s i t i v e integers m and n are r a m i f i c a t i o n indices. A Associated to any discr e t e valuation v: L -*• Z i s a s u r j e c t i v e homomorphism 3 : K„(L) v 2 -> K^LCv)) = L(v) , where L(v) i s the residue c l a s s f i e l d of v, which i s defined on a representative x®y by 3 (x®y) = ( - l ) v ^ x ^ v ^ y ^ [ x v ^ y ) / y v ^ x ^ ] . There i s a commutative diagram K 2(F(x)) v F(x)(v) n -* K 2(L) •* L(v) - K 2(L') L'(v) , m for L and L' as above, where, for example, p (x) = x 1 1. Observe n that F(x)(v) = F. D i v i d i n g by the roots of unity from the groups L(v) gives groups L(v) and induces i n j e c t i o n s p^, etc. Passing to the d i r e c t l i m i t determines maps K 2(k) F c L , -A where L = l i m L ( v ) . Not every element of Q or F (y) i s a root of unity, —A so that F * 0, whence K 2(k) * 0. QED 86 Thus, K 2(k) = 0 i f <5(k) = 0, and K 2(k) * 0 i f <S (k) > 1. This leaves the case <5(k) =1, i n which k i s eit h e r a union of number f i e l d s i n c h a r a c t e r i s t i c 0, or a union of function f i e l d s i f char(k) = p. K 2 of such f i e l d s has been studied extensively (see [35], [4], [3], [16], [14]). Central to that study i s the homomorphism 3^: K 2(L) »- K ^ L C O ) , which i s associated to a di s c r e t e valuation v of L, and which was introduced i n the proof of Lemma 3.2. If L i s a global f i e l d i n the sense that i t i s e i t h e r a number f i e l d or a function f i e l d , then the residue c l a s s f i e l d L(v) i s f i n i t e f o r any d i s c r e t e valuation v. Moreover, L i n h e r i t s the property from e i t h e r Q or F p ( y ) that a l l but f i n i t e l y many di s c r e t e valuations vanish on a f i x e d element x e L. Thus, by adding up the homomorphisms 3^, we may form a homomorphism 3: ,K 2(L) > ® K ^ L t v ) ) , v where the sum i s taken over a l l d i s c r e t e valuations v of L. Putting together well-known r e s u l t s of Bass, Tate, and Garland (see the discussion [4, p. 396] and [3, p. 243]), one sees that the kernel of 3 i s a f i n i t e group, and i t follows that K 2(L) i s a tors i o n group. But then K 2(k) i s a d i r e c t l i m i t of such K 2 ( L ) , so that K 2(k) i s a t o r s i o n group as well as being uniquely d i v i s i b l e . I t follows that K 2(k) = 0. We have,.in e f f e c t , shown Theorem 3.3: Suppose that S l ^ and s P 2 m a r e defined over an a l g e b r a i c a l l y closed f i e l d k. Then the groups TT, (SI ) and TT. (Sp. ) are t r i v i a l i f I n i zm 6(k) < 2, and are n o n - t r i v i a l uniquely d i v i s i b l e abelian groups otherwise. 87 Bibliography: [1] D.W. Anderson: Relationship among K-theories, Lecture Notes i n Mathematics, Vol. 341, Springer.-Verlag (1973), 52-67. [2] M. A r t i n and B. Mazur: Etale homotopy, Lecture Notes i n Mathematics, Vol. 100, Springer-Verlag (1969). [3] H. Bass: K des corps globaux (d'apres J . Tate, H. Garland."...), Sem. BourbaEi n°394 (1971), 233-255. [4] H. Bass and J . Tate: The Milnor r i n g of a global f i e l d , Lecture Notes i n Mathematics, Vol. 342, Springer-Verlag (1973), 349-428. [5] A.K. Bousfield and V.K.A.M. Guggenheim: On PL deRham theory and r a t i o n a l homotopy type, Memoir Am. Math. Soc. 179 (1976). [6] A. Borel: Linear Algebraic Groups. W.A. Benjamin';. 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