COHOMOLOGY OF GROUPS


 Darcy Preston
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1 Actes, Congrès intern. Math., Tome 2, p. 47 à 51. COHOMOLOGY OF GROUPS by Daniel QUILLEN * This is a report of research done at the Institute for Advanced Study the past year. It includes some general results on the structure of the ring H* (BG, Z/pZ) when G is a compact Lie group, a theorem computing this ring for a large number of interesting finite groups, and applications to algebraic Aftheory consisting of a definition of ATgroups K t A for / > 0 agreeing with those of Bass and Milnor and their computation when A is a finite field. 1. The spectrum of H*(BG, Z/pZ). Let G be a compact Lie group (e.g. a finite group) and let H*(BG) be the cohomology ring of its classifying space with coefficients in Z/pZ where p is a fixed prime number. According to Venkov (and Evens for finite G) the ring #*(5G) is finitelygenerated, hence its Poincaré series 2 (dim z/pz H n (BG)) f 1 is a rational function of t and one may define the dimension dim H*(BG) to be the order of the pole of this function at t = 1. For example if A = (Z/pZ) r is an elementary abelian pgroup ([p]group for short) of rank r 9 then dim H*(BA) = r. The following for finite G has been conjectured independently by Atiyah and Swan. PROPOSITION 1.  dim H*(BG) = the maximum rank of a [p]subgroup G. To prove this one follows the method used by AtiyahSegal to prove the completion theorem in equivariant ^theory and first generalizes it to Gspaces, which for the sake of simplicity I suppose to be smooth compact Gmanifolds with boundary. Let X G be the associated fibre space over BG with fibre X and set H*(X) = H*(X G ). PROPOSITION 1'  dim H G (X) = the maximum rank of a [p]subgroup G fixing some point of X. To prove this one can replace the pair (G, X) by (U, Y) where U is a unitary group containing G and Y = U x G X ; then one can reduce to the case (A, Y) where A is the subgroup of elements of order p in a maximal torus of U, because H%(Y) is a finitelygenerated free //y(y>module. Hence one can of of (*) Supported by The Institute for Advanced Study and the Alfred P. Sloan Foundation.
2 48 D. QUILLEN C 1 suppose that G is a [p]group, in which case the result can be checked by using the spectral sequence E% = H S (X/G, Gx > H G (Gx)) => H G +t (X). The same technique can be used to prove the following result. THEOREM  Consider the [p]subgroups A of G as the objects of a category in which a morphism from A to A 1 is a component of the set of g such that gag~ l CA' 9 and let u : H*(BG)*]imH*(BA) be the homorphism induced by restriction. Then every element of Ker (u) is nilpotent and if z is an element of the inverse limit then z pn G Im (u) for large n. In other words 'up to extraction of pth roots' a cohomology class of BG is the same as a family of cohomology classes for each [p]subgroup compatible with conjugation and restriction. One should compare this result with Brauer's theorem asserting that the analogous map with character rings and the category of elementary subgroups is an isomorphism when G is finite. This theorem and some commutative algebra permit one to deduce the following description of the space Spec H*(BG) of prime ideals in H*(BG) (i.e. inverse images of prime ideals in the commutative ring H*(BG) nd = H*(BG)/ideal of nilpotent elements). If A is a [p]subgroup of G, let h = Ker {H*(BG) + H*(BA) ted }. Then ^4 > ^^ gives an orderreversing bijection between conjugacy classes of [p]subgroups and those homogeneous prime ideals of H*(BG) which are closed under the Steenrod operations. In particular the irreducible components of Spec H*(BG) are in oneone correspondence with maximal [p]subgroups up to conjugacy. If T A is the v )set of prime ideals containing $ A but not fc 4, for A 1 < A, then there is a s tification Spec H*(BG) =L1T A into irreducible locally closed subspaces indexed by the conjugacy classes of [p]subgroups. Moreover T A = (Spec S(A v )[e l A ])/N(A) where N(A) is the finite group of components of the normalizer of A in G, where S(A y ) = H*(BA) Kd is the symmetric algebra of the dual of A over Z/pZ, and e A is the product of the nonzero elements of A. 2. Computations using etale cohomology and the Lang isomorphism. One knows (Chevalley, Steinberg) that a large number of interesting finite groups occur as the group G of fixpoints of an endomorphism a of a connected algebraic group G defined over an algebraically closed field k. For example if G is defined over a finite subfied k 0 of k then the group of rational points G(k 0 ) is the group of fixpoints of the Frobenius endomorphism associated to this finite field of definition. Since G a is finite there is an inseparable isogeny
3 COHOMOLOGY OF GROUPS 49 G/G a > G gg'tgiog) 1 (the Lang isomorphism when a is a Frobenius endomorphism), hence G/G a and G are homeomorphic for the etale topology. This suggests that H*(BG a ) (coefficients in Z//Z where / is a prime number different from the characteristic of k) might be computed by using the analogue in etale cohomology of the Leray spectral sequence of the "fibration" (G/G a, BG a, BG) 9 because the rings H*(BG) and H*(G) are usually known, e.g. by lifting G to characteristic zero. Before going on I should explain what is meant by BG in this context. Let % be the topos of sheaves for the etale topology on the category of all algebraic fcschemes. Identifying a fcscheme with the sheaf it represents, G becomes a group object of % and so it has a "classifying topos" % G consisting of objects of ^ endowed with Gaction (Grothendieck, réédition of SGAA). If X is a fcscheme endowed with a Gaction, let X G be the object of *S G it gives rise to, and denote by H G (X) the cohomology of X G with coefficients in the constant sheaf Z//Z ; write BG instead of e G where e = Spec k. The Leray spectral sequence for the map X G +BG 9 or as I shall say of the fibration (X, X G, BG) takes the form (1) E 2 =H*(BG) H*(X)=*H*(X) provided the map X > e is cohomologically proper, which is the case for X = G because the map factors into a sequence of principal G a and G m bundles and the proper map G/B * e. Taking X to be G acting on itself by left translations gives a spectral sequence (2) E 2 = H* (BG) H* (G) => H* (e). Assume that this spectral sequence has the nice form studied by Borei in his thesis, namely H*(G) has a simple system of transgressive generators, whence H*(BG) = S(V) is a polynomial ring and the transgression sets up an isomorphism of the primitive subspace P of H*(G) and V[ 1] (the [ 1] means degrees are shifted down by one). When X = G t 9 the Gscheme obtained by letting G act on itself by the rule g(g x ) = gg x (og)~ l 9 (1) takes the form (3) E 2 = H*(BG) 9 H*(G) ** H*(G/G a ) = H*(BG a ). on account of the Lang isomorphism. To determine the differentials in (3), let G s be the (G x G)scheme obtained by letting G x G act on G by the rule (#i. 2) = SiStfi" 1 an d consider the map of spectral sequences associated to the map (G, (G%,5G)> (G, (G s ) GxG,B(G x G )). In the latter spectral sequence a primitive element z of H* (G) transgresses to v 1 1 v if z transgresses to v in (2), consequently in (3) z transgresses to v o*(v). Thus the spectral sequence (3) can be determined completely and it yields the following. THEOREM  Let G be a connected algebraic group defined over an algebraically closed field k, and let a be an endomorphism of G such that G a is finite. Assume that the etale cohomology 77* (G) (coefficients in Z/ Z,J2 prime = char (k)) has a simple system of transgressive generators for the spectral sequence (2) (e.g. if H*(G) is an exterior algebra with odd degree generators), and that the
4 50 D. QUILLEN C 1 subspace V of generators for the polynomial ring H* (BG) can be chosen so as to be stable under a*. Let V a and V a be the kernel and cokernel of the endomorphism id a* of V. Then there is an isomorphism of graded Z/fiZ vector spaces H*(BG a )=S(V a )*h (V a [ 1]) which is an algebra isomorphism if I is odd. This theorem may be used to determine the mod cohomology rings of the classical groups over a finite field k Q (at least additively when = 2) provided fi is different from the characteristic. For example if k 0 has q elements and r is the order of q mod fi, then H*(BGL n (k Q ), Z/fiZ) is the tensor product of a polynomial ring with one generaror of degree 2/V and an exterior algebra with one generator of degree 2/> 1 for / < / < [n/r], except that this is only true additively when fi = Applications to algebraic Xtheory Let A be a ring, GL (A) its infinite general linear group, and E (A) the subgroup generated by elementary matrices. As E (A) is perfect, by attaching 2 and 3cells to BE (A) to kill its kundamental group without changing homology one can construct a map / : BGL(A) > BGL(A) + such that ir t (f) kills E(A) and such that / as a map in the homotopy category of pointed spaces is universal with this property. Set K t A = 7r,BGL(A) + for / > 1 ; it is not hard to show that this definition agrees with those of Bass and Milnor. The representable functor on the homotopy category K(X ;A) = [X, K 0 A x BGL(A) + ] deserves to be called ^theory with coefficients in A, because it enjoys many of the properties of topological ATtheory. For example it is the degree zero part of a connected generalized cohomology theory. Indeed Graeme Segal has recently associated such a cohomology theory to any category with a coherent commutative associative composition law, and the cohomology theory in question come from the additive category of finitelygenerated projective ^4modules. Also K(X \A) is naturally a Xring when A is commutative. If one wants to compute the groups K t A by standard techniques of homotopy theory (e.g. unstable Adams spectral sequence for the //space BGL(A) + ) it is necessary to know the homology of BGL(A) +, which is the same as that of BGL(A). For example K t A Q is isomorphic to the primitive subspace of H t (BGL(A) 9 Q). For a finite field enough is known about the homology to do the computation : Let k Q be a finite field with q elements, let k be an algebraic closure of k Q, and let 0 : k* * C* be an embedding. By modular character theory one knows how to associate to a representation of a group G over k 0 a virtual complex representation fixed under the Adams operation ty q by using 0 to lift eigenvalues. Lifting the standard representation of GL n (k 0 ) on k% 9 one obtains a map BGL n (k 0 ) > E^! q 9 where the latter space is the fibre of the endomorphism ^q id of BU. This map kills elementary matrices, hence gives rise to a map in the pointed homotopy category
5 COHOMOLOGY OF GROUPS 51 BGL(k 0 ) + which depends only on the choice of 0. *EV q THEOREM 1. The map (*) is a homotopy equivalence. This is proved by showing that the map induces an isomorphism on homology and using the Whitehead theorem. The homotopy groups of E^q may be computed by using Bott periodicity, so one obtains the formulas K 2i (k 0 ) = 0 / > 1 The functorial behavior of these groups as the finite field varies may be determined in similar fashion, and it leads to the following : THEOREM 2. If k is an algebraic closure of F p, then K 2i (k) = 0 i>\ V,W= Q fi /Zg i> 1 fifp and the Frobenius automorphism of k over F p acts on K 2i _ x (k) by multiplying by p*. If k t is any subfield of k, then the extensionofscalars homomorphism induces an isomorphism Gal (*/*,) K 2i _ x (k,) ~ K 2i _,(k) l/l> in terms of which the restrictionofscalars homomorphism u* : K 2i _ x (k 2 \+ Ku^kJ associated to a finite extension u : k x * k 2 is given by the norm from Ga\(k/k 2 ) invariants to GdX (kik^invariants. Massachusetts Institute of Technology Dept. of Mathematics Cambridge, Massachusetts U.S.A.
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