HOMOLOGY OF LINEAR GROUPS VIA CYCLES IN BG x X KEVIN P. KNUDSON AND MARK E. WALKER 1. Introduction Homology theories for algebraic varieties are often constructed using sim- plicial sets of algebraic cycles. For example, Bloch's higher Chow groups and motivic cohomology, as defined by Suslin and Voevodsky [17], are given in this fashion. In this paper, we construct homology groups Hi(X, G), where G is an algebraic group and X is a variety, by considering cycles on the simplicial scheme BG x X, an idea first suggested by Andrei Suslin. If X = Spec(R) is an affine scheme, then there is a natural map Hi(G(R), A) -! Hi(X, G; A), whose source is the usual group homology of the discrete group G(R) of R-points of the algebraic group G. Moreover, this map is an isomorphism if R = k and k is algebraically closed, so that these groups capture the homology of the discrete group G(k). The functors Hi(-, G; A) are naturally equipped with transfer maps in the sense of Suslin-Voevodsky [16]. We also have cohomological versions Hi(X, G; A) and by applying these to the standard cosimplicial variety okwe deduce a spectral sequence Es,t1= Hs( t, G; Z=n) =) Hs+t'et(BGk, Z=n) for k algebraically closed with char(k) not dividing n. Since Hi( 0, G; A) ~= Hi(G(k), A), the bottom row of the spectral sequence is just Ho(G(k), Z=n). Thus, this spectral sequence provides a mechanism for comparing the group cohomology of the discrete group G(k) with the 'etale cohomology of the simplicial scheme BGk. We recall the following conjecture of E. Friedlander [4]. Conjecture 1.1 (Friedlander's Generalized Isomorphism Conjecture). If G is an algebraic group over an algebraically closed field k, then the natural map of simplicial schemes BG(k) ! BGk (with the first one being "discrete") induces an isomorphism Ho'et(BGk, Z=n) -! Ho(BG(k), Z=n) = Ho(G(k), Z=n) ____________ Date: November 19, 2003. 1991 Mathematics Subject Classification. Primary 14F42; Secondary 14C25 19E15 20G10 55N91. First author partially supported by NSF grant nos. DMS-0070119, DMS-0242906. Second author partially supported by the NSA. 1 2 KEVIN P. KNUDSON AND MARK E. WALKER provided char(k) does not divide n. If k = C, then the Isomorphism Conjecture reduces to the claim that the identity map Gffi! Gtop, where Gffiis the Lie group Gtopviewed as a discrete group, induces an isomorphism in cohomology with finite coefficients: Ho(BGtop, Z=n) ! Ho(BGffi, Z=n) = Ho(Gffi, Z=n). In this context, it also makes sense to consider the field k = R and assert this isomorphism for real Lie groups. The Isomorphism_Conjecture is known to hold in the following cases: (1) k = Fp, G arbitrary [4]; (2) G solvable [4],[7],[11]; (3) G is one of the stable groups GL, SL, Sp, O [7],[8],[14]; (4) H2(G, Z=n) for G a real Chevalley group [13]; (5) H3(GLm (k), Z=n) (k arbitrary), H3(SL2(C), Z=n), [9],[12],[15]. Using the spectral sequence described above, we are able to deduce all the known cases of the Isomorphism Conjecture immediately (except for the results about real Lie groups). Moreover, we can explicitly identify the group H4'et(BGLm , Z=n). Corollary 3.5. There is an exact sequence 0 -! H4'et(BGLm , Z=n) -! H4(BGLm (k), Z=n) -d! H4( 1, GLm ; Z=n) where d is the difference of the maps induced by the two face maps 0 ! 1. Conjecturally, the map d is the zero map, of course. This paper is organized as follows. In Section 2 we construct the functors Hi(X, G; A) and discuss their basic properties. Section 2.3 contains some relatively easy examples. The spectral sequence is constructed in Section 3. There is a_relationship_between the_Hi(Spec (k), G; A) and the homology groups Hi(BG(k)= ; A), where = Gal(k=k). We discuss this in Section 4 and prove the following results. Theorem 4.1. Let G be an algebraic group over R and let A = Q or Z=n where n is odd. Then there are canonical isomorphisms Hi(Spec (R), G; A) ~=Hi(G(C), A)Z=2 for all i 0. If A = Z=2, then the structure of Hi(Spec (R), G; A) is much more com- plicated. However, we can deal with the case G = Gm . Theorem 4.8. For all k 1, H2k(Spec (R), Gm ; Z=2) ~=H2k+1(Spec (R), Gm ; Z=2) ~=(Z=2)k-1. This is proved by noting that an equivariant version of Friedlander's con- jecture holds for Gm , and then using the calculation of the Bredon cohomol- ogy of BGm (C)top' CP1 given in [3]. HOMOLOGY OF LINEAR GROUPS 3 We are also able to handle unipotent groups over R. We discuss this at the end of Section 4. Notation. Throughout, k denotes a field and p denotes the exponential characteristic of k; that is, p = 1 if char(k) = 0 and p = char(k) if char(k) > 0. Acknowledgments. The construction of the Hi(X, G; A) was suggested to us by Andrei Suslin. We thank him for putting us on this path. Conventions: Throughout this paper, given a field k, a k-scheme is a sepa- rated scheme of finite type over k. 2.Construction of the functors Hi(X, G; A) Let G be a linear algebraic group over k. Consider the simplicial classi- fying scheme BG: oo___ oo___ oo___. oo___. oo___. *oo___ G oo___G2oo_o..oG3_. .o...oGn_oo_..._. .. For any k-scheme X, we may form the product BG x X: oo___ oo___ oo___. oo___. n oo___. X oo___G x X oo___G2oxoX_ oo.._G3 x X . .o.o.._G x X oo.._. ... If U and V are smooth k-schemes, define C0(U x V=V ) as 8 9 < closed integral subschemesZ U x V such = C0(U x V=V ) = Z that Z -! V is finite and surjective over . : some connected component of V ; This construction was first studied by Suslin-Voevodsky in [16]. The functor C0(U x V=V ) is covariant in U via pushforward of cycles. In more detail, if p : U ! U0 is a morphism of k-schemes and if Z U x V is a closed integral subscheme that is finite and surjective over some connected component of V , then Z0 = (p x idV)(Z) is finite and surjective over some component of V . The map p* is defined by sending [Z] (the cycle associated to Z) to d[Z0], where d is the degree of the finite, dominant map Z ! Z0, and then extending p* to all of C0(U x V=V ) by linearity. The functor C0(U x V=V ) is contravariant in V via pullback of cycles. Given a morphism f : V 0! V of smooth k-schemes, for a closed, integral subscheme Z of U x V that is finite and surjective over a component of V , the pullback f-1 (Z) = V 0xV Z is a closed subscheme of U xV 0each integral component of which is finite and surjective over a component of V 0. One defines f*([Z]) to be the cycle in U x V 0associated to f-1 (Z) by taking multiplicities of its integral components in the usual fashion. The definition of f* is then extended to all of C0(U x V=V ) by linearity. In particular, we apply the covariant functor C0(- x X=X) degreewise to the simplicial scheme BG to obtain a simplicial abelian group C0(BG x 4 KEVIN P. KNUDSON AND MARK E. WALKER X=X): oo___ oo___. oo___. n oo___. C0(* x X=X) oo___ C0(G x X=X) oo_.._. .o...oC0(G_ x X=X) oo.._. ... Definition 2.1. Let A be an abelian group. The group Hi(X, G; A) is de- fined to be the i-th homotopy group of the simplicial abelian group C0(BG x X=X) A; that is, Hi(X, G; A) = hi(C0(BG x X=X) A, d), P where d = (-1)kpk* and pk : Gi! Gi-1 is the map 8 ><(g1, . .,.gi-1) k = 0 pk(g0, g1, . .,.gi-1) = (g0, . .,.gk-1gk, . .g.i-1)0 < k < i - 1 >: (g0, . .,.gi-2) k = i - 1. We also define groups Hi(X, G; A) by Hi(X, G; A) = hi(Hom (C0(BG x X=X), A)). 2.1. Variance. The notation Hi(X, G; A) was chosen to suggest the same variance in X and G as the bi-functor Hom (-, -). Namely, if f : G ! H is a morphism of linear algebraic groups over k, then there is an induced map f* : Hi(X, G; A) ! Hi(X, H; A) for any scheme X. Indeed, the push-forward maps fxn*: C0(Gn x X=X) ! C0(Hn x X=X), n 0, are compatible with the simplicial structures so that we obtain a map of simplicial abelian groups f* : C0(BGxX=X) ! C0(BHxX=X) and hence a map on homotopy groups as desired. Likewise, if ' : X ! Y is a morphism of smooth k-schemes, then pull-back of cycles determines maps '* : C0(Gn x X=X) ! C0(Gn x Y=Y ), n 0, that are again compatible with simplicial structures so that we have an induced map '* : Hi(Y, G; A) ! Hi(X, G; A) on homotopy groups. Of course, we have the opposite variance for the functors Hi(X, G; A) _ these are contravariant in G and covariant in X. 2.2. Sheaf-theoretic interpretation of C0(BGxX=X). We now present an equivalent definition, due to Suslin-Voevodsky, [16] of Hi(X, G) in terms of sheaves in the qfh topology. Recall [16] that a morphism q : X ! Y is a topological epimorphism if the underlying Zariski topological space of Y is a quotient space of the underlying Zariski topological space of X. The map q is a universal topological epimorphism if for any Z ! Y the morphism qZ : X xY Z ! Z is a topological epimorphism. An h-covering of a scheme X`is a finite`family of morphism of finite type {pi : Xi ! X} such that pi : Xi ! X is a universal topological epimorphism. A qfh-covering of X is an h-covering {pi} such that all the morphisms pi are quasi-finite. If X is a smooth separated scheme of finite type over k, let Z[1=p]qfh(X) denote the sheaf in the qfh topology associated to the presheaf T 7! Z[1=p]Hom Sch=k(T, X), HOMOLOGY OF LINEAR GROUPS 5 where Z[1=p]S denotes the free Z[1=p]-module on the set S and p is the exponential characteristic of k. Theorem 2.2 ([16],6.7). If Y is a separated scheme, then there is an iso- morphism C0(Y x X=X) Z[1=p] ~= (X, Z[1=p]qfh(Y )) that is natural in both X and Y . Note that this theorem implicitly asserts that C0(Y x-=-)[1=p] is a sheaf in the qfh-topology. As shown in [16, x5], any sheaf F in the qfh topology admits transfer maps; that is, for any finite surjective map f : X ! S, where X is reduced and irreducible and S is irreducible and regular, there is a transfer homomorphism TrX=S : F(X) -! F(S) satisfying certain expected properties (cf. [16, 4.1]). In the case F = C0(Y x -=-), the transfer homomorphism is defined by pushforward of cycles in the evident manner. By Theorem 2.2, if p is invertible in A, the complex (C0(BGx-=-) A, d) is a complex of qfh sheaves and so is equipped with degreewise transfer maps. These commute with d and hence for any finite surjective map X ! S, we have transfer maps TrX=S : Hi(X, G; A) -! Hi(S, G; A). Thus we have proved the following proposition. Proposition 2.3. Suppose that p is invertible in the abelian group A. Then the functor Hi(-, G; A) is a presheaf with transfers. 2.3. First examples. Proposition 2.4. Let G be a finite group. For all i 0, Hi(X, G; Z) = Hi(G, Z) for any integral scheme X. Proof.An irreducible subscheme Z Gix X that is finite and surjective over X must be isomorphic to X since the scheme Gi x X is simply the disjoint union of copies of X. It follows that the group C0(Gix X=X) is the free abelian group on the closed points of Gi. Thus, Hi(X, G; A) is the i-th homology of the standard complex for computing Ho(G, Z). In this case the pullback and transfer maps are easy to describe. For any morphism f : X ! S, f* : Hi(S, G; Z) ! Hi(X, G; Z) is induced by pullback of cycles, which in this case is the map sending [{g} x S] to [{g} x X]. Thus f* is simply the identity map (after making the identification of Proposition 2.4). Suppose f : X ! S is a finite surjective morphism between integral schemes, and set d = deg(f) = [k(X) : k(S)]. 6 KEVIN P. KNUDSON AND MARK E. WALKER Then f* is induced by pushforward of cycles, which in this case is the map sending [{g} x X] to d[{g} x S]. Thus f* : Hi(X, G; Z) ! Hi(S, G; Z) is simply multiplication by d. Proposition 2.5. Let k be an algebraically closed field and let A be an abelian group. Then we have natural (in G and A) isomorphisms Hi(Spec (k), G; A)~=Hi(G(k), A) Hi(Spec (k), G; A)~=Hi(G(k), A), for all i 0, where G(k) is the discrete group of k-rational points of G. Proof.This follows from the observation that C0(Gi= Spec(k)) is the free abelian group on k-points of Gi. If k is not algebraically closed, then the situation is more complicated. __ Proposition 2.6. Let k be a field and_k_be an algebraic closure of k. Let denote the absolute Galois group Gal(k=k). Then for all i 0, __ Hi(Spec (k), G; Z) ~=Hi(BG(k)= ). Proof.Observe that C0(Gi= Spec(k)) is the free abelian group on the closed points of Gi, which_are in one-to-one correspondence with the orbits of the action on Gi(k). We thus have __ C0(Gi= Spec(k)) ~=Z{Gi(k)= }. This point of view allows us to make some calculations in Section 4. Observe that a map X ! Gi naturally defines an element of C0(Gi x X=X), and hence there is a map of chain complexes Z Hom (X, BG) ! C0(BG x X=X). For example, if X = Spec(R) is affine, then a morphism X ! Gi is simply an R-point of Gi and hence we have the map of chain complexes Co(G(R)) -! C0(BG x X=X). Proposition 2.7. Let X = Spec(R) be an affine scheme. Then there is a natural map Hi(G(R), Z) -! Hi(X, G; Z) for each i 0. These maps are isomorphisms in the contexts of Propositions 2.4 and 2.5, but in general they need not be either injective or surjective. In fact, we have the following result. Proposition 2.8. There is an isomorphism H1(Spec (R), Gm ; Z) ~=Rx>0 and the map H1(Gm (R); Z) ! H1(Spec (R), Gm ; Z) is surjective with kernel { 1}. HOMOLOGY OF LINEAR GROUPS 7 Proof.The abelian group H1(Spec (R), Gm ; Z) is generated by classes of non- zero complex numbers, [z] for z 2 Cx , modulo the relations [z] = [__z] and [zw] = [z] + [w], for all z, w 2 Cx . The homomorphism Z(Cx ) ! Rx>0 induced by sending z 2 Cx to kzk annihilates these relations and hence induces a map H1(Spec (R), Gm ; Z) ! Rx>0. Likewise, these relations show that the function Rx>0! H1(Spec (R), Gm ; Z) given by r 7! [r] is a homomorphism. The composition Rx>0! H1(Spec (R), Gm ; Z) ! Rx>0 is clearly the identity. For w 2 Cx , we have w = z2, for some z, and hence [w] = [z2] = 2[z] = [z] + [__z] = [z__z] = [kwk]. This shows that the composition H1(Spec (R), Gm ; Z) ! Rx>0! H1(Spec (R), Gm ; Z) is also the identity. Finally, the composition Rx ~= H1(Gm (R); Z) ! H1(Spec (R), Gm ; Z) ! Rx>0is given by r 7! |r|. 3.The spectral sequence and Friedlander's Isomorphism Conjecture Let q denote the linear hypersurface in Aq+1 defined by the equation t0 + t1 + . .+.tq = 1. There are obvious coface and codegeneracy maps ffii: q-1 -! q oei : q -! q-1 making o a cosimplicial scheme. Fix s 0. Applying the functor C0(Gs x -=-) to o yields a simplicial abelian group C0(Gs x o= o). We therefore have a double complex Aoo with As,t= C0(Gs x t= t). Applying the functor Hom (-, Z=n) to Aoo yields a double cochain complex Eoo with Es,t= Hom (C0(Gs x t= t), Z=n). As usual, we have two first quadrant spectral sequences converging to the cohomology of the total complex. Taking horizontal cohomology first, we obtain Es,t1= Hs( t, G; Z=n). Using the work of Suslin-Voevodsky and the second spectral sequence, we may identify the abutment. If F is a presheaf with transfers, define a presheaf Ft by Ft(X) = F(X x t). Then Fo is a simplicial presheaf of abelian groups. Also, we let Co(F) denote the simplicial abelian group F( o). There are obvious maps of simplicial presheaves Co(F) ! Fo and F ! Fo (where Co(F) is regarded as a degree-wise constant presheaf). 8 KEVIN P. KNUDSON AND MARK E. WALKER Corollary 7.7 of [16] asserts that both of these maps induce isomorphisms upon applying the functor Extoqfh(-, Z=n), provided k is algebraically closed and n is relatively prime to the exponential characteristic of k. Moreover, these qfh Ext groups coincide with the 'etale Ext groups Exto'et(-, Z=n) by [16, 10.10]. Now consider the case F = C0(Gs x -=-). Taking vertical homology in the double complex yields s-th column Extoqfh(C0(Gs x -=-), Z=n) = Exto'et(C0(Gs x -=-), Z=n). In turn, the latter is Exto'et(ZHom (-, Gs), Z=n) since C0(Gs x -=-)[1=p] is the qfh sheafification of Z[1=p]Hom (-, Gs). But these groups are precisely the 'etale cohomology groups Ho'et(Gs, Z=n) (Corollary 7.8 of [16]). What we conclude, then, is that the second spectral sequence is simply the usual spec- tral sequence for computing the 'etale cohomology of the simplicial scheme BGk: Es,t1= Ht'et(Gs, Z=n) =) Hs+t(BGk, Z=n) (see [5], p. 16). We have thus proved the following result. Theorem 3.1. Let k be an algebraically closed field and n a positive integer relatively prime to the exponential characteristic of k. Then there is a first quadrant spectral sequence (1) Es,t1= Hs( t, G; Z=n) =) Hs+t'et(BGk, Z=n). Remark 3.2. The results of [16] are stated only for fields of characteristic zero. Work of de Jong [2] allows us to replace this assumption with the assumption that n is relatively prime to the exponential characteristic of k. Observe that we have isomorphisms Es,01= Hs(Spec (k), G; Z=n) ~=Hs(G(k), Z=n) = Hs(BG(k), Z=n) by Proposition 2.5, and under this isomorphism, the edge homomorphism Hs'et(BGk, Z=n) ! Es,01~=Hs(BG(k), Z=n) is the map conjectured to be an isomorphism by Friedlander. Consequently, we have: Corollary 3.3. Friedlander's Isomorphism Conjecture for an algebraically closed field k, an algebraic group G defined over k, and a positive integer n relatively prime to the exponential characteristic of k is equivalent to the assertion that the map of complexes (2) Tot(Hom (C0(BG x o= o), Z=n)) ! Hom (C0(BG= Spec(k)), Z=n) is a quasi-isomorphism. HOMOLOGY OF LINEAR GROUPS 9 For example, one might optimistically conjecture that for each fixed t 0 the map Hi( t, G; Z=n) ! Hi(Spec (k), G; Z=n) is an isomorphism, a result which would imply that (2) is a quasi-isomorphism and hence the Isomorphism Conjecture. Though we have no counter-example to this stronger assertion, it seems unlikely to hold. The evident generalization to arbitrary coefficients of the above assertion _ i.e., the assertion that the map of complexes (3) C0(BG= Spec(k)) ! Tot(C0(BG x o= o)) is a quasi-isomorphism _ turns out to be false: Take G = Gm and for each fixed t let C0(G^omx t= t) denote the normalized chain complex obtained from C0(BGm x t= t) by modding out degeneracies. Then the bicomplexes C0(BG^omx o= o) and C0(BGm x o= o) are quasi-isomorphic, and by [17], we have that for each fixed s, the complex C0(G^smx o= o) computes the weight s motivic cohomology of the field k: ^s o o s-i hi C0(Gm x = ) ~= HM (k, Z(s)). It follows that h2 of the total complex associated to C0(Gomx o= o) is the second K-group of the field k: ^ 2 ff h2(C0(Gomx o= o)) ~=K2(k) = (kx )= a ^ (1 - a) | a, 1 - a 2.kx V2 By contrast, we have h2(C0(BGm = Spec(k))) ~= (kx ), so that (3) has a kernel at h2 (given by the Steinberg relations). The map (2) is a quasi-isomorphism for G = Gm (with Z=n coefficients), however, since the Isomorphism Conjecture holds for this algebraic group (see Corollary 3.5 below). This is connected to the fact that the motivic co- homology of k with Z=n coefficients coincides with the singular cohomology of a point: ( Z=n, if s = i, hi(C0(G^smx o= o)) ~=Hs-iM(k, Z=n(s)) ~= 0, otherwise. The spectral sequence also allows us to deduce the following criterion for verifying Friedlander's Isomorphism Conjecture: Theorem 3.4. Let G be an algebraic group over an algebraically closed field k and let n be a positive integer relatively prime to the exponential characteristic of K. Assume that for all smooth k-varieties X and closed points x of X, the map __ (4) Hi(G(k), Z=n) -! Hi(G(R ), Z=n) __ is an isomorphism for all i r, where R is the absolute integral closure of R = OhenX,x, the henselization of the local ring of X at x. Then the map Hi'et(BGk, Z=n) -! Hi(BG(k), Z=n) 10 KEVIN P. KNUDSON AND MARK E. WALKER is an isomorphism for all i r and, moreover, there is an exact sequence 0 -! Hr+1'et(BGk, Z=n) -! Hr+1(BG(k), Z=n) -d! Hr+1( 1, G; Z=n), where d is the difference of the maps induced by the two face maps 0 ! 1. Proof.The first implication is well-known_under the stronger hypothesis that the map (4) is an isomorphism with R replaced by R (see e.g. [7]). Our machinery allows us to use this weaker hypothesis. Consider the spectral sequence Es,t1= Hs( t, G; Z=n) =) Hs+t'et(BGk, Z=n) __ of Theorem 3.1. Since Spec(R ) is a limit of qfh neighborhoods of x 2 X, our hypothesis implies that the functors Hi(-, G; Z=n) are locally constant for the qfh topology. If F is a presheaf on the category of schemes over k, let Fqfhdenote the sheafification of F in the qfh topology. We then have the following chain of natural isomorphisms (where Ab denotes the category of abelian groups): Ext oAb(Hi(Spec (k), G; Z=n), Z=n)~=Ext oqfh(Hi(-, G; Z=n)qfh, Z=n) ~= Ext oqfh(Hi(- x o, G; Z=n)qfh, Z=n) ~= Ext oqfh(Hi( o, G; Z=n), Z=n) ~= Ext oAb(Hi( o, G; Z=n), Z=n) where the first holds because the functor Hi(-, G; Z=n) is locally constant in the qfh topology, the middle two follow from Theorem 7.6 of [16], and the last holds by definition. Since the functors Hi(-, G; Z=n) are n-torsion, the map Hi( o, G; Z=n) ! Hi(Spec (k), G; Z=n) must be a weak equivalence for i r and hence the same is true of the map Hi(Spec (k), G; Z=n) -! Hi( o, G; Z=n) for i r. Thus, for all s r, ( Hs(BG(k), Z=n) t = 0 Es,t2= 0 t > 0. So we see that the map Hi'et(BGk, Z=n) ! Hi(BG(k), Z=n) is an isomor- phism for i r. Moreover, we have Er+1,01= Er+1,02and therefore we obtain the exact sequence 0 -! Hr+1'et(BGk, Z=n) -! Hr+1(BG(k), Z=n) -d! Hr+1( 1, G; Z=n). Corollary 3.5. The natural map Hi'et(BGk, Z=n) -! Hi(BG(k), Z=n) is an isomorphism in the following cases: (1) G finite, solvable, or the normalizer of a maximal torus in a reductive group; HOMOLOGY OF LINEAR GROUPS 11 (2) G = GLm in cohomological degrees i 3. Moreover, there is an exact sequence 0 -! H4'et(BGLm , Z=n) -! H4(BGLm (k), Z=n) -! H4( 1, GLm ; Z=n). __ Proof.Let X be a smooth k-variety, x 2 X a closed point, and R the absolute integral closure of OhenX,x. If G is finite, then clearly ~= __ Hi(G(k), Z=n) -! Hi(G(R ), Z=n) for all i 0. If G is solvable, then G has a descending central series whose graded quotients are either Gm or Ga. Clearly, __ Hi(Ga(k), Z=n) -! Hi(Ga(R ), Z=n) is an isomorphism for i 0. An easy application of Hensel's lemma shows that the same is true for Gm . By iterated use of the Hochschild-Serre spectral sequence we see that the map __ Hi(G(k), Z=n) -! Hi(G(R ), Z=n) is an isomorphism for all i 0. If T is a maximal torus in a reductive group S, then there is a short exact sequence 1 -! T -! NS(T ) -! W -! 1 where W is finite. Again, the Hochschild-Serre spectral sequence shows that the map __ Hi(NS(T )(k), Z=n) -! Hi(NS(T )(R ), Z=n) is an isomorphism for i 0. (All the preceding facts may be found in [7].) Finally, if G = GLm , then for n prime to the characteristic of k the map __ Hi(G(k), Z=n) -! Hi(G(R ), Z=n) is an isomorphism for i 3 ([9], p. 146). Therefore, the isomorphism con- jecture holds in this case as well and we obtain the above mentioned exact sequence. 4. Calculations For an arbitrary field k, Proposition 2.6 relates the groups Ho(Spec (k), G; * *Z) with a construction_using a quotient of the action of the absolute Galois group = Gal(k=k): __ Hi(Spec (k), G; Z) ~=Hi(BG(k)= ). __ __ Now consider the field k = R. We have R = C and Gal(k=k) = Z=2. If G is an algebraic group over R, then we have an isomorphism Ho(Spec (R), G; A) = Ho(BG(C)= ; A). We therefore have the following result. 12 KEVIN P. KNUDSON AND MARK E. WALKER Theorem 4.1. Let G be an algebraic group over R and let A = Q or Z=n, where n is odd. Then there is a canonical isomorphism ~= Z=2 Hi(Spec (R), G; A) -! Hi(G(C), A) for each i 0. Proof.This is a standard fact in the homology of quotient spaces, proved using the transfer map. See [1], p. 38. If A = Z=2, the calculation is more difficult and we are only able to handle two cases. First, consider the case G = Gm . We are interested in calculating Ho(BGm (C)= ; Z=2). Since homology and cohomology are dual with field coefficients, it suffices to compute Ho(BGm (C)= ; Z=2). While it is possible to do this directly using various standard techniques, we proceed as follows. Suppose Q is a finite group acting on a CW-complex X in such a way that if an element of Q fixes a cell of X, then it fixes it pointwise. There is as- sociated to this a cohomology theory, called (ordinary) Bredon cohomology, HoQ(X; M), where M is a Mackey functor, with the property that HoQ(X; Z_) ~=Ho(X=Q; Z). Here, A_is the constant Mackey functor associated to A. For a thorough dis- cussion of Bredon cohomology, we refer the reader to [10]. What is relevant for us is the following result. Proposition 4.2. Suppose f : X ! Y is a map of Q-CW-complexes such that for each subgroup H of Q the induced map fH : XH ! Y H of fixed point spaces induces an isomorphism Ho(XH ; Z=p) -! Ho(Y H; Z=p). Then the induced map f* : HoQ(Y ; Z=p_) -! HoQ(X; Z=p_) is an isomorphism. Proof.See [10], p. 26. This suggests the following. Equivariant Isomorphism Conjecture 4.3. Let G be an algebraic group over R. Let = Gal(C=R), and let p be a prime number. Then the identity map G(C) ! G(C)topinduces an isomorphism Ho(BG(C)top; Z=p_) -! Ho(BG(C); Z=p_). Proposition 4.4. Friedlander's isomorphism conjecture for G(C) and G(R) implies the equivariant isomorphism conjecture for G(C). HOMOLOGY OF LINEAR GROUPS 13 Proof.Note that for = Z=2, the only subgroups are and the trivial subgroup {1}, and the corresponding fixed point spaces are BG(R) and BG(C), respectively. Thus, if the maps Ho(BG(R); Z=p) -! Ho(BG(R)top; Z=p) and Ho(BG(C); Z=p) -! Ho(BG(C)top; Z=p) are both isomorphisms, then Proposition 4.2 implies that the map Ho(BG(C)top; Z=p_) -! Ho(BG(C); Z=p_) is an isomorphism. Corollary 4.5. Let G be a solvable Lie group. Then the equivariant iso- morphism conjecture holds for G. Proof.Friedlander's isomorphism conjecture holds for G(C) and G(R) [11]. In the case G = Gm , we see that there is an isomorphism Ho(BGm (C)top; Z=2_) -! Ho(BGm (C); Z=2_). We are trying to calculate the latter; these are the groups Ho(Spec (R), Gm ; Z* *=2). Note, however, that BGm (C)top is (equivariantly) homotopy equivalent to CP1 . So we must compute the groups Ho(CP1 ; Z=2_). Associated to a Z=2-CW-complex X is its Z=2-equivariant cohomology, which forms a bigraded ring Ho,o(X; A_), and extends the Bredon cohomology ring in the sense that we have Hk,0(X; Z_) ~=HkZ=2(X; Z_). We shall not need the detailed definition of this theory, but the interested reader may consult [10]. What is important for us is the following result. Proposition 4.6. The cohomology of CP1 is given by Ho,o(CP1 ; Z=2_) ~=Ho,o(pt; Z=2_)[c], where deg c = (2, 1). Proof.See [3], 5.4, p. 18. The Z=2-equivariant cohomology of a point with Z_coefficients has been calculated (see [3], Appendix B), and from it one deduces ( Z=2 q p 0 or q + 2 p 0 Hp,q(pt; Z=2_) ~= 0 otherwise. 14 KEVIN P. KNUDSON AND MARK E. WALKER Moreover, there is a commutative product on Ho,o(pt) with the property that the product of any element in degree (p, q), q p 0 with an element of degree (i, j), j + 2 i 0 is zero. Proposition 4.7. The cohomology groups Hs,0(CP1 ; Z=2_) satisfy H1,0= 0 and for k 1, H2k,0~=H2k+1,0~=(Z=2)k-1. Proof.Observe that elements of Hs,0(CP1 ) arise only as products of powers of the generator c (of degree (2, 1)) with elements of Ho,o(pt) in degrees (p, * *q) with q + 2 p 0. As there are no elements in degrees (-1, -1), (0, -1), or (-1, -2), we see that Hs,0vanishes for s = 1, 2, 3. Denote the generator of Hs,t(pt) by x(s,t). Then one sees easily that for k 2, we have __group__|____________________generators______________________ | H2k,0 || x(0,-k)ck, x(-2,-(k+1))ck+1, . .,.x(4-2k,2-2k)c2k-2 | H2k+1,0 ||x(-1,-(k+1))ck+1, x(-3,-(k+2))ck+2, . .,.x(3-2k,1-2k)c2k-1 This completes the proof. Recall that we have isomorphisms Ho(CP1 ; Z=2_) ~=Ho(BGm (C)= ; Z=2) ~=Ho(Spec (R), Gm ; Z=2). Proposition 4.7 therefore gives us the following result. Theorem 4.8. For all k 1, H2k(Spec (R), Gm ; Z=2) ~=H2k+1(Spec (R), Gm ; Z=2) ~=(Z=2)k-1. The same is therefore true for H2k and H2k+1. Now suppose G is a unipotent group over R. Let X = BG(C) and Y = BG(R) = X . According to [10], p. 35, there is a long exact sequence (with Z=2 coefficients) ~Hn((X=Y )= ) ! Hn (X) ! ~Hn((X=Y )= ) Hn (Y ) ! ~Hn+1((X=Y )= ). Note, however, that (X=Y )= = (X= )=Y and ~Ho((X= )=Y ) ~=Ho(X= , Y ). Thus, this sequence becomes Hn (X= , Y ) ! Hn (X) ! Hn (X= , Y ) Hn (Y ) ! Hn+1 (X= , Y ). Since ~Hn(X) = ~Hn(Y ) = 0 in this case, we see that Hn (X= , Y ) = 0 for all n 0. The long exact sequence of the pair (X= , Y ) then shows that H~n(X= ) = 0 for all n 0. We therefore have the following result. HOMOLOGY OF LINEAR GROUPS 15 Theorem 4.9. Let G be a unipotent group over R. Then for all i > 0, Hi(Spec (R), G; Z=2) = 0. The same is therefore true for Hi. References [1]A. Borel, et. al., Seminar on transformation groups, Annals of Mathematics * *Studies 46, Princeton University Press, 1960. [2]A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes 'Et* *udes Sci. Publ. 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Friedlander, Cycles, transfers, and motiv* *ic ho- mology theories, Ann. of Math. Stud., 143, Princeton Univ. Press, Princeton,* * NJ, 2000. Department of Mathematics and Statistics, Mississippi State University, P. O. Drawer MA, Mississippi State, MS 39762 E-mail address: knudson@math.msstate.edu Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Ne- braska 68588 E-mail address: mwalker@math.unl.edu