Contemporary Mathematics Volume 00, 0000 On the Cohomology of the Steenrod algebra CHARIYA PETERSON October 18, 1993 Abstract. Let A* be the dual of the mod p Steenrod algebra. A* ~= Fp[1; : :]:. Let A(n)* be the subalgebra generated by 1; : :;:n. We show that there exists a family of finite p-groups G(n; r) whose group algebra Fp(G(n; r)) is a quotient Hopf al- gebra of A(n) and the restriction map on the cohomology is an isomorphism up to certain level depending on r. 1. Introduction One of the main goal in homotopy theory is to compute the homotopy group of sphere, ss*(S0). An important tool is the mod p Steenrod algebra A and its cohomology Ext A(Z=p; Z=p), which forms the E2 term of the Adams' spectral sequence converging to the p component of sss*(S0). However, Ext A(Z=p; Z=p) is itself very difficult to compute. The dualPalgebra,iA* is a polynomial alge- bra Fp[1; : :]:with comultiplication 4n = pn-i i. It is isomorphic to the algebra of functions on the group scheme G = Aut s(Fad(x; y)) of strict au- tomorphisms of the additive formal group law over Fp, [1 ], [3 ] and [11 ]. In this paper, we apply some results in the theory of algebraic groups to analyze Ext A(Z=p; Z=p). We made two approximations for Ext *;*A(Z=p; Z=p). First, we replace A* by its finitely generated subalgebra A(n)*, or equivalently, we replace the group sche* *me G by an algebraic quotient group G(n). We then conjecture that Quillen's theo- rem on F -isomorphism holds for certain class of algebraic groups including G(n* *). This is done in Section 2. Next, we approximate G(n) further by considering ______________ 1991 Mathematics Subject Classification. Primary 55S10, 20G10; Secondary 20G* *400. cO00000American0Mathematical0So* *ciety0-0000/00 $1.00 + $.25 per page 1 2 C. PETERSON a family of finite p-subgroups G(n; r) of Fpr rational points of G(n). For each bidegree (i; j) we have Ext i;jA(Z=p; Z=p) - Hi;j(G(n); Z=p) -OE!Hi;j(G(n; r); Z=p); where j j (mod pr - 1) and the finite group cohomology Hi(G(n; r); Z=p) is Z=(pr-1) graded. The analysis of which is neither surjective nor injective and of G(n; r) are still under investigation. The main result of this paper is abo* *ut the map OE. It is a modification of a result of Cline, Parshall, Scott and van * *der Kallen [2 ]. Theoremh1.1. Let m; j, e and f be positive integers with f logp(j + 1)+ 2 n-1-1)m-1 i m and e (p__________p-1and for p > 2 also assume e > [logp( __d)] + n - 1. Then the restriction map Exti;jA(n)(Z=p; Z=p) = Hi;j(G(n); Z=p) ! Hi;j(G(n; e + f); Z=p) is an isomorphism for i m and an injection for i = m + 1. We have the following weak form of Quillen's theorem Corollary 1.1. Let m; j; e and f be as in Theorem. Then any element x in Ext i;jA(n)(Z=p; Z=p) is nilpotent if and only if it is nilpotent in Hi(G(n; r)* *; Z=p) for large enough r. The map OE following by Quillen's F-isomorphism for the finite group cohomology of G(n; r) yields a result related to that of C. Wilkerson in [12 ] concerning * *the ext group of a finite dimensional graded connected Hopf subalgebra of the Steenrod algebra at prime 2, and its elementary Hopf subalgebras. More precisely, in [9 ] Quillen gave a theorem comparing the mod p cohomology of a finite group with the limit of the mod p cohomology of its elementary abelian subgroups. Roughly speaking, the two cohomology groups are isomorphic modulo nilpotent elements as unstable A-modules. Such theorem does not hold if we replace finite group by finite dimensional graded connected Hopf algebra and replace elementary abelian subgroups by any family of abelian Hopf subalgebras. In [12 ], Clarence Wilkerson introduced the notion of elementary Hopf algebra and showed that for every finite dimensional graded connected Hopf subalgebra B of the mod 2 Steenrod algebra, Ext*;*B(Z=2; Z=2) satisfies the theorem of Quill* *en and Venkov [10 ] with respect to the family of all elementary subalgebras: i.e. an element x 2 Ext *;*B(Z=2; Z=2) is non nilpotent if and only if there exists * *an elementary subalgebra V such that res|BV(x) 6= 0. The second condition for F isomorphism is however, not satisfied in general. This is because the dual of a* *ny finite dimensional graded connected Hopf algebra is nilpotent and the transfer argument which is one of the main ingredients for Quillen's theorem fails. ON THE COHOMOLOGY OF THE STEENROD ALGEBRA 3 2. Quillen's theorem for unipotent algebraic groups It is by now a wellknown fact that the dual of the (reduced) Steenrod alge- bra A* is isomorphic to the coordinate algebra of the (infinite dimensional) group scheme G = Aut s(Fa(x; y)) of the strict automorphism of the additive formal group law defined over Fp. See [1 ], [11 ] and [3 ]. The group scheme G is in a natural way an inverse limit of unipotent algebraic groups G(n), as- sociated to the Hopf subalgebra A(n)* = Fp[1; : :;:n-1 ] A*, see [3 ]. We have the restriction map on the rational cohomology : H*(G(n); Z=p) = Ext *;*A(n)(Z=p; Z=p) -! Ext*;*A(Z=p; Z=p). is neither surjective nor injec* *tive. However, Ext*;*A(Z=p; Z=p) ~=lim H*(G(n); Z=p) when n goes to infinity. In this Section, we discuss the comparison between H*(G(n); Z=p) and the cohomology of its (algebraic) elementary abelian subgroups. An algebraic group is reduced (resp. connected) if its associated algebra has no nilpotent element (resp. is connected). A reduced connected unipotent group is precisely the group whose associated algebra is a polynomial algebra. By a vect* *or group we mean a finite product of the additive algebraic group Ga. Write OE for the category of vector subgroups of E with inclusions following by conjugations as morphisms. For any algebraic group E we write the rational cohomology of E with trivial coefficients as H*(E). Conjecture 2.1. Let E be a reduced unipotent algebraic group defined over the field Fp and let Q be the map induced by the restriction homomorphisms: (1) Q : H*(E) -! lim H*(V ): O(E) Then Q is an isomorphism modulo nilpotent of unstable modules over the uni- versal Steenrod algebra. The conjecture is true if E is a finite p-group. Recall that, for any positive * *integer r, the finite group E(Fpr) is a p-group and as a (constant) algebraic group, it is a finite, reduced algebraic subgroup of E, denoted by E(r). The Fpr rational points of the map Q in (1) is precisely Quillen's F-isomorphism. Conversely, any finite p-group can be realized as a subgroup of Fpr rational points of some unipotent algebraic group for some r. The Conjecture is true for any commutative reduced connected unipotent al- gebraic group. Any such group is a product of Witt groups W (d) at prime p of finite length, so it suffices to consider W (d). W (d) has a unique vector subgroup V of rank 1. H*(V ) and H*(W (d)) are computed in [2 ] and [8 ] re- spectively. Forgetting the higher Bocksteins, both cohomology groups are of the form Fp[xi|0 i] E(yi|1 i), where the symbol E stands for exterior algebra. The cokernel of Q : H*(W (d)) -! H*(V ) is exterior algebra and the kernel is 4 C. PETERSON generated by exterior generators. Hence Q is clearly an F-isomorphism. Let cW (n) be the algebraic group of big Witt vectors of length n - 1. Then Y Wc (n) ~= W (ri); p-i where ri is the smallest integer such that ipri n. Our group G(n) is in some sense the twisted version of cW(n). It is a (semidirect) product of smaller gro* *ups and it has a unique vector subgroup of maximal rank [ n_2]. We believe that the Conjecture holds for G(n). Conjecture 2.1 can be stated in terms of reductive groups as follows. Let B be a Borel subgroup of a reductive group G with maximal torus T and unipotent radical U. Then for any G-module M, H*(G; M) ~=H*(B; M) ~=H*(U; M)T . In particular, if M is the induced module Ind|GB1 associated to a dominant weight , then H*(G; Ind|GB1 ) ~= H*(U) . When = 0, Hi(G; Ind|GB0) = Hi(G; Z=p) = 0 for i > 0. We have the following consequence of the Conjecture. Precorollary 2.1. Let G be a connected reductive group or any parabolic subgroup. Then for any G-module M, the restriction map Q : H*(G; M) -! limO(U) H*(V ; M|V )T is an F -isomorphism. 3. Some preliminaries We assume some familiarity with the theory of reductive algebraic groups. Let G(n) denote the algebraic group represented by the finitely generated Steenrod P pi algebra A(n)* = Fp[1; 2; : :;:n-1 ] with 4m = m-i i where 0 = 1. We may impose an action of a rank one torus on G(n) as follows. Let B(n) be the group of upper triangular n x n matrices and let U(n) be the subgroup of strict upper triangular matrices. Their associated algebras are Fp[B(n)] = Fp[ai;j; a1ii|1 i j n]; Fp[U(n)] = Fp[ai;j|1 i < j n] j-iX 4(aij) = ai;i+k ai+k;j; 0 4(aii) = aii aii; where ai;i= 1 in Fp[U(n)]. There is a natural inclusion G(n) U(n) B(n) i-1 given by the surjection of Hopf algebras defined by j(aij) = pj-i. Let T n be the maximal torus in B(n). As a subgroup of B(n), G(n) is stable undernthe-1 conjugation of a rank 1 torus f : T 1,! T n; f(a) = (a; ap; : :;:ap ). We h* *ave ON THE COHOMOLOGY OF THE STEENROD ALGEBRA 5 the following injective map of group extensions: 1 -! G(n) -! bG(n) -! T 1 -! 1 # i # j # f 1 -! U(n) -! B(n) -! T n -! 1 where bG(n) is represented by the Hopf algebra Fp[10 ; 1; : :;:n-1 ] with 4d = P pi d-i i. We note here that this algebra is closely related to the dual of the unstable Steenrod algebra referred to in section 5. The character groups X(T 1) and X(T n) are free Abelian groups of rank 1 and n respectively. Let ffi (resp. {ei|1 i n}) be the canonical basis for X(T 1) (r* *esp. X(T n).) The homomorphism f induces a map of Abelian groups f* : X(T n) -! X(T 1) which is the projection f*(e1+. .+.pn-1 en) = ffi. The conjugation action of T 1provides a pseudo root system of type A1 for G(n). Lemma 3.1. The action of T 1on G(n) can be described in terms of Hopf algebras as the algebra homomorphism ae : Fp[1; : :;:n-1 ] -! Fp[1; : :;:n-1 ] Fp[t; t-1 ] with ae(i) = i tpi-1. Proof. Check the conjugation action of T 1on G(n) as subgroups of B(n). Then translate it to coaction of Hopf algebras. __|_ | This action gives A(n)* a weight decomposition which can be identified with the topological grading in A*. More over, it extends naturally to an action on the rational cohomology H*(G(n)) and make it a bigraded object. To avoid possible confusion with the cohomological grading, we refer to the degree of th* *is "internal" grading algebraically as weight. More generally, any G(n) module M becomes X(T 1) = ffiZ graded; M = Md, where we write Mdffias Md and call Md the weight space of M of weight dffi or simply d. r For any positive integer r, let q = pr. The ideal Jr A(n)* generated by pi-i for 1 i < n is a Hopf ideal and the quotient A(n; r)* = A(n)*=Jr is a reduced finite dimensional Hopf algebra. It represents the subgroup of Fq rational poin* *ts G(n; r) = G(n)(Fq). As an abstract group, G(n; r) is a finite p-group and the dual A(n; r) is isomorphic to the group algebra of G(n; r). Similarly, we write T 1(r) for T 1(Fq). As an abstract group T 1(r) is isomorphic to the cyclic gro* *up Z=(q - 1). The action of T 1on G(n) restricts to an action of T 1(r) on G(n; r) and gives any G(n; r)-module a Z=(q - 1) graded structure. Definition 3.1. Let Ga be the additive algebraic group. The associated algebra of Ga is Fp[x] with 4(x) = x 1 + 1 x. An action of a 1-torus on Ga is given 6 C. PETERSON by the coaction OE : Fp[x] -! Fp[x] Fp[t; t-1 ]; OE(x) = x ti: We said T 1acts on Ga with weight i and denote Ga with this action by Gai. The cohomology of Ga is computed by Cline, Parshall, Scott and van der Kallen in [2 ]. We quote two of their results here for reference. Theorem 3.1. Let T 1acts on Ga with weight 2 X(T 1). Let V (resp. W ) be the vector space spanned by elements ai = -pi, 0 i (resp. bi = -pi+1, 0 < i). Then ( S*(V ); for p = 2 (2) H*(Ga; Z=p) = S*(W ) E*(V ); for p odd, where S* and E* denote the symmetric algebra and the exterior algebra respec- tively. For q = pr, H*(Ga(Fq); k) is as in ( 2) but with V (resp. W ) replaced * *by the subspace V (p) (resp. W (p)) spanned by ai (resp. bi+1) for 0 i < r. 4. Main results In what follows, we fix a power q = pr of p and a weight = dffi 2 X(T 1). G(n) admits a filtration by normal subgroups: (3) Ga = G(2) ,! G(3) ,! . .,.! G(n) Lemma 4.1. The conjugation action of T 1on G(n) induces and action of T 1on G(i)=G(i - 1) ~=Ga with weight (pn-i+1 - 1): We identify G(i)=G(i - 1) with Gaj, where j = n - i + 1. By Theorem 3.1, ( Fp[x0j; x1j; : :]: for p = 2; H*(Gaj) ~= Fp[x0j; x1j; : :]: E(y1j; y2j; : :):for p >;2 where the weight of xij and yij is pi(pj - 1) and pi+1(pj - 1) respectively. The E2 term of Serre's spectral sequence for the extension 1 -! Gan-1 -! G(n) -! G(n - 1) -! 1 is Ep;q2= Hp(G(n - 1)) Hq(Gan-1 ) =) Hp+q (G(n)): The action of T 1induces the third grading by weight on E2. Since the different* *ial preserves weights, the = dffi weight space of E2 also forms a spectral sequence i j d i+j;d Ei;j2;d= H (G(n - 1); k) H (Gan-1 ; k) =) H (G(n); k): ON THE COHOMOLOGY OF THE STEENROD ALGEBRA 7 Ei;j2;dcould be refined using the filtration (3), to Es1;:::;sn-12;d= (Hs1(Ga1) . . .Hsn-1(Gan-1 )) d=) Hsi;d (G(n)): The homogeneous subspace of E*;*2;dof elements of total degree m and weight d M s ;:::;s L(m; d) = E21;d n-1~=Hm;d (Ga1 x . .x.Gan-1 ); si=m Q m is spanned by monomials k=1 aikjk, with multiplicity, where aij stands for xij for pQ= 2 and either xij or yi+1j for p > 2, such that i 0, 1 j < n. Let f = mk=1aikjk2 L(m; d), then the weight of f, for p = 2 and for p odd are respectively Xm d = pik(pjk - 1); k=1 Xm1 m2X 0 d = pik(pjk - 1) + pil(pjl- 1) k=1 l=1 where ik 0, 1 jk < n, i0l> 0 and m1 + 2m2 = m. As for the finite subgroup G(n; r), there exists a similar spectral sequence __ E(Fq)s1;:::;sn-12;_d= (Hs1(Ga1(Fq)) . . .Hsn-1(Gan-1 (Fq))) d _ =) Hsi;d (Gr(n)); __ __ where d is the congruent class of d in X(Tr1) = Z=(q - 1). Write_R(m;_d) for the homogeneous subspace of E(Fq)*;*2ofQdegree m and weight d. R(m; d) has a basis consisting of monomials mk=1a_ikjkwith the same notation as for L(m; d), _ P m __ but with 0 ik < r and pik(pjk - 1) d (mod q - 1): The inclusion G(n; r) G(n) induces the (degree 0) restriction map of spectral sequences __ m OEmd : L(m; d) -! R(m; d); OEd (xikjk) = x_ikjk: If OEidis an isomorphism for i_ m and an injection for i = m + 1, then the induced map Hi;d(G(n)) -! Hi;d(G(n; r)) is also an isomorphism when i m and an injection when i = m + 1. To show this, one has to consider what OE*d does to the basis elements. For each monomial in L(m; *), we may rewrite its weight as Xm Xt d = pik(pjk - 1) = pli; k=1 i=1 where li = ik for some k and t m(pn-1 - 1). Even if we are in a different situation from that of [2 ], the proof of the main theorem is basically the same as that given in [2 ]. We may apply their injecti* *vity, 8 C. PETERSON isomorphism conditions and their digit counting lemma, which we quote here with minor change. Q m Condition 4.1. OEmd is injective if any monomial k=1 aikjkin L(m; d) satisf* *ies, for p = 2 and for p odd respectively Xm d = pik(pjk - 1); k=1 m1X m2X 0 d = pik(pjk - 1) + pil(pjl- 1); k=1 l=1 with m1 + m2 = m and 1 jk < n. Then 0 ik < r; 0 < i0l r for all k. Q m Condition 4.2. OEmd is an isomorphism if any monomial k=1 aikjkin L(m; *) congruent to a monomial in R(m; d) must be in L(m; d). More precisely, every congruent relation, for p = 2 and for p odd respectively, mX pik(pjk - 1) d (mod pr - 1); k=0 m1X m2X 0 pik(pjk - 1) + pil(pjl- 1) d (mod pr - 1); k=0 l=0 where 0 ik < r, 1 jk < n, 0 < i0l r and m1 + m2 = m, is an equality. Proposition 4.1. [2 ] Proposition 6.5. Let t be a positive integer. Consider * *an expression Xt (4) pik = L + (pe+f - 1)M k=1 h i for integers L, M, e t-1_p-1, f logp(|L| + 1)+ 2 and 0 ik < e + f, where [ ] denotes the largest integer function. Then there are no term with L < pik <* * pf and the terms on the left hand side with pik L sum to L - M. Proposition 4.2. Let m,hanand-b1be non-negativeiintegers. Let e and f be integers satisfying es (p___-1)m-1_p-1and f logp(a + 1) + 2. Suppose Xm pik(pjk - 1) = a + (pe+f - 1)b; k=1 where 0 ik < e + f, and 1 jk < n. Then a b 0, and the terms in the sum with pik a sum to a - b. P m P t Proof. Write k=1pik(pjk - 1) = i=1pli = ah+ (pe+fi- 1)b; with t (pn-1 - 1)m, li = ik < e + f for some k. Hence e _t-1_p-1. Lemma 4 now finishes the proof. __|_ | ON THE COHOMOLOGY OF THE STEENROD ALGEBRA 9 Theorem 4.1. Let m be a non-negativehinteger and = dffi a weight in X(T 1). n-1-1)m-1 i Let e; f be integers with e (p__________p-1, f logp(d + 1) + 2. For p odd assume also that e > [logp( m_d)] + n - 1. Then the restriction map ed i;___ped OEiped: Hi;p (G(n); k) -! H (Ge+f (n); k) is an isomorphism for i m and an injection for i = m + 1. Proof. For monomorphism,Pwe0check ConditionP4.1.0Let0f0be a monomial in L(i; d) of weight ik=1pik(pjk- 1) + il=1pil(pjl- 1) = ped, with i0+ 2i00* *= i, 0 ik, 0 < i0land 0 < jk < n, where for p = 2 we set i00= 0. Need to show ik < e + f for all 0 k i. Assume otherwise, for p = 2 we have pe+f (p - 1) = pe+f ped and hence pf d < pf, which is a contradiction. For p odd, we have pe+f+1 (p - 1) ped and hence pf+1 < pf+1 (p - 1) d < pf+1 , again a contradiction. Note that this argument does not depend on i. For isomorphism part we check condition 4.2. Given a congruent relation Xi0 i00X 0 0 (5) pik(pjk - 1) + pik(pjk - 1) = ped + (pe+f - 1)b k=1 k=1 ped (mod pe+f - 1); with i0 + 2i00= i and set i00= 0 for p = 2, i m, 0 ik < e + f,P0 < i0k e + f, 1 jk; j0k< n. Write the left hand side of (5) as tu=1plu with t (pn-1 -1)m. For p = 2, the hypotheses on e and f together with Proposition 4 and the discussion in [2 ] section 6 implies b = 0 and the restriction map is* * an isomorphism. For p odd, assume the second sum in the expression (5) is in the increasing ord* *er with respect to i00k. Let i000is the smallest number i000 i00such that i0i000< * *e + f, we have Xi0 i000X0 0 Xi0 0 pik(pjk - 1) + pik(pjk - 1) + (pjk - 1) = ped; k=1 k=1 k=i000+1 We may assume both m and d are greater than zero. By the digit counting lemma of [2 ], pn-1 m > (pn-1 - 1)m t dpe. Thus pe-n+1 < m_dwhich is a contradiction. __|_ | Corollary 4.1. Let V be a finite dimensional bG(n) module and let m; e and f be positive integers satisfying the conditions given in the theorem. The restri* *ction map Hi(Gb(n); V (e)) -! Hi(Gb(n; r); V (e)) is an isomorphism for i m and an injection for i = m + 1. 10 C. PETERSON It is clear that the notion of generic cohomology makes sense. Since a G(n; r)- module is cyclically graded by Z=(pr-1), there is an isomorphism Hi(Gb(n; r); V* * ) -! Hi(Gb(n; r); V (e)) for each i, r, e and finite dimensional bG(n) module V , an* *d thus Hi(Gb(n; r); V ) is stable for large r. By definition, Higen(Gb(n); V ) is thi* *s group in the stable range. The restriction map now becomes H*(Gb(n); V ) -! H*gen(Gb(n); V ) and similarly, _ H*;*(G(n); k) -! H*;*gen(G(n); k) which is an isomorphism under the conditions of Theorem 4.1 or Corollary 4.1. It is interesting to compare ExtA (Z=p; Z=p) with H*gen(G(Fp1 )). Remark 4.1. The bound for e and f are minimal for n = 2. This follows directly from Cline et. al's result. However, for n > 2 it is not minimal, in particul* *ar, the extra assumption for e when p > 2 can be improved. Remark 4.2. Corollary 4.1 applies to any finite dimensional module V over the universal Steenrod algebra, on which P 0acts invertibly. In particular, if P 0a* *cts as identity, in which case V is just a module over the Steenrod algebra. 5. Application to the unstable extension groups Let G be a finite group, or more generally a finite v.c.d. group or a compact L* *ie group, an let OG be the category of finite elementary abelian subgroups of G, then, by a theorem of Quillen, the induced map on cohomology H*(G; Fp) -! lim-OG H*(V; Fp) is an F -isomorphism, in the sense that Ker(F ) is the submodu* *le of the nilpotent elements and Coker(F ) is nilpotent as unstable module over the Steenrod algebra. Theorem 5.1. With the same notations and conditions as in Theorem 4.1, an element x 2 Exti;jA(k; k) is nilpotent if and only if it is nilpotent in Hi(G(n* *; r))j. Conjecture 2.1 or Theorem 4.1 might be useful, for example, in the "algebraic homotopy theory", a program introduced by John Palmeiri in [5 , 6], in which he applied Wilkerson's result to obtain a family of nilpotent "algebraic self maps* *", i.e. nilpotent elements in Ext*;*A(M; M), for some (stable) A-module M. We end this subsection with an application of Theorem 4.1 to the extension group of unstable modules. The link between the theory of algebraic groups and stable mod-p theory exists in the unstable theory as well. The algebra of (reduced) mod p unstable cohomology cooperations is a polynomial algebra P pi B* = Fp[0; 1; : :]:with comultiplication 4n = n-i i. B* represents the multiplicative monoid (scheme) End Fp(Fad(x; y)). This connection occurs in any ON THE COHOMOLOGY OF THE STEENROD ALGEBRA 11 complex oriented cohomology theory, for more detail about this see [7 ]. We have an extension of monoids (6) 1 -! Aut s(Fad(x; y)) -! End (Fad(x; y)) -! L -! 1; where L is the algebraic monoid represented by Fp[x] with 4(x) = x x. The inclusion Aut s(Fad(x; y)) End (Fad(x; y)) induces the full embedding U ~=B-Mod A-Mod. Now, the higher cohomology Hi(L; M) vanishes for any unstable module M, for the same reason as the vanishing of the cohomology of the torus. Therefore, the Serre's spectral sequence for the extension (6) yi* *elds an isomorphism Ext *U(k; M) = H*(End (Fad(x; y); M) ~=H*(G; M)L ; for any unstable module M and where the action of L on H*(G; M) is induced by the conjugation action of L on G in End (Fad(x; y)). For any positive integer n, let Bn* be the sub bialgebra of B* generated by 0; : :;:n-1 . Bn* represents an algebraic monoid G(n) o L = Gb(n). A Bn - module becomes an unstable module by restriction. Moreover, the notion of a submonoid bGr(n) of Fpr rational points, makes sense. Corollary 5.1. Let m; e be given as in Theorem 4.1. Let M be a finite di- mensional Bn module with d the largest integer such that Md 6= 0 and let f be a positive integer satisfying f [logp(d + 1)] + 2. Then the restriction map Ext iBn(k; eM) ! Hi(Ge+f (n); eM)Lr is an isomorphism for i m and injection for i = m + 1, where Lr L is the sub monoid of Fpr rational points. Remark 5.1. If E is a complex oriented cohomology theory, then the algebra of E-cohomology cooperations can be realized as the algebra of functions on an appropriate groupoid scheme depending on the associated formal group law of E [4 ]. It is likely that, one can find algebraic finite subgroups of such grou* *poid such that there is a similar results as in Theorem 4.1). References 1. M. F. Atiyah and F. Hirzebruch, Cohomologie-Operationen und charakteris- tische Klasses, 77 (1961), 149-187. 2. E. Cline, B. Parshall, L. Scott and W. van der Kallen, Rational and generic cohomology, Invent. Math. 39 (1977), 143-163. 3. M. Kaneda, N. Shimada, M. Tezuka and N. Yagita, Cohomology of infinites- imal algebraic groups, Math. Z. 205 (1990), 61-95. 4. J. Morava, noetherian localizations of category of cobordism comodules, Ann. Math. 121 (1985), 1-39. 12 C. PETERSON 5. J. Palmeiri, Nilpotence for modules over the mod 2 Steenrod algebra I, II, Preprints 1993. 6. J. Palmeiri, Nilpotence for modules over the mod 2 Steenrod algebra I, II, Preprints 1993. 7. C. Peterson, Unstable cohomology operations, in preparation. 8. C. Peterson and N. Yagita, The rational cohomology of the group of Witt vectors, submitted to the Journal of Pure and Applied Algebra, (1992). 9. D. Quillen The spectrum of an equivariant cohomology ring I, Ann. of Math. 94 (1971), 549-572. 10. D. Quillen and B. B. Venkov, Cohomology of finite groups and elementary abelian subgroups, Topology 11 (1972), 317-318. 11. D. C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Academic Press, New York, 1986. 12. C. Wilkerson, The cohomology algebras of finite dimensional Hopf algebras, Trans. AMS., 264 (1981), 137-150. Mathematics Department, Northwestern University, Evanston IL 60208 E-mail address: chariya@math.nwu.edu