DETECTING RATIONAL COHOMOLOGY OF ALGEBRAIC GROUPS

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1 DETECTING RATIONAL COHOMOLOGY OF ALGEBRAIC GROUPS EDWARD T. CLINE, BRIAN J. PARSHALL AND LEONARD L. SCOTT Let G be a connected, semisimple algebraic group defined over an algebraically closed field k of positive characteristic p. Assume that G is defined and split over the prime field k 0 = GF (p), and for q = p m, let G(q) be the subgroup of GF (g)-rational points. Let V be a rational G-module, and, for a non-negative integer r, let V(r) be the rational G-module obtained by 'twisting' the original G-action on V by the r-th power of the Frobenius endomorphism a of G. In [5] we showed (together with W. van der Kallen) that, if V is finite dimensional and n is a non-negative integer, for sufficiently large m and r (depending on V and n), there are isomorphisms H"(G, V{r)) ~ H"{G(q), V(r)) g* H"(G(q), V). Here the first isomorphism is restriction and the second is the obvious one resulting from the fact that G(q) is invariant under the action of o"'. Parallel results for the groups of twisted type (that is, for arbitrary surjective endomorphisms with finite fixed-point sets) were obtained by Avrunin [1]. The stable value of H"(G{q), V) is called the degree n generic cohomology Hg en (G, V) of G with coefficients in V. Many explicit computations are given in [10]; see also [11]. A question left open in [5, 6.7] concerned the injectivity of the map H"{G, V) -> Hg en (G, V). In this paper we shall prove a stronger result, namely, that the natural map H"(G, V) -* H n (G, V(lj) is injective for any rational G-module V. The analogue in the twisted situation studied by Avrunin is also valid. The paper is organized as follows. In 1 we prove a general result on the injectivity of cohomology. This is applied in 2 to obtain the above mentioned result, and to improve upon a result of Donkin [9], whose paper inspired this one. Finally, in 3, we indicate how the above goes through in the twisted case. 1. The base map and inflation In this section k is an algebraically closed field, and G is an affine algebraic group scheme over k. (That is, G = SpecO(G) for a finitely generated commutative Hopf algebra (9{G).) Let V be a rational G-module (equivalently, a comodule for (9{G)). If N is a normal closed subgroup scheme of G, then by [6, 4.5] we have a Hochschild-Serre spectral sequence for rational cohomology E 5 i l = H S {G/N, H'(N, V)) => H s+t {G, V). Received 5 October, This research was supported by the National Science Foundation. J. London Math. Soc. (2), 28 (1983),

2 294 EDWARD T. CLINE, BRIAN J. PARSHALL AND LEONARD L. SCOTT The classical interpretation [12, Proposition 10.2] in the cohomology of abstract groups of the base map is also valid in this context. (1.1) PROPOSITION. For each integer n ^ 0, the map H n (G/N, V N ) > H"(G, V) obtained from the above spectral sequence as H"(G/N, V N ) = 2 '' > '; : C H"(G, V) is the composite of the inflation map H"(G/N, V N ) -> H"{G, V N ) and the map H n (G, V N ) -> H n (G, V) induced by the inclusion V N V. Proof. By naturality in V we can assume that V = V N. However, the spectral sequence, together with its aboutissement H*(G, V), is also natural in the pair G, N. (If S*(G) denotes the standard injective resolution of the trivial rational G-module, then the Hochschild-Serre sequence may be viewed as arising from the double complex / \C/N (V S*(G) S*(G/N)) C * [{V S*{G)) N S*(G/N)J, which is clearly natural in G, N. One can also give more functorial arguments.) For the case when N = 1, the result is clear, and now it follows in general using the obvious map of pairs ( G, 1) -> (G, N). Henceforth we shall refer to the above map H"{G/N, V N ) -* H"{G, V) as the base map of degree n. We can now give a general criterion under which the base map is injective. (1.2) LEMMA. Let n ^ 0 be an integer. Suppose that there is a G-module map from V into a rational G-module V inducing a split injection V N -> V N of G/N-modules, and suppose that H m (N,V) = 0 for 0 < m < n. Then the base map H n (G/N, V N ) -> H"(G, V) is injective. Proof. If the composition of the base map and the natural map H n {G, V) -+ H"(G, V) is injective, then so is the base map. Since V N -> V N is split this composition is the composite of the injection H n {G/N, V N ) -> H"{G/N, V N ) and the base map for G, N, V. (Note that in the map from the spectral sequence for V to that of V, the map on '2 may (by naturality) be identified with the corresponding map for V N and V N.) Hence it is enough to show that the base map of degree n for G, N, V is injective. This is, of course, obvious, since the hypothesis H"'(N, V) = 0 for 0 < m < n guarantees that all differentials *' -> * +r '- r+l with s + t = n l and r ^ 2 are zero, just because *' ' = 0. This proves the lemma. Of course one way to get a split injection V N -* V N isomorphism V N ^. V N. is to actually get an (1.3) COROLLARY. Let E be a rational G-module which can be mapped to a rational G-module E which is injective as a rational N-module, in such a way that

3 DETECTING RATIONAL COHOMOLOGY OF ALGEBRAIC GROUPS 295 E N ^ E N. Let V be a rational G-module of the form V = E (x) Z where N acts trivially on Z (and thus V N = E N <g> Z). Then the base map H"(G/N, V N ) -> H n (G, V) is injective for each integer n ^ 0. Proof. Take K = Z in the previous lemma, and use the fact that the tensor product of any rational N-module with a rationally injective N-module remains rationally injective [6, 2.1]. As a further corollary, we obtain the following. (1.4) THEOREM. Suppose that the trivial G-module k can be embedded in a rational G-module Q which is rationally injective as an N-module and satisfies Q N = k. Then the inflation map H n (G/N, V) -> H"(G, V) is injective for each integer n ^ 0 and every rational G-module V on which N acts trivially. Proof This is just the case in which E = k of the previous corollary. A standard case in which the injectivity of the inflation map occurs is that of a semidirect product G = N-S. Traditionally injectivity here is a consequence of the fact that the map G -> G/N is split. In our set-up, it follows from the fact that the rational N-module &(N) may be extended to a rational G-module (giving a Q for the above theorem) by letting S act by conjugation. In the next section we shall be discussing situations where the map G -> G/N is not split, but injectivity is nevertheless satisfied. 2. The main results We continue the notation of the previous section and in addition assume that k has characteristic p > 0. We also assume that G is connected and semisimple, as well as defined and split over the prime field k 0 = GF (p). As in the introduction, if V is a rational G-module then V(r) denotes the rational G-module obtained from V by letting G act through the r-th power <j r of the Frobenius endomorphism a. The scheme-theoretical kernel of o r is denoted by G r. For the basic facts concerning this set-up we refer the reader to [6]. In addition, [8] contains an elementary exposition of properties of the Steinberg module. (2.1) THEOREM. Let V be a rational G-module and let n ^ 0 be an integer. Then the natural map H"{G, V) > H"{G, obtained from the Frobenius map a : G -» G is an injection. Proof. Without loss we can assume that G is simply connected (see [5, 2.7]). Now put Q = Hom k (St, St) where St denotes the irreducible Steinberg module of high weight (p \)p. Here p is the sum of all the fundamental dominant weights. Then Q is an injective G-module with Q Gi ^ k, and so the theorem follows from (1.4) and the identification G/G x ^ G.

4 2 9 6 EDWARD T. CLINE, BRIAN J. PARSHALL AND LEONARD L. SCOTT (2.2) COROLLARY. Let G,V,n be as above. Then the natural map H n (G, V) > H n (G, V{r)) obtained from a r : G -* G is an injection for all integers r ^ 0. Proof. This follows inductively from (2.1), since V{r + l) ^ V{r)(i). As an immediate consequence of this corollary and [5] we have the following. (2.3) THEOREM. Let G, V, n be as above and assume that V is finite dimensional. Then the natural map H"(G,v) wycv) is an injection for each integer n ^ 0. This paper was inspired by Donkin's paper [9]. In the remainder of this section we aim at strengthening his main result, which concerned the triviality of certain spectral sequence differentials, to an injectivity along the base. This gives as well a generalization of (2.1). (2.4) THEOREM. Let S be an irreducible rational module with high weight in the restricted range. (That is, S remains irreducible upon restriction to G,.) Let V, W be rational G-modules. Then we have a natural injection for all integers n ^ 0. Ext G (K, W) Ext G (S K(l), S < Before proving the theorem, we need a known result concerning the Steinberg module. Fix a split maximal torus T and a Borel subgroup B containing T. As usual these define the notion of positive root, dominant weight, &c, and the irreducible rational G-modules are parametrized by the dominant weights. The restricted dominant weights X are those satisfying (X, a v ) ^ p 1 for each simple root a. (Here a v denotes as usual the coroot defined by a.) We let S(A) denote the irreducible rational G-module with high weight X. As above, p denotes the sum of all the fundamental dominant weights. Without loss we can assume that G is simply connected. (2.5) LEMMA. Suppose that X x, X 2 are dominant weights with X l + X 2 = (p l)p. Then Hom Cl (St, S(X t ) S(X 2 )) s Hom G (St, Sfa) S(X 2 )) s k. Proof. First of all, Hom G (St, S(X { ) S(X 2 )) is at least one-dimensional, by universal mapping properties of St. (This module is irreducible, self-dual, and induced from the J5-module (p l)p.) Now it suffices to prove that Hom G (St, S^) S{X 2 )) is at most onedimensional. Clearly, T acts on Hom Gl (St, S^) S(A 2 )), with T x, the kernel of a\ T, acting trivially. For any weight \i of T, the corresponding weight space is

5 DETECTING RATIONAL COHOMOLOGY OF ALGEBRAIC GROUPS 297 Hom rgl (St p., S^J S{X 2 )), where p. is regarded as a one-dimensional rational TG^module with trivial G^action. (The group scheme TG X is defined as the pullback of T through a.) If conjugating by the Weyl group, we can assume that p is dominant. The image of a maximal vector under a non-zero homomorphism here must be non-zero, and hence have weight ^ X^ + Xj. This gives that {p-\)p + H ^ X t +X 2 = (p-l)p and forces p = 0. Also our image vector now lies in the X t +X 2 weight space of S(X X ) S{X 2 ) which is one-dimensional. This shows that Hom Gl (St, S(X t ) S(X 2 j) is at most one-dimensional, and completes the proof of the lemma. We have included the above proof for the reader's convenience. For a different argument, see J. Ballard [2]. Proof of (2.4). Let the dual S* of S be S(X) where A is a dominant restricted weight, and put A' = (p l)p A, which is also dominant. Since St is self-dual we obtain that Hom G (S, St S{X')) s Hom G (S St*, S(A')) s Hom G (St, S(A) S(X')) s fe, and also that Hom Gl (S, St S(A')) = k. Put = Hom k (S, S) and = Hom fc (S, St S(A')); these isomorphisms show that embeds in as a G-submodule, and that G = G. Since is a rationally injective G x -module, we can apply (1.3) to obtain that the base map H"(G/G,, Hom Gl (S, S) Z) H n {G, Hom k (S, S) Z) is injective for any rational G-module Z on which G x acts trivially, and in particular for Z = Hom k (K(l), W(l)). Of course Hom Gl (S, S) s k. Translating this injection into the language of Ext and using the identification G/G Y ^ G gives the desired result. This completes the proof. We remark that the case in which n = 1 in the above theorem is a known direct consequence of the Hochschild-Serre sequence, cf. [4, 3.3]. Donkin's result is the fact that d'i'': E 2 '' -> E" 2 +2 ' Q is zero in the spectral sequence involved in the above argument. Of course injectivity of the base map gives d n r'- x = 0 for each r ^ Twisted groups In this section we briefly indicate how the above results work in the twisted cases. We first recall some generalities concerning purely inseparable isogenies, and we take the opportunity to indicate how the method of [7] provides new proofs of certain classical results, due to Steinberg [13, 14], on the representation theory of G.

6 298 EDWARD T. CLINE, BRIAN J. PARSHALL AND LEONARD L. SCOTT Let G be a semisimple, simply connected group over an algebraically closed field of characteristic p. Fix a maximal torus T of G. We suppose that we are given an endormorphism 0 : X*{T) -> X*(T) of the character group of T, a permutation ^ of the root system O of T in G, and p-powers q(ct), a e O. These are to be related by the relations (/>(t^(a)) = q(<x)<x, aeo. Then, according to [3], $ defines a unique endomorphism a:g-*g stabilizing T such that o\ T induces $. Necessarily, a is a purely inseparable isogeny. Conversely, every purely inseparable isogeny arises in this way for some choice of T and $. In addition to [3], the reader will find an exposition of these matters in [14, 11]. Notice that if ip is the identity permutation and each q(<x) = p, then a is the Frobenius endomorphism considered in 2. Now let a be a purely inseparable isogeny of G. We shall denote by G[c] the scheme-theoretic kernel of a. Thus, G[<r] is an infinitesimal subgroup of G, and identifies with the G, of 2 when a is the Frobenius morphism. Also, if V is a rational G-module, V will denote the rational G-module obtained from V by twisting the action of G on V by o. We can assume that a leaves fixed a Borel subgroup B containing T. Let A + be the corresponding set of dominant weights, and let A + (<r) be the set of dominant weights X satisfying (X, a v ) ^ q(ot) 1 for each simple root a. Finally, for X e A +, let S(A) be the irreducible rational G-module of high weight X. We make the following observations. The first and third are infinitesimal analogues of results of Steinberg for finite groups, while the second is actually due to Steinberg. (3.1) Every irreducible rational G[a]-module extends uniquely to a rational G-module. (3.2) For XeA +, let X = A o + 0(/l 1 ) r (/l r ), A,eA + (<7), be the 'twisted p-adic expansion' of X. Then (3.3) The restrictions to G[c] of the S{X), X e A + (<r), are exactly the irreducible rational G[o~\-modules. They are distinct for distinct X. Let us only briefly indicate how the method of [7] provides the proofs of these results. First, let T be a purely inseparable isogeny such that each <j(a) is at most p. It is enough to establish (3.1) (3.3) for a = T", n = 1, 2,... The case in which n = 1 is handled exactly as in [7]. (The main step in deducing (3.2) from (3.1) is the isomorphism Hom C[T] (K, S(X)) V ^ + S(X) of rational G-modules, where V is a G[r]-irreducible submodule of S{X).) The argument is then easily completed by induction on n, and use of the fact that G/G[T] s G and G[T"]/G[T] S G[r n ~ l ~]. (3.4) LEMMA. Let a be a purely inseparable isogeny of G. Then there exists an irreducible rational G-module S{X) which is irreducible and rationally injective as a G[a]-module.

7 DETECTING RATIONAL COHOMOLOGY OF ALGEBRAIC GROUPS 299 Proof. In the case when a is an r-th power of the Frobenius endomorphism we can take S{X) = St St(l)... St(r-l), where, as in 2, St is the Steinberg module (of high weight (p l)p). In general, we can assume that there exists a positive integer n such that a" is a power of the Frobenius morphism on each simple component of G (otherwise a is an automorphism, in which case the lemma is obvious). Then if S(A) is a rationally injective G[(r"]-module, S{X) is also a rationally injective G[ff]-module by [6, 4.2], since G[a"]/G[cr] ^ G[p n ~ l ~\ is affine. The desired result now follows from (3.2) since GO] acts trivially on any V a>, i > 1. (3.5) REMARK. One can in fact show that the S(A) in (3.4) has the property that (A, a v ) = q(a) l for each simple a, that is, S(A) is a 'twisted Steinberg module'. (3.6) THEOREM. Let a be a purely inseparable isogeny of G. Then for V a rational G-module we have a canonical injection for any integer n ^ 0. H"(G, V) > H"(G, V) This follows from (3.4), using 1. We leave the formulation of (2.4) to the interested reader. Now let a be a purely inseparable isogeny such that n#( a ) > 1 Then for each r ^ 1, the subgroup G ar of cr r -fixed points is finite [14]. In addition to the finite Chevalley groups these finite groups include the twisted groups of Steinberg, Ree and Suzuki. Let V be a finite dimensional rational G-module. According to Avrunin [1], the cohomology groups H"(G ar, V) achieve a stable value, H" &en (G, V), as r -> oo. (3.7) THEOREM. With the above notation, there is a canonical injection for each integer n ^ 0. H"{G, V) > Hg en (G, V) Proof. By [1], H\ m {G, V) «H n (G, V(r)) s H"{G, V') for r» 0. Our result thus follows from either (3.6) above or (2.1). References 1. G. AVRUNIN, 'Generic cohomology of twisted groups', Trans. Amer. Math. Soc, 268 (1981), J. BALLARD, 'Injective modules for restricted enveloping algebras', Math. Z., 163 (1978), C. CHEVALLEY, Classification des groupes de Lie algebriques (2 vols.) (Ecole Normale Superieure, Paris, ). 4. E. T. CLINE, 'A second look at Weyl modules for SL 2 ', preprint, Clark University, E. T. CLINE, B. J. PARSHALL, L. L. SCOTT and W. VAN DER KALLEN, 'Rational and generic cohomology', Invent. Math., 39 (1977), E. T. CLINE, B. J. PARSHALL and L. L. SCOTT, 'Cohomology, hyperalgebras, and representations', J. Algebra, 63 (1980), E. T. CLINE, B. J. PARSHALL and L. L. SCOTT, 'On the tensor product theorem for algebraic groups', J. Algebra, 63 (1980), E. T. CLINE, B. J. PARSHALL and L. L. SCOTT, 'A Mackey imprimitivity theory for algebraic groups', Math. Z., 182 (1983),

8 300 DETECTING RATIONAL COHOMOLOGY OF ALGEBRAIC GROUPS 9. S. DONKIN, 'On Ext 1 for semisimple groups and infinitesimal subgroups', Math. Proc. Cambridge Philos. Soc, 92 (1982), E. FRIEDLANDER and B. J. PARSHALL, 'On the cohomology of algebraic and related finite groups', Invent. Math., to appear. 11. W. JONES and B. j. PARSHALL, 'On the cohomology of finite groups of Lie type', Park City conference on finite groups (Academic Press, New York, 1976), pp S. MACLANE, Homology (Springer, New York, 1967). 13. R. STEINBERG, 'Representations of algebraic groups', Nagoya Math. J., 22 (1963), R. STEINBERG, Endomorphisms of linear algebraic groups, Memoir 80 (American Mathematical Society, Providence, 1968). (Cline) (Parshall and Scott) Department of Mathematics, Department of Mathematics, Clark University, University of Virginia, Worcester, Mathematics-Astronomy Building, Massachusetts 01610, Charlottesville, U.S.A. Virginia 22903, U.S.A.

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