Universal Classes for Algebraic Groups

We exhibit cocycles representing certain classes in the cohomology of the algebraic group GLn with coefficients in the representation Γ(gl n ). These classes’ existence was anticipated by van der Kallen, and they intervene in the proof that reductive linear algebraic groups have finitely generated cohomology algebras [18]. Let k be a field of positive characteristic, let A be a finitely generated k-algebra, and let G be a reductive linear algebraic group defined over k and acting rationally on A by algebra automorphisms. Then the rational cohomology H(G,A) is an algebra, and one can wonder if it is finitely generated. In degree 0, the finite generation of the subalgebra AG = H0(G,A) is part of Hilbert’s fourteenth problem and was solved positively by the work of Nagata [13] and Haboush [9]. The finite generation of the whole cohomology algebra remained unsolved in general, though much progress had been made in recent years [7, 20]. In [20], van der Kallen proved (under some restrictions on the characteristic of k which were removed in [15]) that the finite generation of H(G,A) holds under the following condition: the group G embeds in GLn,k for some integer n, and there exist universal cohomology classes in H∗(GLn,k,Γ∗(gl(1) n )) satisfying some divided power algebra relations. He proved the existence of these universal cohomology classes for n = 2 [20, Th 4.4], and for n = 3 in characteristic p = 2 [21]. Later, van der Kallen mentioned that cohomological finite generation holds under a weaker condition, namely the existence of the so-called “lifted universal cohomology classes”. Our main result is the existence of these lifted classes: Theorem 0.1. Let k be a field of positive characteristic and let n > 1 be an integer. There are cohomology classes c[d] ∈ H2d(GLn,k,Γd(gl(1) n )) such that : (1) c[1] ∈ H2(GLn,k, gl n ) is non zero. (2) Let d ≥ 1 and let ∆(1,...,1) : Γ d(gl (1) n ) → (gl (1) n )⊗d be the map induced by the diagonal Γd → ⊗d. Then ∆(1,...,1) ∗c[d] = c[1] ∪d. This actually reproves and significantly extends the famous theorem of Friedlander and Suslin [7, Thm 1.2] on the existence of nonzero universal classes er. Indeed, as indicated in [18, Lemma 6.2], we may draw the following corollary from theorem 0.1: Date: May 5, 2009. 1 2 ANTOINE TOUZÉ Corollary 0.2. ([7, Thm 1.2]) Let k be a field of positive characteristic. For any n > 1 there exist classes er ∈ H 2pr−1(GLn,k, gl n ) which restrict nontrivially to H2p r−1 ((GLn,k)(1), gl n ). The proof of theorem 0.1 may be summarized as follows. In section 1, we remark that theorem 0.1 reduces to a stable cohomology statement, that is, it suffices to prove it for large values of n. Bifunctor cohomology [4] gives access to the stable rational cohomology of GLn,k, and we translate theorem 0.1 in terms of (strict polynomial) bifunctors. More specifically, we show that theorem 0.1 reduces to theorem 1.4, that is, to the computation of some classes in the cohomology of the strict polynomial bifunctors Γd(gl(1)). The proof of theorem 1.4 is given in section 4. We use explicit coresolutions of the bifunctors Γd(gl(1)) to compute cocycles representing the classes c[d]. Section 3 is devoted to building the explicit coresolutions of the Γd(gl(1)). We use the following strategy. First, the Γd(gl(1)) are bifunctors of the form F (gl), obtained by precomposing a functor F by the bifunctor gl(−1,−2) := Homk(−1,−2). We remark in section 3.1 that the cohomology of this kind of bifunctor may be computed via acyclic coresolutions obtained by precomposing an injective coresolution of F by the bifunctor gl. Thus, our seeking of the explicit coresolutions of the bifunctors Γd(gl(1)) reduces to the (combinatorially easier) seeking of injective coresolutions of the functors Γd(I(1)) obtained by precomposing the functors Γd by the Frobenius twist I(1). Second, we define in section 3.2 a class of injective coresolutions of strict polynomial functors called “twist compatible coresolutions”. These coresolutions enjoy the following nice property: we may use an injective twist compatible coresolution JF of F to build an explicit injective coresolution of the functor F (I(1)). Third, we build injective twist compatible coresolutions of the functors Γd in section 3.3. When the characteristic p is odd, the combinatorics of the Frobenius twist bring the notion of p-complex into play. Section 2 contains the recollections about p-complexes needed in section 3.2, as well as a new result (proposition 2.4) which is the key point to identify cup products in section 4.