This is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p > 0. Whereas [SFB] is concerned with detection of cohomology classes, the present paper introduces the graded algebra k[V r (G)] of functions on the scheme of infinitesimal 1-parameter subgroups of height ≤ r on an affine group scheme G and demonstrates that this algebra is essentially a retract ofThis work is a continuation of [F-S] in which the existence of certain universal extension classes was established, thereby enabling the proof of finite generation of H * (G, k) for any finite group scheme G over k. The role of the scheme of infinitesimal 1-parameter subgroups of G was foreshadowed in [F-P] where H * (G (1) , k) was shown to be isomorphic to the coordinate algebra of the scheme of p-nilpotent elements of g = Lie(G) for G a smooth reductive group, G (1) the first Frobenius kernel of G, and p = char(k) sufficiently large. Indeed, p-nilpotent elements of g correspond precisely to infinitesimal 1-parameter subgroups of G (1) . Much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1-parameter subgroups.In §1, we construct the affine scheme V r (G) of homomorphisms from G a(r) to G which we call the scheme of infinitesimal 1-parameter subgroups of height ≤ r in G. In the special case r = 1, this is the scheme of p-nilpotent elements of the p-restricted Lie algebra Lie(G); for various classical groups G, V r (G) is the scheme of r-tuples of p-nilpotent, pairwise commuting elements of Lie(G). More generally, an embedding G ⊂ GL n determines a closed embedding of V r (G) into the scheme of r-tuples of p-nilpotent, pairwise commuting elements of gl n = Lie(GL n ). The relationship between k[V r (G)], the coordinate algebra of V r (G), and H * (G, k) is introduced in Theorem 1.14: a natural homomorphism of graded k-algebrasis constructed, a map which we show in [SFB] to induce a bijective map on associated schemes. The universal classes e r ∈ H 2p r−1 (GL n , gln ) constructed in [F-S] determine characteristic classes e r (G, V ) for any affine group scheme G provided with a finite dimensional representation V . Our central computation is the determination in

The representation theory of a connected smooth affine group scheme over a field k of characteristic p > 0 is faithfully captured by that of its family of Frobenius kernels. Such Frobenius kernels are examples of infinitesimal group schemes, affine group schemes G whose coordinate (Hopf) algebra k[G] is a finite-dimensional local k-algebra. This paper presents a study of the cohomology algebra H * (G, k) of an arbitrary infinitesimal group scheme over k.We provide a geometric determination of the "cohomological support variety" |G| ≡ Spec H ev (G, k) analogous to that given by D. Quillen for the cohomology of finite groups [Q]. We further study finite-dimensional rational G-modules M for arbitrary infinitesimal group schemes G over k. In a manner initiated by J. An interesting aspect of our work is the extent to which infinitesimal 1-parameter subgroups ν : G a(r) → G for infinitesimal group schemes G of height ≤ r play the role of elementary abelian p-subgroups (and their generalizations, shifted subgroups) for finite groups. Indeed, much of our effort is dedicated to proving that cohomology classes are detected (modulo nilpotence) by such 1-parameter subgroups. This is first done in §2 for unipotent infinitesimal group schemes, using an induction argument made possible by a structure theorem presented in §1. This structure theorem is the analogue in our context of a theorem of J.-P. Serre characterizing elementary abelian p-groups [S]. Alperin and L. Evens [A-E] and J. Carlson [C1] for finite groups, we consider the variety |G|

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Let G be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p. Let B be a Borel subgroup of G and U be its unipotent radical. In this paper, we determine the second cohomology groups of B and its Frobenius kernels for all simple B-modules. We also consider the standard induced modules obtained by inducing a simple B-module to G and compute all second cohomology groups of the Frobenius kernels of G for these induced modules. Also included is a calculation of the second ordinary Lie algebra cohomology group of Lie(U ) with coefficients in k.

Let G(Fq) be a finite Chevalley group defined over the field of q = p r elements, and k be an algebraically closed field of characteristic p > 0. A fundamental open and elusive problem has been the computation of the cohomology ring H • (G(Fq), k). In this paper we determine initial vanishing ranges which improves upon known results. For root systems of type An and Cn, the first non-trivial cohomology classes are determined when p is larger than the Coxeter number (larger than twice the Coxeter number for type An