Infinitesimal 1-parameter subgroups and cohomology

Abstract

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[Vr(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 of H (G, k) provided that G is an infinitesimal group scheme of height <Kr. This 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 Vr(G) of homomorphisms from Ga(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, Vr(G) is the scheme of r-tuples of p-nilpotent, pairwise commuting elements of Lie(G). More generally, an embedding G c GLn determines a closed embedding of Vr(G) into the scheme of r-tuples of p-nilpotent, pairwise commuting elements of gl1 = Lie(GLn). The relationship between k[Vr (G)], the coordinate algebra of Vr (G), and H* (G, k) is introduced in Theorem 1.14: a natural homomorphism of graded k-algebras