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 our previous paper 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 of H^{ev}(G,k) provided that G is an
infinitesimal group scheme of height <= r.
This work is a continuation of Cohomology of finite group schemes
over a field (by by E. Friedlander and A. Suslin) 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.