### Infinitesimal 1-Parameter Subgroups and Cohomology, by Andrei Suslin, Eric M. Friedlander, and Christopher P. Bendel

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.

Andrei Suslin <suslin@math.nwu.edu>

Eric M. Friedlander <eric@math.nwu.edu>

Christopher P. Bendel <bendel@math.nwu.edu>