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>