Support Varieties for Infinitesimal Group Schemes, by Andrei Suslin, Eric M. Friedlander, and Christopher P. Bendel

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| \equiv Spec H^ev(G,k) analogous to that given by D. Quillen for the cohomology of finite groups. We further study finite dimensional rational G-modules M for arbitrary infinitesimal group schemes G over k. In a manner initiated by J. Alperin and L. Evens and J. Carlson for finite groups, we consider the variety |G|_M \subset |G| of the ideal I_M = ker H^ev(G,k) --> Ext_G^*(M,M) and provide a geometric description of this variety which is analogous to that given by G. Avrunin and L. Scott for finite dimensional modules for finite groups.

Andrei Suslin <>
Eric M. Friedlander <>
Christopher P. Bendel <>