Support varieties for infinitesimal group schemes

Abstract

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 support CGl _ Spec H (G, k) analogous to that given by D. Quillen for the cohomology of finite groups [Q]. 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 [A-E] and J. Carlson [CI] for finite groups, we consider the variety IGIM c IGI of the ideal IM = ker{Hev(G,k) -* Ext*(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 [A-S]. This paper is a continuation of our recent work establishing the finite generation of H* (G, k) [F-S] and investigating the infinitesimal 1-parameter subgroups of G [S-F-B]. Earlier work of E. Friedlander and B. Parshall [FP1], [FP2], [FP3], [FP4] and J. Jantzen [JI] concerning the cohomology of restricted Lie algebras are forerunners of the results presented here: finite-dimensional restricted Lie algebras are in 1-1 correspondence with infinitesimal group schemes of height < 1. Our main theorems (Theorems 5.2 and 6.7 below) when restricted to infinitesimal group schemes of height < 1 significantly strengthen previously known cohomological information for restricted Lie algebras. An interesting aspect of our work is the extent to which infinitesimal 1-parameter subgroups v : G?a(r) G for infinitesimal group schemes G of height < r play the role of elementary abelian p-subgroups (and their generalizations, shifted subgroups) for finite groups. Indeed, much of our effort is dedicated to proving that cohomology classes are detected (modulo nilpotence) by such 1-parameter subgroups. This is first done in ?2 for unipotent infinitesimal group schemes, using an induction argument made possible by a structure theorem presented in ?1. This structure theorem is the analogue in our context of a theorem of J.-P. Serre characterizing elementary abelian p-groups [S].