DMCA
COHOMOLOGY OF FINITE GROUP SCHEMES OVER A FIELD
| Citations: | 105 - 13 self |
BibTeX
@MISC{Friedlander_cohomologyof,
author = {Eric M. Friedlander and Andrei Suslin},
title = {COHOMOLOGY OF FINITE GROUP SCHEMES OVER A FIELD},
year = {}
}
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Abstract
A finite group scheme G over a field k is equivalent to its coordinate algebra, a finite dimensional commutative Hopf algebra k[G] over k. In many contexts, it is natural to consider the rational (or Hochschild) cohomology of G with coefficients in a k[G]-comodule M. This is naturally isomorphic to the cohomology of the dual cocommutative Hopf algebra k[G] # with coefficients in the k[G] #-module M. In this latter formulation, we encounter familiar examples of the cohomology of group algebras kπ of a finite groups π and of restricted enveloping algebras V (g) of finite dimensional restricted Lie algebras g. In recent years, the representation theory of the algebras kπ and V (g) has been studied by considering the spectrum of the cohomology algebra with coefficients in the ground field k and the support in this spectrum of the cohomology with coefficients in various modules. This approach relies on the fact that H ∗ (π, k) and H ∗ (V (g), k) are finitely generated k-algebras as proved in [G], [E], [V], [FP2]. Rational representations of algebraic groups in positive characteristic correspond to representations of a hierarchy of finite group schemes. In order to begin the process of introducing geometric methods to the study of these other group schemes, finite generation must be proved. Such a proof has proved surprisingly elusive (though partial results can be found in [FP2]). The main theorem of this paper is the following: Theorem 1.1. Let G be a finite group scheme and M a finite dimensional rational G-module. Then H ∗ (G, k) is a finitely generated k-algebra and H ∗ (G, M) is a finite H ∗ (G, k)-module. Work in progress by C. Bendel (and the authors) reveals that Theorem 1.1 and its proof will provide interesting theorems of a geometric nature concerning the representation theory of finite group schemes. In a sense that is made explicit in section 1, our proof of finite generation is quite constructive. We embed G in some general linear group GLn and establish the existence of universal extension classes for GLn of specified degrees. In a direct manner, these classes provide the generators of H ∗ (G, k). In order to construct these universal extension classes, we follow closely the approach of V. Franjou, J. Lannes, and L. Schwartz [FLS]. This entails the study of
Keyphrases
finite group scheme cohomology finite group scheme finite generation representation theory universal extension class finite dimensional rational g-module partial result geometric method schwartz fl coordinate algebra finite group interesting theorem direct manner dual cocommutative hopf algebra rational representation finite dimensional commutative hopf algebra geometric nature specified degree positive characteristic correspond generated k-algebras cohomology algebra various module group scheme latter formulation main theorem algebraic group ground field many context lie algebra general linear group gln recent year familiar example group algebra
