Bounding Cohomology for Finite Groups and Frobenius Kernels

Abstract

Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let σ :G → G be a strict endomorphism (i.e., the subgroup G(σ) of σ-fixed points is finite). Also, let Gσ be the scheme-theoretic kernel of σ, an infinitesimal subgroup of G. This paper shows that the dimension of the degree m cohomology group Hm(G(σ),L) for any irreducible kG(σ)-module L is bounded by a constant depending on the root system Φ of G and the integer m. These bounds are actually established for the degree m extension groups \( Ext^{m}_{G(\sigma )}(L,L^{\prime })\) between irreducible kG(σ)-modules \(L,L^{\prime }\), with a similar result holding for Gσ. In these Extm results, the bounds also depend on the highest weight associated to L, but are, nevertheless, independent of the characteristic p.