Bounding Cohomology for Finite Groups and Frobenius Kernels

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 G_σ 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 ExtG^m_G(σ)(L,^L′) between irreducible kG(σ)-modules L,^L′, with a similar result holding for G_σ. In these Extm results, the bounds also depend on the highest weight associated to L, but are, nevertheless, independent of the characteristic p. We also show that one can find bounds independent of the prime for the Cartan invariants of G(σ) and G_σ, and even for the lengths of the underlying PIMs.