On comparing the cohomology of algebraic groups, finite Chevalley groups and Frobenius kernels

Let G be a semisimple simply connected algebraic group defined and split over the field F-p with p elements, G(F-q) be the finite Chevalley group consisting of the F-q-rational points of G where q = p(r), and G(r) be the rth Frobenius kernel of G. This paper investigates relationships between the extension theories of G, G(F-q), and G(r) over the algebraic closure of F-p. First, some qualitative results relating extensions over G(F-q) and G(r) are presented. Then certain extensions over G(F-q) and G(r) are explicitly identified in terms of extensions over G. (C) 2001 Elsevier Science B.V. All rights reserved.