Rationally injective modules for algebraic linear groups

Abstract

1. Let G be an algebraic linear group over a field F. If G acts by linear automorphisms on some vector space M over F we say that M is a rational G-module if it is the sum of finite-dimensional G-stable subspaces V such that the representation of G on each V is a rational representation of G. The rational G-module M is said to be rationally injective if, whenever U is a rational G-module and q5 is a G-module homomorphism of a G-submodule V of U into M, 4) can be extended to a G-module homomorphism of U into M. This notion is basic for the cohomology theory of algebraic linear groups, as developed in [I1]. If K is a normal algebraic subgroup of G it is of interest to secure the spectral sequence relation between the cohomologies of G, K and G/K. In order to do this, one needs to know that a rationally injective rational G-module is cohomologically trivial as a K-module. Although this remains unsettled over arbitrary fields, we shall see here that an even stronger result holds over fields of characteristic 0. We shall also obtain convenient criteria for injectivity that are valid over arbitrary fields.