A variant of the Reynolds operator

Let G be a linearly reductive group over a field k, and let R be a k-algebra with a rational action of G. Given rational R-G-modules M and N, we define for the induced G-action on Hom R (M,N) a generalized Reynolds operator, which exists even if the action on Hom R (M,N) is not rational. Given an R-module homomorphism M → N, it produces, in a natural way, an R-module homomorphism which is G-equivariaut. We use this generalized Reynolds operator to study properties of rational R-G modules. In particular, we prove that if M is invariantly generated (i.e. M = R.M G ), then M G is a projective (resp. flat) R G -module provided that M is a projective (resp. flat) R-module. We also give a criterion whether an R-projective (or R-flat) rational R-G-module is extended from an R G -module.