Contramodules for algebraic groups: the existence of mock projectives

Abstract Let G be an affine algebraic group over an algebraically closed field of positive characteristic. Recent work of Hardesty, Nakano, and Sobaje gives necessary and sufficient conditions for the existence of so-called mock injective G -modules, that is, modules which are injective upon restriction to all Frobenius kernels of G . In this article, we give analogous results for contramodules, including showing that the same necessary and sufficient conditions on G guarantee the existence of mock projective contramodules. In order to do this, we first develop contramodule analogs to many well-known (co)module constructions.