Contramodules for algebraic groups: the existence of mock projectives

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 paper, 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.