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.