Contramodules for algebraic groups: the existence of mock projectives

TL;DR

This paper extends the theory of mock injective modules for algebraic groups to contramodules, establishing conditions for the existence of mock-projective contramodules and developing related constructions.

Contribution

It provides the first analogous results for contramodules, including necessary and sufficient conditions for mock-projective contramodules, expanding the understanding of module theory over algebraic groups.

Findings

Identifies conditions for the existence of mock-projective contramodules

Develops contramodule analogs of classical (co)module constructions

Establishes a parallel between injective and projective contramodules in this context

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 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.