Infinite dimensional modules for linear algebraic groups

TL;DR

This paper explores infinite dimensional modules for linear algebraic groups over fields of positive characteristic, introducing new concepts like cofinite type and mock injective modules, and establishing their properties within a tensor triangulated framework.

Contribution

It introduces the notion of cofinite type modules and mock injective modules for algebraic groups, extending support theories and developing a stable category framework.

Findings

Defined cofinite type modules using finite dimensional subcoalgebras.

Introduced and analyzed properties of mock injective modules.

Established the stable category of mock modules as a tensor triangulated category.

Abstract

We investigate infinite dimensional modules for a linear algebraic group $\mathbb G$ over a field of positive characteristic $p$. For any subcoalgebra $C \subset \mathcal O(\mathbb G)$ of the coordinate algebra of $\mathbb G$, we consider the abelian subcategory $CoMod(C) \subset Mod(\mathbb G)$ and the left exact functor $(-)_C: Mod(\mathbb G) \to CoMod(C)$ that is right adjoint to the inclusion functor. The class of cofinite $\mathbb G$-modules is formulated using finite dimensional subcoalgebras of $\mathcal O(\mathbb G)$ and the new invariant of "cofinite type" is introduced. We are particularly interested in mock injective $\mathbb G$-modules, $\mathbb G$-modules which are not seen by earlier support theories. Various properties of these ghostly $\mathbb G$-modules are established. The stable category $StMock(\mathbb G)$ is introduced, enabling mock injective $\mathbb G$-modules to fit into the framework of tensor triangulated categories.