Infinite dimensional modules for linear algebraic groups

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.