Infinite dimensional modules for linear algebraic groups

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