  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.
