  We investigate infinite dimensional modules for an affine group scheme
$\mathbb G$ of finite type over a field of positive characteristic $p$. For any
subspace $X \subset \mathcal O(\mathbb G)$ of the coordinate algebra of
$\mathbb G$, we consider the abelian subcategory $Mod(\mathbb G,X) \subset
Mod(\mathbb G)$ of ``$X$-comodules" and the left exact functor $(-)_X:
Mod(\mathbb G) \to Mod(\mathbb G,X)$ which is right adjoint to the inclusion
functor. We employ ``ascending converging sequences" $\{ X_i \}$ of subspaces
of $\mathcal O(\mathbb G)$ to provide functorial filtrations $\{ M_{X_i }\}$ of
each $\mathbb G$-module $M$. A $\mathbb G$-module $M$ is injective if and only
if each $M_{X_i}$ is an injective $X_i$-comodule for some (or, equivalently,
for all) such $\{ X_i \}$.
  We consider the explicit ascending converging sequence $ \{ \mathcal
O(\mathbb G)_{\leq d,\phi} \}$ of finite dimensional subcoalgebras of $\mathcal
O(\mathbb G)$ depending upon a closed embedding $\phi: \mathbb G \
\hookrightarrow \ GL_N$. Of particular interest to us are mock injective
$\mathbb G$-modules, modules whose support varieties are empty. Restrictions of
a $\mathbb G$-module to each $\mathcal O(\mathbb G)_{\leq d,\phi}$ provide new
invariants for $\mathbb G$-modules. For cofinite $\mathbb G$-modules $M$, we
explore the the growth of $d \mapsto M_{\cal O(\mathbb G)_{\leq d,\phi}}$.
