  Let $R$ be a ring essentially of finite type over an $F$-finite field. Given
an ideal $\mathfrak{a}$ and a principal Cartier module $M$ we introduce the
notion of a $V$-filtration of $M$ along $\mathfrak{a}$. If $M$ is $F$-regular
then this coincides with the test module filtration. We also show that the
associated graded induces a functor $Gr^{[0,1]}$ from Cartier crystals to
Cartier crystals supported on $V(\mathfrak{a})$. This functor commutes with
finite pushforwards for principal ideals and with pullbacks along essentially
\'etale morphisms. We also derive corresponding transformation rules for test
modules generalizing previous results by Schwede and Tucker in the \'etale case
(cf. arXiv:1003.4333).
  If $\mathfrak{a} = (f)$ defines a smooth hypersurface and $R$ is in addition
regular then for a Cartier crystal corresponding to a locally constant sheaf on
$\Spec R_{\acute{e}t}$ the functor $Gr^{[0,1]}$ corresponds, up to a shift, to
$i^!$, where $i: V(\mathfrak{a}) \to \Spec R$ is the closed immersion.
