Let $R$ be a ring essentially of finite type over an $F$-finite field. Givenan ideal $\mathfrak{a}$ and a principal Cartier module $M$ we introduce thenotion of a $V$-filtration of $M$ along $\mathfrak{a}$. If $M$ is $F$-regularthen this coincides with the test module filtration. We also show that theassociated graded induces a functor $Gr^{[0,1]}$ from Cartier crystals toCartier crystals supported on $V(\mathfrak{a})$. This functor commutes withfinite pushforwards for principal ideals and with pullbacks along essentially\'etale morphisms. We also derive corresponding transformation rules for testmodules 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 additionregular 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.
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机译：假设$ R $是$ F $有限域上的有限类型环。给出理想的$ \ mathfrak {a} $和主要的Cartier模块$ M $，我们引入了$ M $沿着$ \ mathfrak {a} $进行$ V $过滤的概念。如果$ M $是$ F $ -regular，那么这与测试模块过滤是一致的。我们还显示，关联的渐变会诱导从Cartier晶体到支持在$ V（\ mathfrak {a}）$上的Cartier晶体的函子$ Gr ^ {[[0,1]} $。该函子可对主要理想进行无穷的前推，并沿本质上的童话变态进行回调。我们还导出了测试模块的对应转换规则，这些转换模块将Schwede和Tucker在\'etale情况下的先前结果进行了概括（参见arXiv：1003.4333）。如果$ \ mathfrak {a} =（f）$定义了一个光滑的超曲面，而$ R $是规则的，则对于与局部常数捆对应的Cartier晶体，在$ \ Spec R _ {\ acute {e} t} $上，该仿函数$ Gr ^ {[[0,1]} $最多对应于$ i ^！$，其中$ i：V（\ mathfrak {a}）\ to \ Spec R $是封闭浸入。
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