𝑉-filtrations in positive characteristic and test modules

Stäbler, Axel

doi:10.1090/tran/6632

$V$-filtrations in positive characteristic and test modules
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Trans. Amer. Math. Soc. 368 (2016), 7777-7808 Request permission

Abstract:

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 étale morphisms. We also derive corresponding transformation rules for test modules generalizing previous results by Schwede and Tucker in the étale case.

If $\mathfrak {a} = (f)$ defines a smooth hypersurface and $R$ is in addition smooth, then for a Cartier crystal corresponding to a locally constant sheaf on $\operatorname {Spec} R_{\acute {e}t}$ the functor $Gr^{[0,1]}$ corresponds, up to a shift, to $i^!$, where $i: V(\mathfrak {a}) \to \operatorname {Spec} R$ is the closed immersion.

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