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$ V$-filtrations in positive characteristic and test modules


Author: Axel Stäbler
Journal: Trans. Amer. Math. Soc. 368 (2016), 7777-7808
MSC (2010): Primary 13A35; Secondary 14B05
DOI: https://doi.org/10.1090/tran/6632
Published electronically: January 27, 2016
MathSciNet review: 3546784
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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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Additional Information

Axel Stäbler
Affiliation: Johannes Gutenberg-Universität Mainz, Fachbereich 08, Staudingerweg 9, 55099 Mainz, Germany
Email: staebler@uni-mainz.de

DOI: https://doi.org/10.1090/tran/6632
Received by editor(s): January 21, 2014
Received by editor(s) in revised form: April 1, 2014, and December 4, 2014
Published electronically: January 27, 2016
Article copyright: © Copyright 2016 American Mathematical Society