V-filtrations in positive characteristic and test modules

2013/10/31 by Stäbler, Axel · 1 citation
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13A35 #Secondary 14B05

paper · doi:10.48550/arxiv.1310.8549

Abstract

Let R be a ring essentially of finite type over an F-finite field. Given an ideal \mathfraka and a principal Cartier module M we introduce the notion of a V-filtration of M along \mathfraka. 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(\mathfraka). 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 (cf. arXiv:1003.4333). If \mathfraka = (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_\acuteet the functor Gr^[0,1] corresponds, up to a shift, to i^!, where i: V(\mathfraka) → \Spec R is the closed immersion.

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