$V$-filtrations in positive characteristic and test modules

Abstract

Abstract. Let R be a ring essentially of finite type over an F -finite field. Given an ideal a and a principal Cartier module M we introduce the notion of a V -filtration of M along 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 (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 (cf. [31]).If a = (f ) defines a smooth hypersurface and R is in addition smooth then for a Cartier crystal corresponding to a locally constant sheaf on Spec Ré t the functor Gr [0,1] corresponds, up to a shift, to i ! , where i : V (a) → Spec R is the closed immersion.

Test module filtrations for unit F-modules

“…We claim that f is a test element in the sense of [1,Theorem 3.11] for N and N . By [20,Lemma 4.1] and the above one has Supp N = Supp N C . Moreover, C f coincides with the R f -Cartier algebra generated by κ.…”

“…The following Theorem generalizes [20,Lemma 4.3] and is valid for any F -finite R provided that test modules do exist. It will allow us to define test module filtrations for unit F -modules (cf.…”

“…naturally carries a Cartier structure induced by κf t(p−1) (see [20,Section 4]). A morphism of Cartier modules is a morphism ϕ : M → N of the underlying R-modules such that for every κ e ∈ C e one has κ e F e * ϕ = ϕκ e .…”

“…In [21] Stadnik introduced a notion of V -filtration in the category of unit R[F ]-modules, proved existence in a special case and showed that in this case the zeroth graded piece corresponds to the unipotent part of the (underived) tame nearby cycles. In [20] the author showed that in the category of Cartier modules one may characterize the test module filtration by certain axioms that are akin to a V -filtration and showed that if (M, κ) corresponds to a locally constant sheaf and x defines a smooth hypersurface then Gr τ M (the associated graded of the test module filtration in the range [0, 1]) is naturally isomorphic to i ! M as a crystal, where i : Spec R/(x) → Spec R is the inclusion.…”

“…This is a local problem. Since the étale case was handled in [20] this boils down to understanding how ϕ ! transforms the test module for the map ϕ : A n R → Spec R. Once one has analyzed how the structural map F * ϕ !…”