The V-filtration for tame unit $F$-crystals

TL;DR

This paper develops a unique V-filtration theory for tame unit F-crystals on smooth varieties in characteristic p, linking it to nearby cycles and providing applications in algebraic geometry.

Contribution

It axiomatizes the V-filtration for tame unit F-crystals and proves its existence and uniqueness, connecting it to nearby cycles under the Riemann-Hilbert correspondence.

Findings

Unique V-filtration determined by axioms for tame unit F-crystals.

Existence of V-filtration on j_*M for tame unit F-crystals.

Degree zero component corresponds to nearby cycles functor.

Abstract

Let X be a smooth variety over an algebraically closed field of characteristic p > 0, Z a smooth divisor, and j : U = X\Z --> X the natural inclusion. An axiomatizing of the properties of a V -filtration on a unit F-crystal is proposed and is proven to determine a unique filtration. It is shown that if M is a tame unit F-crystal on U then such a V -filtration along Z exists on j_*M. The degree zero component of the associated graded module is proven to be the (unipotent) nearby cycles functor of Grothendieck and Deligne under the Emerton-Kisin Riemann-Hilbert correspondence. A few applications to A^1 and gluing are then discussed.