Differential Operators, Gauges, and Mixed Hodge Modules

TL;DR

This paper develops a new theory of gauges in mixed characteristic using arithmetic differential operators, extending $ ext{D}$-module theory and applying it to mixed Hodge modules and cohomology of varieties over number fields.

Contribution

It introduces a new sheaf of algebras $oxed{ ext{} ext{ extasciitilde} ext{ extasciitilde} ext{ extasciitilde}}$ as a higher-dimensional analogue of the Dieudonne ring, generalizes Mazur's theorem, and connects mixed Hodge modules with $ ext{D}$-modules in mixed characteristic.

Findings

Defined a new sheaf of algebras $oxed{ ext{}} ext{ extasciitilde} ext{ extasciitilde} ext{ extasciitilde}$ for smooth formal schemes over $W(k)$.

Modules over $oxed{ ext{}} ext{ extasciitilde} ext{ extasciitilde} ext{ extasciitilde}$ admit all usual $ ext{D}$-module operations.

Proved a version of Mazur's theorem for intersection and ordinary cohomology of quasiprojective varieties over number fields.

Abstract

The purpose of this paper is to develop a new theory of gauges in mixed characteristic. Namely, let $k$ be a perfect field of characteristic $p>0$ and $W(k)$ the $p$-typical Witt vectors. Making use of Berthelot's arithmetic differential operators, we define for a smooth formal scheme $\mathfrak{X}$ over $W(k)$, a new sheaf of algebras $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ which can be considered a higher dimensional analogue of the (commutative) Dieudonne ring. Modules over this sheaf of algebras can be considered the analogue (over $\mathfrak{X}$) of the gauges of Ekedahl and Fontain-Jannsen. We show that modules over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ admit all of the usual $\mathcal{D}$-module operations, and we prove a robust generalization of Mazur's theorem in this context. Finally, we show that an integral form of a mixed Hodge module of geometric origin admits, after a suitable $p$-adic completion, the structure of a module over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$. This allows us to prove a version of Mazur's theorem for the intersection cohomology and the ordinary cohomology of an arbitrary quasiprojective variety defined over a number field.