Witt Differential Operators

TL;DR

This paper develops a theory of differential operators on Witt vectors over smooth schemes in positive characteristic, establishing connections with crystals, formalism of pushforward/pullback, and de Rham Witt resolutions.

Contribution

It introduces sheaves of differential operators on Witt vectors, relates modules over these to Berthelot's rings, and extends the formalism of crystals and de Rham-Witt complexes.

Findings

Defined sheaves of differential operators on Witt vectors.

Established an embedding of crystals into modules over these operators.

Developed a relative de Rham Witt resolution for pushforward computations.

Abstract

For a smooth scheme $X$ over a perfect field $k$ of positive characteristic, we define (for each $m\in\mathbb{Z}$) a sheaf of rings $\mathcal{\widehat{D}}_{W(X)}^{(m)}$ of differential operators (of level $m$) over the Witt vectors of $X$. If $\mathfrak{X}$ is a lift of $X$ to a smooth formal scheme over $W(k)$, then for $m\geq0$ modules over $\mathcal{\widehat{D}}_{W(X)}^{(m)}$ are closely related to modules over Berthelot's ring $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$ of differential operators of level $m$ on $\mathfrak{X}$. Our construction therefore gives an description of suitable categories of modules over these algebras, which depends only on the special fibre $X$. There is an embedding of the category of crystals on $X$ (over $W_{r}(k)$) into modules over $\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}$; and so we obtain an alternate description of this category as well. For a map $\varphi:X\to Y$ we develop the formalism of pullback and pushforward of $\mathcal{\widehat{D}}_{W(X)}^{(m)}$-modules and show all of the expected properties. When working mod $p^{r}$, this includes compatibility with the corresponding formalism for crystals, assuming $\varphi$ is smooth. In this case we also show that there is a ``relative de Rham Witt resolution'' (analogous to the usual relative de Rham resolution in $\mathcal{D}$-module theory) and therefore that the pushforward of (a quite general subcategory of) modules over $\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}$ can be computed via the reduction mod $p^{r}$ of Langer-Zink's relative de Rham Witt complex. Finally we explain a generalization of Bloch's theorem relating integrable de Rham-Witt connections to crystals.