A Simpson Correspondence in Positive Characteristic

Abstract

Let $p$ be a prime number, $k$ a perfect field of characteristic $p>0$, $X$ a smooth $k$-scheme and ${\mathcal{D}}_X^{(m)}$ the algebra of (arithmetic) differential operators of level $m\ge 0$. We study the Azumaya nature of this algebra and show how to construct, using an additional data, a splitting of a completion (along some ideal contained in its center) of it.