On the center of the ring of differential operators on a smooth variety over $\bZ/p^n\bZ$

TL;DR

This paper determines the center of the ring of PD differential operators on smooth varieties over $Z/p^nZ$, confirming a conjecture and revealing a structure related to Witt vectors under certain conditions.

Contribution

It proves the structure of the center of the ring of PD differential operators over $Z/p^nZ$ and generalizes to deformations of associative algebras over $F_p$.

Findings

Center of PD differential operators matches Witt vectors under non-degeneracy.

Confirmed Kaledin's conjecture on the center structure.

Established isomorphism between centers in deformation settings.

Abstract

We compute the center of the ring of PD differential operators on a smooth variety over $\bZ/p^n\bZ$ confirming a conjecture of Kaledin. More generally, given an associative algebra $A_0$ over $\bF_p$ and its flat deformation $A_n$ over $\bZ/p^{n+1}\bZ$ we prove that under a certain non-degeneracy condition the center of $A_n$ is isomorphic to the ring of length $n+1$ Witt vectors over the center of $A_0$.