Twisting the Infinitesimal Site

TL;DR

This paper classifies twistings of Grothendieck's differential operators in prime characteristic, linking them to syntomic cohomology, and explores their relationships with crystalline operators and mixed characteristic cases.

Contribution

It establishes a bijection between isomorphism classes of twistings and syntomic cohomology, providing new insights into their structure and relationships in various characteristics.

Findings

Twistings correspond to elements in $H^2(X,\mathbb{Z}_p(1))$.

The classification links differential operator twistings to syntomic cohomology.

Relationships between crystalline and Grothendieck operator twistings are clarified.

Abstract

We classify twistings of Grothendieck's differential operators on a smooth variety $X$ in prime characteristic $p$. We prove isomorphism classes of twistings are in bijection with $H^2(X,\mathbb{Z}_p(1))$, the degree 2, weight 1 syntomic cohomology of $X$. We also discuss the relationship between twistings of crystalline and Grothendieck differential operators. Twistings in mixed characteristic are also analyzed.