Derived Functors of Differential Operators

TL;DR

This paper provides formulas for the ring of differential operators and their derived functors, demonstrating their behavior under reduction to positive characteristic and their ability to detect properties of singularities.

Contribution

It offers explicit formulas for differential operators and their derived functors, advancing understanding of their behavior in positive characteristic and singularity detection.

Findings

Formulas for the ring of differential operators

Demonstration of good behavior under reduction to positive characteristic

Detection of properties of singularities using derived functors

Abstract

In their work on differential operators in positive characteristic, Smith and Van den Bergh define and study the derived functors of differential operators; they arise naturally as obstructions to differential operators reducing to positive characteristic. In this note, we provide formulas for the ring of differential operators as well as these derived functors of differential operators. We apply these descriptions to show that differential operators behave well under reduction to positive characteristic under certain hypotheses. We show that these functors also detect a number of interesting properties of singularities.