Symmetry on rings of differential operators

TL;DR

This paper investigates when the ring of differential operators on a commutative algebra over a field is isomorphic to its opposite, providing new results for Gorenstein local rings and rings of invariants.

Contribution

It establishes conditions under which the differential operator ring is isomorphic to its opposite, including cases for Gorenstein local rings and invariant rings, with new insights into module structures.

Findings

Differential operator rings are isomorphic to their opposites under certain conditions.

Canonical modules can admit right $D$-module structures in many cases.

Some results were previously known in higher generality by Yekutieli.

Abstract

If $k$ is a field and $R$ is a commutative $k$-algebra, we explore the question of when the ring $D_{R|k}$ of $k$-linear differential operators on $R$ is isomorphic to its opposite ring. Under mild hypotheses, we prove this is the case whenever $R$ Gorenstein local or when $R$ is a ring of invariants. As a key step in the proof we show that in many cases of interest canonical modules admit right $D$-module structures. After this work was completed we realized that some of our results were already proved in higher generality by Yekutieli, albeit using more sophisticated methods.