Rings of differential operators as enveloping algebras of Hasse--Schmidt derivations

TL;DR

This paper introduces the enveloping algebra of Hasse--Schmidt derivations for a commutative algebra and proves it is isomorphic to the ring of differential operators under certain conditions, generalizing known characteristic 0 results.

Contribution

It defines the enveloping algebra of Hasse--Schmidt derivations and establishes its isomorphism with the ring of differential operators in a broad setting.

Findings

Enveloping algebra of Hasse--Schmidt derivations is isomorphic to the ring of differential operators.

Generalizes characteristic 0 case to broader settings with smoothness assumptions.

Provides a new algebraic framework connecting derivations and differential operators.

Abstract

Let $k$ be a commutative ring and $A$ a commutative $k$-algebra. In this paper we introduce the notion of enveloping algebra of Hasse--Schmidt derivations of $A$ over $k$ and we prove that, under suitable smoothness hypotheses, the canonical map from the above enveloping algebra to the ring of differential operators $\mathcal{D}_{A/k}$ is an isomorphism. This result generalizes the characteristic 0 case in which the ring $\mathcal{D}_{A/k}$ appears as the enveloping algebra of the Lie-Rinehart algebra of the usual $k$-derivations of $A$ provided that $A$ is smooth over $k$.