Simplicity criteria for rings of differential operators

TL;DR

This paper establishes a criterion for the simplicity of the algebra of differential operators on certain commutative algebras, linking simplicity to the behavior of the Jacobian ideal, and extends this to arbitrary fields.

Contribution

It provides a new simplicity criterion for differential operator algebras on commutative algebras, addressing a longstanding open question in the field.

Findings

The algebra $ ext{D}( ext{A})$ is simple iff $ ext{D}( ext{A}) ext{ga}_r^i ext{D}( ext{A})= ext{D}( ext{A})$ for all $i extgreater 0$ over perfect fields.

A general simplicity criterion is given for $ ext{D}(R)$ over arbitrary fields.

The criterion applies to algebras of essentially finite type and extends to arbitrary commutative algebras.

Abstract

Let $K$ be a field of arbitrary characteristic, $\CA$ be a commutative $K$-algebra which is a domain of essentially finite type (eg, the algebra of functions on an irreducible affine algebraic variety), $\ga_r$ be its {\em Jacobian ideal}, $\CD (\CA )$ be the algebra of differential operators on the algebra $\CA$. The aim of the paper is to give a simplicity criterion for the algebra $\CD (\CA )$: {\em The algebra $\CD (\CA )$ is simple iff $\CD (\CA ) \ga_r^i\CD (\CA )= \CD (\CA )$ for all $i\geq 1$ provided the field $K$ is a perfect field.} Furthermore, a simplicity criterion is given for the algebra $\CD (R)$ of differential operators on an arbitrary commutative algebra $R$ over an arbitrary field. This gives an answer to an old question to find a simplicity criterion for algebras of differential operators.