A characteristic-free proof of a basic result on D-modules

TL;DR

This paper provides the first characteristic-free proof that the localization of a polynomial ring at a non-zero element has finite length as a D-module, generalizing a fundamental result in D-module theory.

Contribution

It introduces a novel characteristic-free proof of a key property of D-modules, extending the understanding beyond characteristic restrictions.

Findings

R_f has finite length as a D-module in any characteristic

The proof is the first to be characteristic-free for this result

Establishes a foundational property of D-modules over polynomial rings

Abstract

Let k be a field, let R be a ring of polynomials in a finite number of variables over k, let D be the ring of k-linear differential operators of R and let f be a non-zero element of R. It is well-known that R_f, with its natural D-module structure, has finite length in the category of D-modules. We give a characteristic-free proof of this fact. To the best of our knowledge this is the first characteristic-free proof.