A Brian\c{c}on-Skoda type result for a non-reduced analytic space

TL;DR

This paper extends the Briançon-Skoda theorem to Cohen-Macaulay non-reduced analytic spaces, providing conditions for ideal membership using differential operators, thus broadening its applicability.

Contribution

It introduces a new analogue of the Briançon-Skoda theorem for non-reduced spaces, generalizing previous results to Cohen-Macaulay germs.

Findings

Established a sufficient condition for ideal membership in non-reduced spaces.

Generalized Huneke's theorem to Cohen-Macaulay, non-reduced analytic spaces.

Utilized size conditions of differential operators for membership criteria.

Abstract

We present here an analogue of the Brian\c{c}on-Skoda theorem for a germ of an analytic space $Z$ at 0, such that $O_{Z,0}$ is Cohen-Macaulay, but not necessarily reduced. More precisely, we find a sufficient condition for membership of a function in a power of an arbitrary ideal $a^l \subset O_{Z,0}$ in terms of size conditions of Noetherian differential operators applied to that function. This result generalizes a theorem by Huneke in the reduced case.