Asymptotic Behavior of Differential Powers

TL;DR

This paper investigates the asymptotic properties of differential powers of ideals, focusing on monomial ideals in characteristic zero, and introduces a differential closure operation related to radical ideals and D-modules.

Contribution

It characterizes when differential powers become principal, analyzes containment relations, and defines a new differential closure operation with connections to radical and D-module theory.

Findings

Differential powers of certain ideals become eventually principal.

Containment relations between ordinary and differential powers are established.

Differential closure coincides with radical in simple D-module rings.

Abstract

In this paper, we study the differential power operation on ideals. We begin with a focus on monomial ideals in characteristic 0 and find a class of ideals whose differential powers are eventually principal. We also study the containment problem between ordinary and differential powers of ideals, in analogy to earlier work comparing ordinary and symbolic powers of ideals. We further define a possible closure operation on ideals, called the differential closure, in analogy with integral closure and tight closure. We show that this closure operation agrees with taking the radical of an ideal if and only if the ambient ring is a simple $D$-module.