Specialization of Integral Closure of Ideals by General Elements

TL;DR

This paper demonstrates that the integral closure of certain ideals remains compatible when specialized by a general element, extending previous results and including specific classes of monomial ideals in polynomial rings.

Contribution

It generalizes the compatibility of integral closure with specialization by a general element to a broader class of ideals and rings, including large powers and specific monomial ideals.

Findings

Integral closure is compatible with specialization by a general element for ideals of height at least two.

Compatibility extends to large powers of the ideal.

Certain squarefree monomial ideals in polynomial rings also exhibit this compatibility.

Abstract

In this paper, we prove a result similar to results of Itoh and Hong-Ulrich, proving that integral closure of an ideal is compatible with specialization by a general element of that ideal for ideals of height at least two in a large class of rings. Moreover, we show integral closure of sufficiently large powers of the ideal is compatible with specialization by a general element of the original ideal. In a polynomial ring over an infinite field, we give a class of squarefree monomial ideals for which the integral closure is compatible with specialization by a general linear form.