Weak subintegral closure of ideals

TL;DR

This paper explores the weak subintegral closure of ideals, providing geometric interpretations, new characterizations, and a valuative criterion, linking algebraic properties with stratification conditions.

Contribution

It introduces new characterizations and a valuative criterion for weak subintegral closure, and defines a novel relative closure operation for modules.

Findings

Characterization of weak subintegral closure of ideals

Introduction of the ideal I_> related to Rees valuations

A new valuative criterion for membership in the closure

Abstract

We describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex-analytic settings. We focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral closure of an ideal. We start by giving a geometric interpretation of the Reid-Roberts-Singh criterion for when an element is weakly subintegral over a subring. We give new characterizations of the weak subintegral closure of an ideal. We associate with an ideal $I$ of a ring $A$ an ideal $I_>$, which consists of all elements of $A$ such that $v(a)>v(I)$, for all Rees valuations $v$ of $I$. The ideal $I_>$ plays an important role in conditions from stratification theory such as Whitney's condition A and Thom's condition $A_f$ and is contained in every reduction of $I$. We close with a valuative criterion for when an element is in the weak subintegral closure of an ideal. For this, we introduce a new closure operation for a pair of modules, which we call relative closure.