Iterated socles and integral dependence in regular rings

TL;DR

This paper investigates the properties of iterated socles of ideals in regular rings, establishing their integral dependence and providing explicit formulas for generators, especially in characteristic zero and for determinantal ideals.

Contribution

It introduces new results on the integral dependence of iterated socles, generalizes Herzog's formulas, and offers explicit generator formulas for specific classes of ideals.

Findings

Iterated socles are integral over the original ideal with reduction number one under certain conditions.

In characteristic zero, explicit formulas for generators of iterated socles are provided.

Simplified formulas for iterated socles of height two ideals in two variables are derived.

Abstract

Let $R$ be a formal power series ring over a field, with maximal ideal $\mathfrak m$, and let $I$ be an ideal of $R$ such that $R/I$ is Artinian. We study the iterated socles of $I$, that is the ideals which are defined as the largest ideal $J$ with $J\mathfrak m^s\subset I$ for a fixed positive integer $s$. We are interested in these ideals in connection with the notion of integral dependence of ideals. In this article we show that the iterated socles are integral over $I$, with reduction number one, provided $s \leq \text{o}(I_1(\varphi_d))-1$, where $\text{o}(I_1(\varphi_d))$ is the order of the ideal of entries of the last map in a minimal free $R$-resolution of $R/I$. In characteristic zero, we also provide formulas for the generators of iterated socles whenever $s\leq \text{o}(I_1(\varphi_d))$. This result generalizes previous work of Herzog, who gave formulas for the socle generators of any ${\mathfrak m}$-primary homogeneous ideal $I$ in terms of Jacobian determinants of the entries of the matrices in a minimal homogeneous free $R$-resolution of $R/I$. Applications are given to iterated socles of determinantal ideals with generic height. In particular, we give surprisingly simple formulas for iterated socles of height two ideals in a power series ring in two variables. These generators are suitable determinants obtained from the Hilbert-Burch matrix.