Quasi-socle ideals in local rings with Gorenstein tangent cones

TL;DR

This paper investigates quasi-socle ideals in local rings with Gorenstein tangent cones, focusing on their integral properties and Cohen-Macaulay and Gorenstein conditions of associated graded and Rees rings.

Contribution

It provides new criteria for when quasi-socle ideals are integral over parameter ideals and when their graded rings are Cohen-Macaulay or Gorenstein.

Findings

Criteria for integrality of quasi-socle ideals over parameter ideals.

Conditions under which the associated graded rings are Cohen-Macaulay.

Conditions for the Gorenstein property of the graded and Rees rings.

Abstract

Quasi-socle ideals, that is the ideals $I$ of the form $I= Q : \mathfrak{m}^q$ in a Noetherian local ring $(A, \mathfrak{m})$ with the Gorenstein tangent cone $\mathrm{G}(\mathfrak{m}) = \bigoplus_{n \geq 0}{\mathfrak{m}}^n/{\mathfrak{m}}^{n+1}$ are explored, where $q \geq 1$ is an integer and $Q$ is a parameter ideal of $A$ generated by monomials of a system $x_1, x_2, ..., x_d$ of elements in $A$ such that $(x_1, x_2, ..., x_d)$ is a reduction of $\mathfrak{m}$. The questions of when $I$ is integral over $Q$ and of when the graded rings $\mathrm{G}(I) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ and $\mathrm{F}(I) = \bigoplus_{n \ge 0}I^n/\mathfrak{m} I^n$ are Cohen-Macaulay are answered. Criteria for $\mathrm{G} (I)$ and $\mathcal{R} (I) = \bigoplus_{n \geq 0}I^n$ to be Gorenstein rings are given.