Quasi-socle ideals in a Gorenstein local ring

TL;DR

This paper investigates the structure of quasi-socle ideals in Gorenstein local rings, focusing on conditions for reduction and Cohen-Macaulayness of the associated graded ring, with examples illustrating the theory.

Contribution

It provides new criteria for when a parameter ideal reduces a quasi-socle ideal and when the associated graded ring is Cohen-Macaulay in Gorenstein local rings.

Findings

Criteria for reduction of quasi-socle ideals by parameter ideals

Conditions for Cohen-Macaulayness of the associated graded ring

Examples illustrating the theoretical results

Abstract

This paper explores the structure of quasi-socle ideals I=Q:m^2 in a Gorenstein local ring A, where Q is a parameter ideal and m is the maximal ideal in A. The purpose is to answer the problem of when Q is a reduction of I and when the associated graded ring G(I) = \bigoplus_{n \geq 0}I^n/I^{n+1} is Cohen-Macaulay. Wild examples are explored.