Trimming a Gorenstein ideal

TL;DR

This paper introduces a method to modify Gorenstein ideals in a 3-dimensional regular local ring, producing quotient rings with near-Gorenstein properties characterized by specific algebraic structures.

Contribution

It provides a novel trimming technique for Gorenstein ideals and explicitly constructs an infinite family of near-Gorenstein rings with detailed algebraic properties.

Findings

Constructed an infinite family of near-Gorenstein rings

Demonstrated the algebraic structure of their Koszul homology

Showed the trimming process preserves certain duality properties

Abstract

Let Q be a regular local ring of dimension 3. We show how to trim a Gorenstein ideal in Q to obtain an ideal that defines a quotient ring that is close to Gorenstein in the sense that its Koszul homology algebra is a Poincare duality algebra P padded with a non-zero graded vector space on which P_{\ge 1} acts trivially. We explicitly construct an infinite family of such rings.