The Gorenstein Colength of an Artinian Local Ring

TL;DR

This paper introduces the concept of Gorenstein colength to measure how closely an Artinian local ring can be approximated by a Gorenstein Artin local ring, and characterizes cases where this colength is at most two.

Contribution

It formalizes the notion of Gorenstein colength and provides criteria for when this measure is at most two, advancing understanding of Gorenstein approximations.

Findings

Defined Gorenstein colength for Artinian local rings.

Characterized when Gorenstein colength is at most two.

Provided new insights into Gorenstein approximations.

Abstract

In this paper, we make the notion of approximating an Artinian local ring by a Gorenstein Artin local ring precise using the concept of Gorenstein colength. We also answer the question as to when the Gorenstein colength is at most two.