Computing Gorenstein Colength

TL;DR

This paper introduces the Gorenstein colength as a measure of how closely an Artinian local ring can be approximated by a Gorenstein ring, providing bounds and constructions in specific algebraic settings.

Contribution

It establishes bounds for Gorenstein colength in certain cases and constructs explicit Gorenstein rings to analyze linkage of parameter ideals.

Findings

G(R) is bounded above by λ(R/ soc(R)) in specified cases.

Constructs explicit Gorenstein rings mapping onto R in the power series case.

Shows ideals generated by powers of parameters are linked via Gorenstein ideals.

Abstract

Given an Artinian local ring $R$, we define its Gorenstein colength $g(R)$ to measure how closely we can approximate $R$ by a Gorenstein Artin local ring. In this paper, we show that $R = T/I$ satisfies the inequality $g(R) \leq \lambda(R/\soc(R))$ in the following two cases: (a) $T$ is a power series ring over a field of characteristic zero and $I$ an ideal that is the power of a system of parameters or (b) $T$ is a 2-dimensional regular local ring with infinite residue field and $I$ is primary to the maximal ideal of $T$. In the first case, we compute $g(R)$ by constructing a Gorenstein Artin local ring mapping onto $R$. We further use this construction to show that an ideal that is the $n$th power of a system of parameters is directly linked to the $(n-1)$st power via Gorenstein ideals. A similar method shows that such ideals are also directly linked to themselves via Gorenstein ideals. Keywords: Gorenstein colength; Gorenstein linkage.