On the canonical ideals of one-dimensional Cohen-Macaulay local rings

TL;DR

This paper investigates explicit methods for finding canonical ideals in one-dimensional Cohen-Macaulay local rings, including Gorenstein ideals and those derived from Hilbert-Burch resolutions, linking the classification of singularities to Artin Gorenstein rings.

Contribution

It provides explicit constructions of canonical ideals in Cohen-Macaulay rings and connects singularity classification to Gorenstein Artin rings.

Findings

Gorenstein ideals in high powers of the maximal ideal are canonical.

Construction of canonical ideals for curve singularities via Hilbert-Burch resolutions.

Classification of curve singularities relates to classifying certain Artin Gorenstein rings.

Abstract

In this paper we consider the problem of finding explicitly canonical ideals of one-dimensional Cohen-Macaulay local rings. We show that Gorenstein ideals contained in a high power of the maximal ideal are canonical ideals. In the codimension two case, from a Hilbert-Burch resolution, we show how to construct canonical ideals of curve singularities. Finally, we translate the problem of the analytic classification of curve singularities to the classification of local Artin Gorenstein rings with suitable length.