The canonical trace of Cohen-Macaulay algebras of codimension 2

TL;DR

This paper proves Herzog's conjecture on the canonical trace of Cohen-Macaulay codimension 2 algebras in certain cases, providing a criterion for specialization and classifying nearly Gorenstein monomial ideals of height two.

Contribution

It confirms Herzog's conjecture in specific cases and introduces a criterion for the specialization of the canonical trace, advancing understanding of Cohen-Macaulay algebras.

Findings

Herzog's conjecture holds in several cases.

A criterion for the specialization of the canonical trace is established.

Nearly Gorenstein monomial ideals of height two are classified.

Abstract

In the present paper, we investigate a conjecture of J\"urgen Herzog. Let $S$ be a local regular ring with residue field $K$ or a positively graded $K$-algebra, $I\subset S$ be a perfect ideal of grade two, and let $R=S/I$ with canonical module $\omega_R$. Herzog conjectured that the canonical trace $\text{tr}(\omega_R)$ is obtained by specialization from the generic case of maximal minors. We prove this conjecture in several cases, and present a criterion that guarantees that the canonical trace specializes under some additional assumptions. As the final conclusion of all of our results, we classify the nearly Gorenstein monomial ideals of height two.