The trace of the canonical module

TL;DR

This paper investigates the properties of the canonical trace in Cohen--Macaulay rings, exploring its behavior under algebraic constructions and classifying nearly Gorenstein Hibi rings, revealing connections with almost Gorenstein rings.

Contribution

It introduces the concept of nearly Gorenstein rings based on the canonical trace and analyzes their structure in various algebraic contexts, including tensor and Segre products.

Findings

Canonical trace determines non-Gorenstein locus.

Nearly Gorenstein rings include almost Gorenstein rings in dimension one.

Classification of nearly Gorenstein Hibi rings.

Abstract

The trace of the canonical module (the canonical trace) determines the non-Gorenstein locus of a local Cohen--Macaulay ring. We call a local Cohen--Macaulay ring nearly Gorenstein, if its canonical trace contains the maximal ideal. Similar definitions can be made for positively graded Cohen--Macaulay $K$-algebras. We study the canonical trace for tensor products and Segre products of algebras, as well as of (squarefree) Veronese subalgebras. The results are used to classify the nearly Gorenstein Hibi rings. We study connections between the class of nearly Gorenstein rings and that of almost Gorenstein rings. We show that in dimension one, the former class includes the latter.