# Measuring the non-Gorenstein locus of Hibi rings and normal affine semigroup rings

## TL;DR

This paper investigates the non-Gorenstein locus of Hibi rings and normal affine semigroup rings by analyzing the trace of the canonical module, providing criteria for Gorenstein properties on the punctured spectrum.

## Contribution

It introduces new methods to compute the non-Gorenstein locus of toric rings and establishes conditions for Hibi and semigroup rings to be Gorenstein on the punctured spectrum.

## Key findings

- Criteria for Gorenstein on the punctured spectrum of Hibi rings
- Conditions for normal semigroup rings to be Gorenstein
- Explicit computation methods for the non-Gorenstein locus

## Abstract

The trace of the canonical module of a Cohen-Macaulay ring describes its non-Gorenstein locus. We study the trace of the canonical module of a Segre product of algebras, and we apply our results to compute the non-Gorenstein locus of toric rings. We provide several sufficient and necessary conditions for Hibi rings and normal semigroup rings to be Gorenstein on the punctured spectrum.