A Canonical module characterization of Serre's $(\mathrm{R}_1)$

TL;DR

This paper characterizes domains satisfying Serre's condition (R_1) using their canonical modules, extending known results for toric rings to a broader class of domains.

Contribution

It provides a new characterization of Serre's (R_1) condition via canonical modules, generalizing previous results specific to toric rings.

Findings

Characterization of (R_1) domains via canonical modules

Extension of toric ring results to general domains

Connection between normality and the shape of canonical modules

Abstract

In this short note, we give a characterization of domains satisfying Serre's condition $(\mathrm{R}_1)$ in terms of their canonical modules. In the special case of toric rings, this generalizes a result of the second author (K. Yanagawa, Dualizing complexes of seminormal affine semigroup rings and toric face rings, J. Algebra 425 (2015).) where the normality is described in terms of the "shape" of the canonical module.