Local log-regular rings vs. toric rings

TL;DR

This paper investigates local log-regular rings within commutative algebra, providing new ring-theoretic proofs for properties like canonical modules and divisor class groups, bridging logarithmic geometry and algebra.

Contribution

It offers a purely commutative ring theoretic approach to properties of local log-regular rings, previously studied via logarithmic geometry.

Findings

Explicit description of a canonical module

Finite generation of the divisor class group

Bridging logarithmic geometry with commutative algebra

Abstract

Local log-regular rings are a certain class of Cohen-Macaulay local rings that are treated in logarithmic geometry. Our paper aims to provide purely commutative ring theoretic proof of some ring-theoretic properties of local log-regular rings such as an explicit description of a canonical module, and the finite generation of the divisor class group.