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Local log-regular rings vs. toric rings

Shinnosuke Ishiro

TL;DR

The paper develops a purely commutative-ring-theoretic framework for local log-regular rings, delivering an explicit description of the canonical module and proving the finite generation of the divisor class group. It establishes a structure theorem reducing completions to complete monoid algebras and shows that the divisor class group of a local log-regular ring is isomorphic to that of its monoid, hence finitely generated and computable from the associated monoid algebra. The results unify and extend toric-type phenomena to the logarithmic setting, with applications to Gorenstein criteria, pseudo-rationality, and mixed-characteristic behavior, while connecting to recent work on BCM-regularity and splinters. Overall, the work provides a robust, algebraic treatment of local log-regular rings, enabling concrete computations and deeper structural insights in both equal and mixed characteristic contexts.

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.

Local log-regular rings vs. toric rings

TL;DR

The paper develops a purely commutative-ring-theoretic framework for local log-regular rings, delivering an explicit description of the canonical module and proving the finite generation of the divisor class group. It establishes a structure theorem reducing completions to complete monoid algebras and shows that the divisor class group of a local log-regular ring is isomorphic to that of its monoid, hence finitely generated and computable from the associated monoid algebra. The results unify and extend toric-type phenomena to the logarithmic setting, with applications to Gorenstein criteria, pseudo-rationality, and mixed-characteristic behavior, while connecting to recent work on BCM-regularity and splinters. Overall, the work provides a robust, algebraic treatment of local log-regular rings, enabling concrete computations and deeper structural insights in both equal and mixed characteristic contexts.

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.
Paper Structure (7 sections, 33 theorems, 17 equations)

This paper contains 7 sections, 33 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on monoids
  3. Properties on monoids
  4. Krull monoids and their divisor class groups
  5. Local log-regular rings
  6. Canonical modules of local log-regular rings
  7. Divisor class groups of local log-regular rings

Key Result

Proposition 2.7

Let $\mathcal{Q}$ be a finitely generated and cancellative monoid. Then $\operatorname{Spec}(Q)$ is a finite set.

Theorems & Definitions (87)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Example 2.4: Associated reduced monoids
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Lemma 2.8
  • proof
  • ...and 77 more