$\mathbb{Z}^r$-graded rings and their canonical modules

TL;DR

The paper generalizes the canonical module from $\mathbb{Z}$-grading to $\mathbb{Z}^r$-gradings and proves that the multigraded canonical module localizes, enabling a divisorial proof of the Danilov--Stanley description for normal affine monoid rings, with the canonical module denoted by $\omega$. It develops the multigraded theory, including dimension theory, MR-type results, and graded homological invariants such as $^*\operatorname{Ext}$ and $^*\operatorname{Hom}$, to support localization arguments. As an application, it provides a divisorial proof of Danilov and Stanley's theorem for $K[M]$, describing $\omega$ as a divisorial intersection corresponding to the interior of the defining cone. Together, the work lays a foundational framework for multigraded duality and canonical-module theory with broad relevance to affine monoid rings and combinatorial commutative algebra.

Abstract

In ``Cohen--Macaulay rings'' Bruns and Herzog define the graded canonical module for $\mathbb{Z}^r$-graded rings. We generalize the definition to multigradings and prove that the canonical module ``localizes''. As an application, we give a divisorial proof of the theorem of Danilov and Stanley on the canonical module of affine normal monoid rings. Along the way, we develop the basic theory of multigraded rings and modules.

Theorems & Definitions (10)

• Lemma 1.1
• proof
• Lemma 1.2
• Lemma 1.3
• proof
• Lemma 1.4
• Lemma 2.1
• proof
• Theorem 2.2
• proof