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

TL;DR

This paper generalizes the concept of canonical modules to multigraded rings, proves their localization property, and applies these results to affine normal monoid rings, enriching the theory of multigraded algebra.

Contribution

It extends the definition of canonical modules to multigraded rings and proves their localization, providing new tools for studying multigraded algebraic structures.

Findings

Canonical modules are generalized to multigraded rings.

Localization of canonical modules is established.

Divisorial proof of Danilov and Stanley's theorem is provided.

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.