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$\mathbb{Z}^r$-graded rings and their canonical modules (2505.11402v1)
Published 16 May 2025 in math.AC
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 modulelocalizes''. 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.
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