Commonly Graded Algebras and Their Homological Properties

Abstract

In this article, we study bounded-below locally finite $\mathbb{Z}$-graded algebras, which are referred to as commonly graded algebras in literature. Commonly graded algebras have almost similar theory as that of connected graded algebras, but sometimes the results need different methods of proof. We give several characterizations of commonly graded AS-Gorenstein algebras, and show that any noetherian commonly graded AS-Gorenstein algebra admits a balanced dualizing complex. We then study (skew) Calabi-Yau properties of commonly graded algebras, and give an example of graded algebra which is skew Calabi-Yau in ungraded sense but not in graded sense. We demonstrate that a noetherian commonly graded algebra is AS-regular if and only if the bounded derived category of its finite-dimensional graded modules constitutes a ``twisted" Calabi-Yau category. At the end of the article, we prove that the Auslander-Buchsbaum formula, along with the Bass theorem and the No-Hole theorem hold for commonly graded algebras under appropriate conditions.