Cohen-Macaulay representations of Artin-Schelter Gorenstein algebras of dimension one

Abstract

Tilting theory is one of the central tools in modern representation theory, in particular in the study of Cohen-Macaulay representations. We study Cohen-Macaulay representations of $\mathbb N$-graded Artin-Schelter Gorenstein algebras $A$ of dimension one, without assuming the connectedness condition. This framework covers a broad class of noncommutative Gorenstein rings, including classical $\mathbb N$-graded Gorenstein orders. We prove that the stable category $\underline{\mathsf{CM}}_0^{\mathbb Z}A$ admits a silting object if and only if $A_0$ has finite global dimension. In this case we give such a silting object explicitly. Assuming that $A$ is ring-indecomposable, we further show that $\underline{\mathsf{CM}}_0^{\mathbb Z}A$ admits a tilting object if and only if either $A$ is Artin-Schelter regular or the average Gorenstein parameter of $A$ is non-positive. These results generalize those of Buchweitz, Iyama, and Yamaura. We give two proofs of the second result: one via Orlov-type semiorthogonal decompositions, and the other via a direct calculation. As an application, we show that for a Gorenstein tiled order $A$, the category $\underline{\mathsf{CM}}^{\mathbb Z}A$ is equivalent to the derived category of the incidence algebra of an explicitly constructed poset. We also apply our results and Koszul duality to study smooth noncommutative projective quadric hypersurfaces $\mathsf{qgr}\,B$ of arbitrary dimension. We prove that $\mathsf{D}^{\mathrm b}(\mathsf{qgr}\,B)$ admits an explicitly constructed tilting object, which contains the tilting object of $\underline{\mathsf{CM}}^{\mathbb Z}B$ due to Smith and Van den Bergh as a direct summand via Orlov's semiorthogonal decomposition.