Cohen-Macaulay representations of Artin-Schelter Gorenstein algebras of dimension one
Osamu Iyama, Yuta Kimura, Kenta Ueyama
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.