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

The existence of a tilting or silting object is an important feature for an algebraic triangulated category since it gives an equivalence with the derived category of a ring. By applying tilting theory, we study Cohen-Macaulay representations of N-graded Artin-Schelter Gorenstein algebras A=₈ ₍Aᵢ, where do not assume that A₀ is a field. This is a large class of noncommutative Gorenstein rings containing Gorenstein orders. In this paper, we concentrate on the case where A has dimension one. Under the assumptions that A is ring-indecomposable and A₀ has finite global dimension, we show that the stable category {CM}₀^ ZA always admits an (explicitly constructed) silting object. We also show that {CM}₀^ ZA admits a tilting object if and only if either A is Artin-Schelter regular or the average Gorenstein parameter pAₐv Q of A is non-positive. These results are far-reaching generalizations of the results of Buchweitz, Iyama, and Yamaura. We give two different proofs of the second result; one is based on Orlov-type semiorthogonal decompositions, and the other is based on a more direct calculation. We apply our results to a Gorenstein tiled order A to prove that {CM}^ ZA is equivalent to the derived category of the incidence algebra of an (explicitly constructed) poset. We also apply our results and Koszul duality to prove that the derived category Dᵇ (qgr A) of a smooth noncommutative projective quadric hypersurface qgr A admits an (explicitly constructed) tilting object, which contains the tilting object of {CM}^ ZA due to Smith and Van den Bergh as a direct summand.