Tilting theory for Gorenstein rings in dimension one

Abstract

For a Z-graded Gorenstein ring R = i≥0 R i such that R 0 is a field, we study the stable category CM Z R of Z-graded maximal Cohen-Macaulay R-modules, which is canonically triangle equivalent to the singularity category of Buchweitz and Orlov. Its thick subcategory CM Z 0 R is central in representation theory since it enjoys Auslander-Reiten-Serre duality and has almost split triangles. It is important to understand when the triangulated category CM Z 0 R admits a tilting (respectively, silting) object. In this paper, we give a complete answer to this question in the case dim R = 1: We prove that CM Z 0 R always admits a silting object, and that CM Z 0 R admits a tilting object if and only if either R is regular or the a-invariant of R is nonnegative. We also show that, if R is reduced and non-regular, then its a-invariant is non-negative and the above tilting object gives a full strong exceptional collection in CM Z 0 R = CM Z R.

Auslander Correspondence for Triangulated Categories

“…Let k be an algebraically closed field and Λ = n≥0 Λ n be a positively graded CM-finite Iwanaga-Gorenstein algebra such that each Λ n is finite dimensional over k and Λ 0 has finite global dimension. Then, the stable category CM Z Λ is [1]-finite and therefore, it is triangle equivalent to D b (mod kQ) for a disjoint union Q of some Dynkin quivers of type A, D, and E. This partially recovers [KST], [BIY,2.1] in a quite different way. Note that our result is more general, but less explicit in the sense that Corollary 1.6 does not give the type of Q from given Λ.…”

Yoneda algebras and their singularity categories

“…Moreover, there exists a canonical equivalence CM Z R ≃ − → D Z sg (R) of triangulated categories [Bu, KV, Ric], by which we will identify these categories. The studies of Cohen-Macaulay modules over Gorenstein rings have attracted enormous attention [CuR,Yo,Si,LW,I3] and recently, various Gorenstein algebras have been discovered and their Cohen-Macaulay representation theory is investigated, for example, in [AIR,BIRS,BIY,DL,GLS,IO,IT,JKS,KST,KMV,KR,Ki1,LZ,MU,MYa,SV,U,Ya].…”