Tilting theory for Gorenstein rings in dimension one

TL;DR

This paper investigates the stable categories of graded maximal Cohen-Macaulay modules over one-dimensional Gorenstein rings, establishing conditions for the existence of tilting objects and exploring their duality properties.

Contribution

It proves that the stable category always has a silting object in dimension one and characterizes when a tilting object exists based on regularity and the $a$-invariant.

Findings

Stable category admits a silting object in dimension one.

Tilting object exists iff the ring is regular or has non-negative $a$-invariant.

Provides insights into the duality and representation theory of Gorenstein rings.

Abstract

For a $Z$-graded Gorenstein ring $R$, we study the stable category $CM^ZR$ of $Z$-graded maximal Cohen-Macaulay $R$-modules, which is canonically triangle equivalent to the singularity category of Buchweitz and Orlov. Its thick subcategory given as the stable category of $CM_0^ZR$ is central in representation theory since it enjoys Auslander-Reiten-Serre duality and has almost split triangles. In the case $dim R=1$, we prove that the stable category of $CM_0^ZR$ always admits a silting object, and that it admits a tilting object if and only if either $R$ is regular or the $a$-invariant of $R$ is non-negative.