Relative singularity categories and Gorenstein‐projective modules

Abstract

AbstractWe introduce the notion of relative singularity category with respect to a self‐orthogonal subcategory ω of an abelian category. We introduce the Frobenius category of ω‐Cohen‐Macaulay objects, and under certain conditions, we show that the stable category of ω‐Cohen‐Macaulay objects is triangle‐equivalent to the relative singularity category. As applications, we rediscover theorems by Buchweitz, Happel and Beligiannis, which relate the stable categories of (unnecessarily finitely‐generated) Gorenstein‐projective modules to the (big) singularity categories of rings. For the case where ω is the additive closure of a self‐orthogonal object, we relate the category of ω‐Cohen‐Macaulay objects to the category of Gorenstein‐projective modules over the opposite endomorphism ring of the self‐orthogonal object. We prove that for a Gorenstein ring, the stable category of Gorenstein‐projective modules is compactly generated and its compact objects coincide with finitely‐generated Gorenstein‐projective modules up to direct summand. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim