Relative singularity categories and Gorenstein-projective modules

We introduce the notion of relative singularity category with respect to a self-orthogonal subcategory omega of an abelian category. We introduce the Frobenius category of omega-Cohen-Macaulay objects, and under certain conditions, we show that the stable category of omega-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 omega is the additive closure of a self-orthogonal object, we relate the category of omega-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.(C) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim