Premium
Relative singularity categories and Gorenstein‐projective modules
Author(s) -
Chen XiaoWu
Publication year - 2011
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200810017
Subject(s) - mathematics , subcategory , pure mathematics , abelian category , projective module , endomorphism ring , derived category , projective test , singularity , flat module , ring (chemistry) , object (grammar) , abelian group , mathematical analysis , functor , computer science , chemistry , organic chemistry , artificial intelligence
We 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