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Homological dimensions of complexes related to cotorsion pairs

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Let (A, B) be a cotorsion pair in R-Mod. We define and study notions of A dimension and B dimension of unbounded complexes, which is given by means of dg - projective resolution and dg - injective resolution, respectively. As an application, we extend the Gorenstein flat dimension of complexes, which was defined by Iacob. Gorenstein cotorsion, FP-projective, FP-injective, Ding projective, and Ding injective dimension are also extended from modules to complexes. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2014) 38: 401 – 418 ¨ ITAK ˙ c TUB ⃝ doi:10.3906/mat-1211-29 Homological dimensions of complexes related to cotorsion pairs Chongqing WEI∗, Limin WANG, Husheng QIAO Department of Mathematics, Northwest Normal University, Lanzhou, P.R. China Received: 15.11.2012 • Accepted: 14.06.2013 • Published Online: 14.03.2014 • Printed: 11.04.2014 Abstract: Let (A, B) be a cotorsion pair in R -Mod. We define and study notions of A dimension and B dimension of unbounded complexes, which is given by means of dg -projective resolution and dg -injective resolution, respectively. As an application, we extend the Gorenstein flat dimension of complexes, which was defined by Iacob. Gorenstein cotorsion, FP-projective, FP-injective, Ding projective, and Ding injective dimension are also extended from modules to complexes. Moreover, we characterize Noetherian rings, von Neumann regular rings, and QF rings by the FP-projective, FP-injective, and Ding projective (injective) dimension of complexes, respectively. Key words: Cotorsion pairs, A dimension of complexes, B dimension of complexes 1. Introduction and preliminaries In the classical book by Cartan and Eilenberg [4], concepts of projective, injective, and weak (flat) dimensions were defined for left R -modules over arbitrary rings. The extension of homological algebra from modules to complexes of modules, which had started already in the last chapter of [4], has produced a theory of homological dimensions. In that chapter, the projective (or flat) dimension was only defined for complexes that are homologically bounded below, while the injective dimension was introduced only for those that are homologically bounded above. However, in [2], Avramov and Foxby defined injective, projective, and flat dimensions for arbitrary complexes of left R -modules over associative rings in terms of dg -injective, dg -projective, and dg .