Cut cotorsion pairs

We present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander–Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander–Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes, and quasi-coherent sheaves, as well as to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-t-structures.

Ding w-Flat Modules and Dimensions

The introduction of w-operation in the class of flat modules has been successful. Let R be a ring. An R-module M is called a w-flat module if [Formula: see text] is GV-torsion for all R-modules N. In this paper, we introduce the w-operation in Gorenstein homological algebra. An R-module M is called Ding w-flat if there exists an exact sequence of projective R-modules … → P1 → P0 → P0 → P1 → … such that M ≅ Im(P0 → P0) and such that the functor HomR(−, F) leaves the sequence exact whenever F is w-flat. Several wellknown classes of rings are characterized in terms of Ding w-flat modules. Some examples are given to show that Ding w-flat modules lie strictly between projective modules and Gorenstein projective modules. The Ding w-flat dimension (of modules and rings) and the existence of Ding w-flat precovers are also studied.

G-Gorenstein Complexes

We present in the context of Gorenstein homological algebra the notion of a "G-Gorenstein complex" as the counterpart of the classical notion of a Gorenstein complex. In particular, we investigate equivalences between the category of G-Gorenstein complexes of fixed dimension and the G-class of modules.

PERIODIC SHORT EXACT SEQUENCES AND PERIODIC PURE-EXACT SEQUENCES

Benson and Goodearl [Periodic flat modules, and flat modules for finite groups, Pacific J. Math.196(1) (2000) 45–67] proved that if M is a flat module over a ring R such that there exists an exact sequence of R-modules 0 → M → P → M → 0 with P a projective module, then M is projective. The main purpose of this paper is to generalize this theorem to any exact sequence of the form 0 → M → G → M → 0, where G is an arbitrary module over R. Moreover, we seek counterpart entities in the Gorenstein homological algebra of pure projective and pure injective modules.