Resolution Dimension Relative to Resolving Subcategories in Extriangulated Categories

Let (C,E,s) be an extriangulated category with a proper class ξ of E-triangles and X a resolving subcategory of C. In this paper, we introduce the notion of X-resolution dimension relative to the subcategory X in C, and then give some descriptions of objects with finite X-resolution dimension. In particular, we obtain Auslander-Buchweitz approximations for these objects. As applications, we construct adjoint pairs for two kinds of inclusion functors, and construct a new resolving subcategory from a given resolving subcategory which reformulates some known results.

Gorenstein flat modules relative to injectively resolving subcategories

Let [Formula: see text] be an injectively resolving subcategory of left [Formula: see text]-modules. We introduce and study [Formula: see text]-Gorenstein flat modules as a common generalization of some known modules such as Gorenstein flat modules (Enochs, Jenda and Torrecillas, 1993), Gorenstein AC-flat modules (Bravo, Estrada and Iacob, 2018). Then we define a resolution dimension relative to the [Formula: see text]-Gorensteinflat modules, investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, stability of the category of [Formula: see text]-Gorensteinflat modules is discussed, and some known results are obtained as applications.

Resolving resolution dimensions in triangulated categories

Let T {\mathcal{T}} be a triangulated category with a proper class ξ \xi of triangles and X {\mathcal{X}} be a subcategory of T {\mathcal{T}} . We first introduce the notion of X {\mathcal{X}} -resolution dimensions for a resolving subcategory of T {\mathcal{T}} and then give some descriptions of objects having finite X {\mathcal{X}} -resolution dimensions. In particular, we obtain Auslander-Buchweitz approximations for these objects. As applications, we construct adjoint pairs for two kinds of inclusion functors and characterize objects having finite X {\mathcal{X}} -resolution dimensions in terms of a notion of ξ \xi -cellular towers. We also construct a new resolving subcategory from a given resolving subcategory and reformulate some known results.

Homological Properties Relative to Injectively Resolving Subcategories

Let ε be an injectively resolving subcategory of left R-modules. A left R-module M (resp. right R-module N) is called ε-injective (resp. ε-flat) if Ext1R (G,M) = 0 (resp. TorR1 (N, G) = 0) for any G ∊ ε. Let ε be a covering subcategory. We prove that a left R-module M is E-injective if and only if M is a direct sum of an injective left R-module and a reduced E-injective left R-module. Suppose ℱ is a preenveloping subcategory of right R-modules such that ε+ ⊆ ℱ and ℱ+ ⊆ ε. It is shown that a finitely presented right R-module M is ε-flat if and only if M is a cokernel of an ℱ-preenvelope of a right R-module. In addition, we introduce and investigate the ε-injective and ε-flat dimensions of modules and rings. We also introduce ε-(semi)hereditary rings and ε-von Neumann regular rings and characterize them in terms of ε-injective and ε-flat modules.