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.

Serre subcategories and the cofiniteness of generalized local cohomology modules with respect to a pair of ideals

We study some properties of the generalized local cohomology modules [Formula: see text] of [Formula: see text] with respect to a pair of ideals [Formula: see text] in Serre subcategories. Some results concerning to [Formula: see text]-cofinite modules are also given in this paper.

Local Cohomology Modules, Serre Subcategories and Derived Functors of Torsion Functors

In this research, by using filter regular sequences, we obtain some exact sequences of right or left derived functors of local cohomology modules. Then we use them to gain some conditions under which a right or left derived functor of some special functors over local cohomology modules belongs to a Serre subcategory. These results can conclude some generalizations of previous results in this context or regain some of them.