Cut cotorsion pairs

TL;DR

This paper introduces the concept of cut cotorsion pairs in abelian categories, generalizing complete cotorsion pairs and providing a framework for restricted approximations with applications across homological algebra and triangulated categories.

Contribution

It develops the theory of cut cotorsion pairs, linking them to existing approximation theories and applying them to various algebraic and categorical contexts.

Findings

Generalization of complete cotorsion pairs through cut cotorsion pairs

Connections established with Auslander-Buchweitz approximation theory

Applications to Gorenstein homological algebra, chain complexes, and triangulated categories

Abstract

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, but also 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.