Relative tilting theory in abelian categories I: Auslander-Buchweitz-Reiten approximations theory in subcategories and cotorsion pairs

TL;DR

This paper develops a generalized approximation theory in abelian categories using relative (co)resolutions, introducing new concepts like $\\\mathcal{X}$-complete pairs to facilitate $n$-$\mathcal{X}$-tilting theory.

Contribution

It introduces relative (co)resolutions and new pair concepts, extending Auslander-Buchweitz theory for tilting applications in abelian categories.

Findings

Existence theorem for relative approximations.

Closure properties of relative (co)resolution classes.

Results on relative homological dimensions.

Abstract

In this paper we introduce a special kind of relative (co)resolutions associated to a pair of classes of objects in an abelian category $\mathcal{C}.$ We will see that, by studying these relative (co)resolutions, we get a possible generalization of a part of the Auslander-Buchweitz approximation theory that is useful for developing $n$-$\mathcal{X}$-tilting theory in [4]. With this goal, new concepts as $\mathcal{X}$-complete and $\mathcal{X}$-hereditary pairs are introduced as a generalization of complete and hereditary cotorsion pairs. These pairs appear in a natural way in the study of the category of representations of a quiver in an abelian category [5]. Our main results will include an existence theorem for relative approximations, among other results related with closure properties of relative (co)resolution classes and relative homological dimensions which are essential in the development of $n$-$\mathcal{X}$-tilting theory in [4].