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

Abstract

AbstractIn this paper, we introduce a special kind of relative (co)resolutions associated with a pair of classes of objects in an abelian category $$\mathcal {C}.$$ C . We will see that, by studying this relative (co)resolutions, we get a possible generalization of a part of the Auslander–Buchweitz approximation theory that is useful for developing n-$$\mathcal {X}$$ X -tilting theory in Argudin Monroy and Mendoza Hernández (relative tilting theory in abelian categories II: n-$$\mathcal {X}$$ X tilting theory. arXiv:2112.14873, 2021). With this goal, new concepts as $$\mathcal {X}$$ X -complete and $$\mathcal {X}$$ 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 (Argudin Monroy and Mendoza Hernández in categories of quiver representations and relative cotorsion pairs. arXiv:2311.12774v1, 2023). 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-$$\mathcal {X}$$ X -tilting theory in Argudin Monroy and Mendoza Hernández (2021).