Frobenius pairs in abelian categories

We revisit Auslander–Buchweitz approximation theory and find some relations with cotorsion pairs and model category structures. From the notion of relative generators, we introduce the concept of left Frobenius pairs \(({\mathcal {X}},\omega )\) in an abelian category \({\mathcal {C}}\). We show how to construct from \(({\mathcal {X}},\omega )\) a projective exact model structure on \({\mathcal {X}}^\wedge \), the subcategory of objects in \({\mathcal {C}}\) with finite \({\mathcal {X}}\)-resolution dimension, via cotorsion pairs relative to a thick subcategory of \({\mathcal {C}}\). We also establish correspondences between these model structures, relative cotorsion pairs, Frobenius pairs, and Auslander–Buchweitz contexts. Some applications of this theory are given in the context of Gorenstein homological algebra, and connections with perfect cotorsion pairs, covering subcategories and cotilting modules are also presented and described.