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In this work, we revisit Auslander-Buchweitz Approximation Theory and findsome relations with cotorsion pairs and model category structures. From thenotions of relatives generators and cogenerators in Approximation Theory, weintroduce the concept of left Frobenius pairs $(\mathcal{X},\omega)$ in anabelian category $\mathcal{C}$. We show how to construct from$(\mathcal{X},\omega)$ a projective exact model structure on$\mathcal{X}^\wedge$, as a result of Hovey-Gillespie Correspondence applied totwo compatible and complete cotorsion pairs in $\mathcal{X}^\wedge$. Thesepairs can be regarded as examples of what we call cotorsion pairs relative to athick subcategory of $\mathcal{C}$. We establish some correspondences betweenFrobenius pairs, relative cotorsion pairs, exact model structures andAuslander-Buchweitz contexts. Finally, some applications of these results aregiven in the context of Gorenstein homological algebra by generalizing someexisting model structures on the categories of modules over Gorenstein andDing-Chen rings, and by encoding the stable module category of a ring as acertain homotopy category. We also present some connections with perfectcotorsion pairs, covering classes, and cotilting modules.

Frobenius pairs in abelian categories: correspondences with cotorsion pairs, exact model categories, and Auslander-Buchweitz contexts