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Frobenius pairs in abelian categories. Frobenius pairs in abelian categories, correspondences with cotorsion pairs, exact model categories, and Auslander-Buchweitz contexts. (English) Zbl 1435.18011

The study of relative homological dimensions, obtained by replacing the projective or injective modules by certain subcategories of an abelian category, was initiated by S. Eilenberg and J. C. Moore [Foundations of relative homological algebra. Providence, RI: American Mathematical Society (AMS) (1965; Zbl 0129.01101)]. An important branch of relative homological algebra was developed by M. Auslander and R.-O. Buchweitz [Mém. Soc. Math. Fr., Nouv. Sér. 38, 5–37 (1989; Zbl 0697.13005)]. The concepts and results in this seminal work comprise what is usually known as Auslander-Buchweitz approximation theory.
The main purpose of this paper is to use Auslander-Buchweitz approximation theory in order to develop, in the general setting provided by an abelian category \(\mathcal{C}\), the theory of left and right Frobenius pairs. The authors introduce the concept of left Frobenius pairs \((\mathcal{X}, \omega)\) in an abelian category \(\mathcal{C}\) and show how to construct from \((\mathcal{X}, \omega)\) a projective exact model structure on \(\mathcal{X}^\vee\), the subcategory of objects in \(\mathcal{C}\) with finite \(\mathcal{X}\)-resolution dimension, via cotorsion pairs relative to a thick subcategory of \(\mathcal{C}\). Correspondences between these model structures, relative cotorsion pairs, Frobenius pairs, and Auslander-Buchweitz contexts is established. 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.

MSC:

18G10 Resolutions; derived functors (category-theoretic aspects)
18G20 Homological dimension (category-theoretic aspects)
16E10 Homological dimension in associative algebras
18N40 Homotopical algebra, Quillen model categories, derivators
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