Frobenius pairs in abelian categories

Correspondences with cotorsion pairs, exact model categories, and Auslander–Buchweitz contexts

Abstract

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.

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Acknowledgements

The authors want to thank the anonymous referees whose corrections, suggestions and remarks have improved the presentation and quality of the contents of this paper.

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Correspondence to Octavio Mendoza.

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The authors thank PAPIIT-Universidad Nacional Autónoma de México projects # IN102914 and # IN103317. The third author was supported by a Dirección General de Asuntos del Personal Académico DGAPA-UNAM Postdoctoral Fellowship.

Communicated by Claude Cibils.

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Becerril, V., Mendoza, O., Pérez, M.A. et al. Frobenius pairs in abelian categories. J. Homotopy Relat. Struct. 14, 1–50 (2019). https://doi.org/10.1007/s40062-018-0208-4

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Keywords

  • Frobenius pairs
  • Relative cotorsion pairs
  • Auslander–Buchweitz model structures
  • Auslander–Buchweitz contexts

Mathematics Subject Classification

  • 18G10
  • 18G20
  • 18G25
  • 18G55
  • 16E10