Relative Gorenstein flat modules and Foxby classes and their model structures
Abstract
A model structure on a category is a formal way of introducing a homotopy theory on that category, and if the model structure is abelian and hereditary, its homotopy category is known to be triangulated. So a good way to both build and model a triangulated category is to build a hereditary abelian model structure. Given a ring $R$ and a (non necessarily semidualizing) left $R$-module $C$, we introduce and study new concepts of relative Gorenstein cotorsion and cotorsion modules: $\rm G_C$-cotorsion and (strongly) $\mathcal{C}_C$-cotorsion. As an application, we prove that there is a unique hereditary abelian model structure on the category of left $R$-modules, in which the cofibrations are the monomorphisms with $\rm G_C$-flat cokernel and the fibrations are the epimorphisms with $\mathcal{C}_C$-cotorsion kernel belonging to the Bass class $\mathcal{B}_C(R)$. In the second part, when $C$ is a semidualizing $(R,S)$-bimodule, we investigate the existence of abelian model structures on the category of left (resp., right) $R$-modules where the cofibrations are the epimorphisms (resp., monomorphisms) with kernel (resp., cokernel) belonging to the Bass (resp., Auslander) class $\mathcal{B}_C(R)$ (resp., $\mathcal{A}_C(R)$). We also study the class of $\rm G_C$-flat modules and the Bass class from the Auslander-Buchweitz approximation theory point of view. We show that they are part of weak AB-contexts. As the concept of weak AB-context can be dualized, we also give dual results that involve the class of $\rm G_C$-cotorsion modules and the Auslander class.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2022
- DOI:
- 10.48550/arXiv.2205.02032
- arXiv:
- arXiv:2205.02032
- Bibcode:
- 2022arXiv220502032B
- Keywords:
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- Mathematics - Rings and Algebras;
- Primary: 18G25. Secondary: 18N40
- E-Print:
- To appear in Journal of Algebra and Its Applications