Cotorsion pairs and model structures on Morita rings

TL;DR

This paper explores cotorsion pairs and abelian model structures on Morita rings, constructing multiple cotorsion pairs and model structures, and analyzing their properties and relations.

Contribution

It introduces new cotorsion pairs and model structures on Morita rings, including explicit constructions and conditions for their equivalences.

Findings

Four distinct cotorsion pairs constructed in Morita rings.

Explicit Hovey triples and Quillen homotopy categories derived.

Conditions for cotorsion pair equivalences analyzed.

Abstract

This paper is to study cotorsion pairs and abelian model structures on some Morita rings \ $\Lambda =\left(\begin{smallmatrix} A & {}_AN_B {}_BM_A & B\end{smallmatrix}\right)$. From cotorsion pairs $(\mathcal U, \mathcal X)$ and $(\mathcal V, \mathcal Y)$, respectively in $A$-Mod and $B$-Mod, one constructs 4 kinds of cotorsion pairs in $\Lambda$-Mod. There even exists an algebra $\Lambda$ such that the four cotorsion pairs above are pairwise different. The heredity and completeness of these cotorsion pairs are studied. The problem of identifications, namely, when the first two cotorsion pairs are the same, and when the second two cotorsion pairs are the same, is investigated. Various model structures on $\Lambda$\mbox{-}{\rm Mod} are obtained, by explicitly giving the corresponding Hovey triples and Quillen's homotopy categories. In particular, cofibrantly generated Hovey triples, and the Gillespie-Hovey triples induced by compatible generalized projective (respectively, injective) cotorsion pairs, are explicitly contructed.