Cotorsion pairs and model structures on Morita rings

Abstract

We study cotorsion pairs and abelian model structures on Morita rings Λ=(ANBAMABB) which are Artin algebras. Given cotorsion pairs (U,X) and (V,Y) in A-Mod and B-Mod, respectively, one can construct four cotorsion pairs in Λ-Mod:((XY)⊥,(XY)),(Δ(U,V),Δ(U,V)⊥),((UV),(UV)⊥),(∇⊥(X,Y),∇(X,Y)). These cotorsion pairs have relations:Δ(U,V)⊥⊆(XY),∇⊥(X,Y)⊆(UV). An important feature is that they are not equal, in general. In fact, there even exists a Morita algebra Λ, such that the four cotorsion pairs are pairwise different. The problem of identifications, i.e., when these inclusions are the same, are studied; the heredity and completeness of these cotorsion pairs are investigated; and finally, various model structures on Λ-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 constructed. All these Hovey triples obtained are pairwise different and “new” in some sense. Some results are even new when M=0 or N=0.