Bimodule monomorphism categories and RSS equivalences via cotilting modules

Abstract

The monomorphism category M(A,M,B) induced by a bimodule MBA is the subcategory of Λ-mod consisting of [XY]ϕ such that ϕ:M⊗BY→X is a monic A-map, where Λ=[AM0B], and A, B are Artin algebras. In general, M(A,M,B) is not the monomorphism category induced by quivers. It could describe the Gorenstein-projective Λ-modules. This monomorphism category is a resolving subcategory of Λ-mod if and only if MB is projective. In this case, it has enough injective objects and Auslander–Reiten sequences, and can be also described as the left perpendicular category of a unique basic cotilting Λ-module. If M satisfies the condition (IP) (see Subsection 1.6), then the stable category of M(A,M,B) admits a recollement of additive categories, which is in fact a recollement of singularity categories if M(A,M,B) is a Frobenius category. Ringel–Schmidmeier–Simson equivalence between M(A,M,B) and its dual is introduced. If M is an exchangeable bimodule, then an RSS equivalence is given by a Λ–Λ bimodule which is a two-sided cotilting Λ-module with a special property; and the Nakayama functor NΛ gives an RSS equivalence if and only if both A and B are Frobenius algebras.