2017/10/01 by Xiong, Bao-Lin, Zhang, Pu, Zhang, Yue-Hui
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1710.00314
The monomorphism category \mathscrS(A, M, B) induced by a bimodule AMB is the subcategory of Λ-mod consisting of [\beginsmallmatrix X Y\endsmallmatrix]ϕ such that ϕ: M⊗B Y→ X is a monic A-map, where Λ=[\beginsmallmatrix A&M 0&B \endsmallmatrix]. In general, it is not the monomorphism categories induced by quivers. It could describe the Gorenstein-projective \m-modules. This monomorphism category is a resolving subcategory of \modcatΛ 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 \rm (IP), then the stable category of \mathscrS(A, M, B) admits a recollement of additive categories, which is in fact a recollement of singularity categories if \mathscrS(A, M, B) is a \rm Frobenius category. Ringel-Schmidmeier-Simson equivalence between \mathscrS(A, M, B) and its dual is introduced. If M is an exchangeable bimodule, then an \rm 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 \mathcal N_\m gives an \rm RSS equivalence if and only if both A and B are Frobenius algebras.