Bimodule monomorphism categories and RSS equivalences via cotilting modules

Abstract

The monomorphism category $\mathscr{S}(A, M, B)$ induced by a bimodule $_AM_B$ is the subcategory of $\Lambda$-mod consisting of $\left[\begin{smallmatrix} X\\ Y\end{smallmatrix}\right]_{\phi}$ such that $\phi: M\otimes_B Y\rightarrow X$ is a monic $A$-map, where $\Lambda=\left[\begin{smallmatrix} A&M\\0&B \end{smallmatrix}\right]$. 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 $\modcat{\Lambda}$ if and only if $M_B$ 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 $\Lambda$-module. If $M$ satisfies the condition ${\rm (IP)}$, then the stable category of $\mathscr{S}(A, M, B)$ admits a recollement of additive categories, which is in fact a recollement of singularity categories if $\mathscr{S}(A, M, B)$ is a {\rm Frobenius} category. Ringel-Schmidmeier-Simson equivalence between $\mathscr{S}(A, M, B)$ and its dual is introduced. If $M$ is an exchangeable bimodule, then an {\rm RSS} equivalence is given by a $\Lambda$-$\Lambda$ bimodule which is a two-sided cotilting $\Lambda$-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.