Monomorphism operator and perpendicular operator

Abstract

For a quiver $Q$, a $k$-algebra $A$, and a full subcategory $\mathcal{X}$ of $A$-mod, the monomorphism category $\mathrm{Mon}(Q,\mathcal{X})$ is introduced. The main result says that if $T$ is an $A$-module such that there is an exact sequence $0\to T_m\to ...\to T_0\to D(A_A)\to 0$ with each $T_i\in \mathrm{add}(T)$, then $\mathrm{Mon}(Q,{}^{\perp }T)={}^{\perp }(kQ{\otimes }_{k}T)$; and if $T$ is cotilting, then $kQ{\otimes }_{k}T$ is a unique cotilting $\m$-module, up to multiplicities of indecomposable direct summands, such that $\mathrm{Mon}(Q,{}^{\perp }T)={}^{\perp }(kQ{\otimes }_{k}T)$. As applications, the category of the Gorenstein-projective $(kQ{\otimes }_{k}A)$-modules is characterized as $\mathrm{Mon}(Q,\mathcal{G}\mathcal{P}(A))$ if $A$ is Gorenstein; the contravariantly finiteness of $\mathrm{Mon}(Q,\mathcal{X})$ can be described; and a sufficient and necessary condition for $\mathrm{Mon}(Q,A)$ being of finite type is given.