Monomorphism operator and perpendicular operator

TL;DR

This paper introduces the monomorphism category for quivers and algebras, establishing connections with perpendicular categories, cotilting modules, and characterizations of Gorenstein-projective modules, advancing the understanding of module categories.

Contribution

It provides new characterizations of monomorphism categories related to perpendicular categories and cotilting modules, and explores their properties and applications in Gorenstein homological algebra.

Findings

Characterization of monomorphism categories as perpendicular categories.

Identification of a unique cotilting module related to monomorphism categories.

Conditions for monomorphism categories to be of finite type.

Abstract

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