Injective Objects of Monomorphism Categories

TL;DR

This paper characterizes injective objects in monomorphism categories over acyclic quivers and finite-dimensional algebras, showing their structure, existence of enough injectives, and applications to tilting theory and singularity categories.

Contribution

It provides a unified description of indecomposable injectives in monomorphism categories and establishes their enough-injective property, with applications to tilting and singularity categories.

Findings

Explicit description of indecomposable injectives in ${ m Mon}(Q,A)$

Proof that ${ m Mon}(Q, A)$ has enough injectives

Realization of the singularity category as a stable monomorphism category

Abstract

For an acyclic quiver $Q$ and a finite-dimensional algebra $A$, we give a unified form of the indecomposable injective objects in the monomorphism category ${\rm Mon}(Q,A)$ and prove that ${\rm Mon}(Q, A)$ has enough injective objects. As applications, we show that for a given self-injective algebra $A$, a tilting object in the stable category $\underline{A}$-mod induces a natural tilting object in the stable monomorphism category $\underline{\rm Mon}(Q,A)$. We also realize the singularity category of the algebra $kQ\otimes_k A$ as the stable monomorphism category of the module category of $A$.