Different exact structures on the monomorphism categories

TL;DR

This paper explores different exact structures on monomorphism categories derived from a subcategory of modules, classifies their indecomposable objects, and establishes triangle equivalences with derived categories.

Contribution

It introduces two new exact structures on the subcategory of monomorphisms and classifies their indecomposable projective and injective objects, enriching the framework for derived category analysis.

Findings

Classification of indecomposable projective and injective objects

Construction of a triangle functor linking derived categories

Establishment of triangle equivalences involving Verdier quotients

Abstract

Let $\mathcal{X}$ be a resolving and contravariantly finite subcategory of $\rm{mod}\mbox{-}\Lambda$, the category of finitely generated right $\Lambda$-modules. We associate to $\mathcal{X}$ the subcategory $\mathcal{S}_{\mathcal{X}}(\Lambda)$ of the morphism category $\rm{H}(\Lambda)$ consisting of all monomorphisms $(A\stackrel{f}\rightarrow B)$ with $A, B$ and $\rm{Cok} \ f$ in $\mathcal{\mathcal{X}}$. Since $\mathcal{S}_{\mathcal{X}}(\Lambda)$ is closed under extensions then it inherits naturally an exact structure from $\rm{H}(\Lambda)$. We will define two other different exact structures else than the canonical one on $\mathcal{S}_{\mathcal{X}}(\Lambda)$, and the indecomposable projective (resp. injective) objects in the corresponding exact categories completely classified. Enhancing $\mathcal{S}_{\mathcal{X}}(\Lambda)$ with the new exact structure provides a framework to construct a triangle functor. Let $\rm{mod}\mbox{-}\underline{\mathcal{X}}$ denote the category of finitely presented functors over the stable category $\underline{\mathcal{X}}$. We then use the triangle functor to show a triangle equivalence between the bounded derived category $\mathbb{D}^{\rm{b}}(\rm{mod}\mbox{-}\underline{\mathcal{X}})$ and a Verdier quotient of the bounded derived category of the associated exact category on $\mathcal{S}_{\mathcal{X}}(\Lambda)$. Similar consideration is also given for the singularity category of $\rm{mod}\mbox{-}\underline{\mathcal{X}}$.