# Different exact structures on the monomorphism categories

Research paper by **Rasool Hafezi, Intan Muchtadi-Alamsyah**

Indexed on: **05 Feb '21**Published on: **06 Oct '19**Published in: **arXiv - Mathematics - Representation Theory**

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#### 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
$\rm{mod}\mbox{-}\underline{\mathcal{X}}$.