Different exact structures on the monomorphism categories

Abstract

Let $\mathcal{X}$ be a resolving and contravariantly finite subcategory of $\rm{mod}\mbox{-}Λ$, the category of finitely generated right $\Lambda$-modules. We associate to $\mathcal{X}$ the subcategory ${\mathcal{S}}_{\mathcal{X}}(\Lambda )$ of the morphism category $\mathrm{H}(\mathrm{\Lambda })$ consisting of all monomorphisms $(A\stackrel{f}{\to }B)$ with $A,B$ and $\mathrm{Cok}\mathrm{f}$ in $\mathcal{X}$. Since ${\mathcal{S}}_{\mathcal{X}}(\Lambda )$ is closed under extensions then it inherits naturally an exact structure from $\mathrm{H}(\mathrm{\Lambda })$. 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 $\rm{mod}\mbox{-}\underline{\mathcal{X}}$.

37 pages, any comments are welcome. arXiv admin note: text overlap with arXiv:1802.03683 by other authors