Different Exact Structures on the Monomorphism Categories

Abstract

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