2019/10/06 by Hafezi, Rasool, Muchtadi-Alamsyah, Intan
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1910.03403
Let X be a resolving and contravariantly finite subcategory of \rmmod-Λ, the category of finitely generated right Λ-modules. We associate to X the subcategory SX(Λ) of the morphism category \rmH(Λ) consisting of all monomorphisms (A\stackrelf→ B) with A, B and \rmCok f in X. Since SX(Λ) is closed under extensions then it inherits naturally an exact structure from \rmH(Λ). We will define two other different exact structures else than the canonical one on SX(Λ), and the indecomposable projective (resp. injective) objects in the corresponding exact categories completely classified. Enhancing SX(Λ) with the new exact structure provides a framework to construct a triangle functor. Let \rmmod-\underlineX denote the category of finitely presented functors over the stable category \underlineX. We then use the triangle functor to show a triangle equivalence between the bounded derived category \mathbbD^\rmb(\rmmod-\underlineX) and a Verdier quotient of the bounded derived category of the associated exact category on SX(Λ). Similar consideration is also given for the singularity category of \rmmod-\underlineX.