Localization of Triangulated Categories with Respect to Extension-Closed Subcategories

The aim of this paper is to develop a framework for localization theory of triangulated categories C , that is, from a given extension-closed subcategory N of C , we construct a natural extriangulated structure on C together with an exact functor Q : C → C ~ N satisfying a suitable universality, which unifies several phenomena. Precisely, a given subcategory N is thick if and only if the localization C ~ N corresponds to a triangulated category. In this case, Q is nothing other than the usual Verdier quotient. Furthermore, it is revealed that C ~ N is an exact category if and only if N satisfies a generating condition Cone ( N , N ) = C . Such an (abelian) exact localization C ~ N provides a good understanding of some cohomological functors C → Ab , e.g., the heart of t -structures on C and the abelian quotient of C by a cluster-tilting subcategory N .