Localization of Triangulated Categories with Respect to Extension-Closed Subcategories

Abstract

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