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