Localization of triangulated categories with respect to extension-closed subcategories
Yasuaki Ogawa
TL;DR
The paper develops a localization theory for triangulated categories by embedding them into an ambient extriangulated framework and constructing a localization $\widetilde{\mathcal{C}}_\mathcal{N}$ with a universal exact functor from $\mathcal{C}$. From an extension-closed subcategory $\mathcal{N}$, a relative extriangulated structure $(\mathcal{C}, \mathbb{E}_{\mathcal{N}}, \mathfrak{s}_{\mathcal{N}})$ is formed and a multiplicative system yields the extriangulated localization $(Q, \mu)$. The work shows that if $\mathcal{N}$ is thick, then $\widetilde{\mathcal{C}}_\mathcal{N}$ is triangulated and $Q$ coincides with the Verdier quotient; if $\operatorname{Cone}(\mathcal{N},\mathcal{N}) = \mathcal{C}$, then $\widetilde{\mathcal{C}}_\mathcal{N}$ is abelian. It relates these localizations to cohomological functors and hearts of cotorsion pairs, providing abelian quotients and heart constructions via the same localization mechanism. The framework thus unifies Verdier/Serre quotients and cohomological phenomena, with concrete implications for cluster-tilting and rigid 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 $Q:\mathcal{C}\to\widetilde{\mathcal{C}}_\mathcal{N}$ 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 $\mathsf{cone}(\mathcal{N},\mathcal{N})=\mathcal{C}$. Such an (abelian) exact localization $\widetilde{\mathcal{C}}_\mathcal{N}$ provides a good understanding of some cohomological functors $\mathcal{C}\to\mathsf{Ab}$, e.g., the heart of $t$-structures on $\mathcal{C}$ and the abelian quotient of $\mathcal{C}$ by a cluster-tilting subcategory $\mathcal{N}$.