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 $Q:\mathcal{C}\to {\tilde{\mathcal{C}}}_{\mathcal{N}}$ satisfying a suitable universality, which unifies several phenomena. Precisely, a given subcategory $\mathcal{N}$ is thick if and only if the localization ${\tilde{\mathcal{C}}}_{\mathcal{N}}$ corresponds to a triangulated category. In this case, $Q$ is nothing other than the usual Verdier quotient. Furthermore, it is revealed that ${\tilde{\mathcal{C}}}_{\mathcal{N}}$ 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 ${\tilde{\mathcal{C}}}_{\mathcal{N}}$ 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 $\mathcal{N}$.

arXiv comment37 pages. v3: Minor improvements due to referee comments. To appear in Algebr. Represent. Theory

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Cited by in corpus (1)

• A resolution theorem for extriangulated categories with applications to the index