Localization theorems for approximable triangulated categories

Yongliang Sun, Yaohua Zhang

Published 2024-02-07Version 1

Approximable triangulated categories, introduced and developed by Neeman, provide a reasonable framework to study localization sequences for triangulated categories. In the paper, we show that an arbitrary recollement of approximable triangulated categories, under mild conditions, induces short exact sequences of triangulated subcategories and Verdier quotient categories. In particular, a recollement of locally finite, noetherian and approximable triangulated categories induces a short exact sequence of bounded closures of compact objects in these categories. If the given recollement extends one step downwards, then we obtain a short exact sequence of the singularity categories of these triangulated categories, which generalizes the localization sequence for the singularity categories of finite-dimensional algebras recently established by Jin-Yang-Zhou.