  Approximable triangulated categories, as introduced and developed by Neeman,
provide a solid framework for studying localization sequences within
triangulated categories. In our paper, we demonstrate that a recollement of
approximable triangulated categories, given certain mild conditions, induces
short exact sequences of both triangulated subcategories and Verdier quotient
categories. Specifically, in the case of a recollement of locally Hom-finite
noetherian approximable triangulated categories, we find that this induces a
short exact sequence of the bounded closures of compact objects within these
categories. Furthermore, if the recollement extends one step downwards, it
yields a short exact sequence of the singularity categories of these
triangulated categories, offering a generalization of the localization sequence
established by Jin-Yang-Zhou for the singularity categories of
finite-dimensional algebras. We also present applications of these results
within the derived categories of finite-dimensional algebras, DG algebras, and
schemes.
