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Localization theorems for approximable triangulated categories

Yongliang Sun, Yaohua Zhang

Abstract

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.

Localization theorems for approximable triangulated categories

Abstract

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.
Paper Structure (17 sections, 22 theorems, 54 equations)

This paper contains 17 sections, 22 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. $t$-structures, short exact sequences and recollements
  5. Approximable triangulated categories
  6. Approximable triangulated categories
  7. Intrinsic subcategories in an approximable triangulated category
  8. Locally Hom-finite approximable triangulated categories
  9. Examples
  10. Restriction of functors and recollements
  11. Localization theorems
  12. A localization theorem for $\mathcal{T}^c$
  13. A localization theorem for $\mathcal{T}^b_c$
  14. Applications
  15. Localization sequences in the derived categories of noetherian algebras
  16. ...and 2 more sections

Key Result

Theorem 1.1

(Theorem thm:exact sequences of quotient)

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Definition 2.6
  • ...and 35 more