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Triangulated categories with a compact silting object, Brown-Comenetz duality and Brown representability theorems

Xiaohu Chen, Yongliang Sun, Yaohua Zhang

TL;DR

The paper addresses dualizing Neeman’s recollement theory by leveraging Brown--Comenetz duality to treat the non-compact side of triangulated categories. It introduces the intrinsic subcategory $T^+_c$ as the formal dual of $T^-_c$ and proves dual representability theorems for $T^+_c$ and $T^b_c$, tying them to locally finite and finite $E$-homological functors. Localization results parallel SZ, yielding short exact sequences on Verdier quotients for the dual side and for the Brown--Comenetz dual $E$, thereby completing a duality between compact and non-compact localization phenomena in recollements. The framework is illustrated with explicit recollements of derived categories of finite-dimensional algebras and is complemented by adjoint-construction results, indicating broad applicability to derived-equivalence contexts and singularity-category settings. Overall, the work provides a coherent, dual perspective on recollements, representability, and localization that complements Neeman’s classical compact-object theory and opens avenues for applications in triangulated and derived-category contexts.

Abstract

In this paper, we establish a dual framework for Neeman's results concerning triangulated categories with compact silting objects by employing Brown--Comenetz duality. This framework introduces an intrinsic non-compact subcategory, provides its characterization, and demonstrates representability theorems for both the low-bounded and bounded subcategories. Additionally, it elucidates how recollements are restricted to short exact sequences on the dual (non-compact) side. Several localization results and specific applications are also derived.

Triangulated categories with a compact silting object, Brown-Comenetz duality and Brown representability theorems

TL;DR

The paper addresses dualizing Neeman’s recollement theory by leveraging Brown--Comenetz duality to treat the non-compact side of triangulated categories. It introduces the intrinsic subcategory as the formal dual of and proves dual representability theorems for and , tying them to locally finite and finite -homological functors. Localization results parallel SZ, yielding short exact sequences on Verdier quotients for the dual side and for the Brown--Comenetz dual , thereby completing a duality between compact and non-compact localization phenomena in recollements. The framework is illustrated with explicit recollements of derived categories of finite-dimensional algebras and is complemented by adjoint-construction results, indicating broad applicability to derived-equivalence contexts and singularity-category settings. Overall, the work provides a coherent, dual perspective on recollements, representability, and localization that complements Neeman’s classical compact-object theory and opens avenues for applications in triangulated and derived-category contexts.

Abstract

In this paper, we establish a dual framework for Neeman's results concerning triangulated categories with compact silting objects by employing Brown--Comenetz duality. This framework introduces an intrinsic non-compact subcategory, provides its characterization, and demonstrates representability theorems for both the low-bounded and bounded subcategories. Additionally, it elucidates how recollements are restricted to short exact sequences on the dual (non-compact) side. Several localization results and specific applications are also derived.
Paper Structure (19 sections, 50 theorems, 174 equations)

This paper contains 19 sections, 50 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. t-structures and homological facts for triangulated categories
  5. Short exact sequneces of triangulated categories
  6. Approximable triangulated categories
  7. Brown--Comenetz duality of compact objects
  8. The intrinsic subcategory $\mathcal{T}^+_c$
  9. Representability theorems for $\mathcal{T}^+_c$ and $\mathcal{T}^b_c$
  10. Homological finite functors
  11. Preliminary representability theorem
  12. Formal representability theorems
  13. Localization theorems for $\mathcal{T}^+_c$ and $\mathcal{E}$
  14. A localization theorem for $\mathcal{T}^+_c$
  15. A localization theorem for $\mathcal{E}$
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let the following diagram be a recollement of compactly generated locally Hom-finite $k$-linear triangulated categories \xymatrix{\mathcal{R}\ar^-{i_*=i_!}[r] &\mathcal{T}\ar^-{j^!=j^*}[r]\ar^-{i^!}@/^1.2pc/[l]\ar_-{i^*}@/_1.6pc/[l] &\mathcal{S}.\ar^-{j_*}@/^1.2pc/[l]\ar_-{j_!}@/_1.6pc/[l]}The where $\mathcal{E}_s, \mathcal{E}_t, \mathcal{E}_r$ are the Brown-Comenetz duality of $\mathcal{S}^

Theorems & Definitions (95)

  • Theorem 1.1: Proposition \ref{['prop:dual neeman']}
  • Theorem 1.2: Theorem \ref{['thm:rep for T+c']} and Theorem \ref{['thm: rep for Tbc']}
  • Theorem 1.3: Theorem \ref{['thm:loc for T+c']} and Theorem \ref{['thm:loc for E']}
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 85 more