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The passage among the subcategories of weakly approximable triangulated categories

Alberto Canonaco, Christian Haesemeyer, Amnon Neeman, Paolo Stellari

Abstract

In this article we prove that all the inclusions between the 'classical' and naturally defined full triangulated subcategories of a weakly approximable triangulated category are intrinsic (in one case under a technical condition). This extends all the existing results about subcategories of weakly approximable triangulated categories. Together with a forthcoming paper about uniqueness of enhancements, our result allows us to generalize a celebrated theorem by Rickard which asserts that if $R$ and $S$ are left coherent rings, then a derived equivalence of $R$ and $S$ is "independent of the decorations". That is, if $D^?(R\text{-}\square)$ and $D^?(S\text{-}\square)$ are equivalent as triangulated categories for some choice of decorations $?$ and $\square$, then they are equivalent for every choice of decorations. But our theorem is much more general, and applies also to quasi-compact and quasi-separated schemes -- even to the relative version, in which the derived categories consist of complexes with cohomology supported on a given closed subscheme with quasi-compact complement.

The passage among the subcategories of weakly approximable triangulated categories

Abstract

In this article we prove that all the inclusions between the 'classical' and naturally defined full triangulated subcategories of a weakly approximable triangulated category are intrinsic (in one case under a technical condition). This extends all the existing results about subcategories of weakly approximable triangulated categories. Together with a forthcoming paper about uniqueness of enhancements, our result allows us to generalize a celebrated theorem by Rickard which asserts that if and are left coherent rings, then a derived equivalence of and is "independent of the decorations". That is, if and are equivalent as triangulated categories for some choice of decorations and , then they are equivalent for every choice of decorations. But our theorem is much more general, and applies also to quasi-compact and quasi-separated schemes -- even to the relative version, in which the derived categories consist of complexes with cohomology supported on a given closed subscheme with quasi-compact complement.
Paper Structure (32 sections, 63 theorems, 64 equations)

This paper contains 32 sections, 63 theorems, 64 equations.

Table of Contents

  1. Reminder of some basic notation
  2. Notation and constructions
  3. Basic properties
  4. The basics of t--structure s
  5. Definitions and basic properties
  6. The more traditional notation for t--structure s
  7. Compactly generated t--structure s
  8. The heart of a compactly generated t--structure
  9. Some technical results
  10. The proof of \ref{['thm:maintstr']}
  11. Weakly approximable triangulated categories and their subcategories
  12. Definitions, examples and the relevant subcategories
  13. Products and coproducts in weakly approximable triangulated categories
  14. The subcategories of ${\mathscr T}^-_c$ and of ${\mathscr T}^c$
  15. Preliminaries to understanding the subcategory ${\mathscr T}^-_c\subseteq{\mathscr T}$
  16. ...and 17 more sections

Key Result

Theorem A

Let $R$ and $S$ be two rings. Then, in the standard notation for the various derived categories associated to the two rings, the following are equivalent: If we assume further that the rings $R$ and $S$ are both left coherent, then the six equivalent conditions above are also equivalent to:

Theorems & Definitions (154)

  • Theorem A: Rickard
  • Theorem B
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 1
  • Remark 1
  • Lemma 3
  • proof
  • ...and 144 more