Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Triangulated categories with a single compact generator and two Brown representability theorems

Amnon Neeman

Abstract

We develop the general formalism of approximable triangulated categories, and prove two representability theorems.

Triangulated categories with a single compact generator and two Brown representability theorems

Abstract

We develop the general formalism of approximable triangulated categories, and prove two representability theorems.

Paper Structure

This paper contains 14 sections, 66 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Basics
  3. The fundamental properties of approximability
  4. Products and homotopy inverse limits in weakly approximable triangulated categories
  5. Examples
  6. An easy representability theorem
  7. Approximating systems
  8. A couple of technical lemmas
  9. The proof of Theorem \ref{['T1.-1']}($\mathrm{i}$)
  10. Implications of the hypothesis ${\mathscr T}=\overline{\langle G'\rangle}_{N}^{}$, with $G'\in{\mathscr T}^b_c\subset{\mathscr T}^b$
  11. Combining the hypothesis ${\mathscr T}=\overline{\langle G\rangle}_{N}^{}$ with approximability
  12. The proof of Theorem \ref{['T1.-1']}($\mathrm{ii}$)
  13. Applications: the construction of adjoints
  14. A criterion for checking that a triangulated functor is an equivalence

Key Result

Theorem 4

Let $R$ be a commutative, noetherian ring, and ${\mathscr T}$ an $R$--linear triangulated category with coproducts. Assume ${\mathscr T}$ is approximable, and suppose further that there exists in ${\mathscr T}$ a compact generator $G$ such that ${\mathop{\rm Hom}}(G,G[n])$ is a finite $R$--module fo defined by the formulas ${\mathscr Y}(B)={\mathop{\rm Hom}}(-,B)$ and $\widetilde{{\mathscr Y}}(A)=

Theorems & Definitions (171)

  • Definition 1
  • Remark 2
  • Remark 3
  • Theorem 4
  • Remark 5
  • Corollary 6
  • Example 7
  • Remark 9
  • Remark 10
  • Remark 11
  • ...and 161 more