Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unbounded Algebraic Derivators

Leovigildo Alonso, Beatriz Álvarez, Ana Jeremías

Abstract

We show that the unbounded derived category of a Grothendieck category with enough projective objects is the base category of a derivator whose category of diagrams is the full 2-category of small categories. With this structure, we give a description of the localization functor associated to a specialization closed subset of the spectrum of a commutative noetherian ring. In addition, using the derivator of modules, we prove some basic theorems of group cohomology for complexes of representations over an arbitrary base ring.

Unbounded Algebraic Derivators

Abstract

We show that the unbounded derived category of a Grothendieck category with enough projective objects is the base category of a derivator whose category of diagrams is the full 2-category of small categories. With this structure, we give a description of the localization functor associated to a specialization closed subset of the spectrum of a commutative noetherian ring. In addition, using the derivator of modules, we prove some basic theorems of group cohomology for complexes of representations over an arbitrary base ring.

Paper Structure

This paper contains 6 sections, 23 theorems, 105 equations.

Table of Contents

  1. The representable derivator of complexes
  2. The basic axioms
  3. The strong derivator of derived categories
  4. Stability
  5. Description of smashing localizations
  6. Derivators and group cohomology

Key Result

Proposition 2.2

The 2-functor $\boldsymbol{\mathcal{D}}_{\!\!\mathsf{A} }$ takes coproducts into products. In other words, the axiom (Der1) in gr13 is satisfied.

Theorems & Definitions (51)

  • Proposition 2.2
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.6
  • proof
  • Lemma 2.9
  • proof
  • Remark
  • Proposition 2.10
  • ...and 41 more