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The resolving completion of an exact category

Marianne Lawson, Janina C. Letz, Julia Sauter

Abstract

For an exact category we provide two constructions of an ambient category in which the initial category is resolving: In the derived category and in the Gabriel--Quillen embedding. For the first construction we describe a pre-aisle and its right orthogonal using different acyclicty conditions. We provide necessary and sufficient conditions when this pair is a t-structure.

The resolving completion of an exact category

Abstract

For an exact category we provide two constructions of an ambient category in which the initial category is resolving: In the derived category and in the Gabriel--Quillen embedding. For the first construction we describe a pre-aisle and its right orthogonal using different acyclicty conditions. We provide necessary and sufficient conditions when this pair is a t-structure.
Paper Structure (20 sections, 39 theorems, 78 equations)

This paper contains 20 sections, 39 theorems, 78 equations.

Table of Contents

  1. Acyclicity and resolving subcategories
  2. Exact categories and exact functors
  3. Acyclicity
  4. Exact subcategories
  5. Resolving subcategories
  6. Ext resolutions
  7. Gluing Ext-acyclic complexes
  8. Morphisms between Ext-resolutions
  9. In the derived category
  10. Candidate for a t-structure
  11. The heart
  12. Maximally non-negative subcategories of triangulated categories
  13. Functorial view on Ext-resolutions
  14. Functors with an Ext-resolution
  15. Localization by deflation-percolating subcategories
  16. ...and 5 more sections

Key Result

Theorem A

Let ${\mathcal{E}}$ be a weakly idempotent complete small exact category. Then both ${\mathcal{LH}}^{}({\mathcal{E}})$ and ${\mathcal{R}}^{}({\mathcal{E}})$ are resolving completions of ${\mathcal{E}}$ and there is an equivalence ${\mathcal{LH}}^{}({\mathcal{E}}) \to {\mathcal{R}}^{}({\mathcal{E}})$

Theorems & Definitions (98)

  • Theorem A
  • Theorem B
  • Definition 1.3
  • Remark 1.4
  • Lemma 1.5
  • proof
  • Remark 1.8
  • Lemma 1.9
  • proof
  • Definition 2.1
  • ...and 88 more