Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Auslander's formula and correspondence for exact categories

Ruben Henrard, Sondre Kvamme, Adam-Christiaan van Roosmalen

Abstract

The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ of admissibly finitely presented functors and use it to give a version of Auslander correspondence for any exact category $\mathcal{E}$. An important ingredient in the proof is the localization theory of exact categories. We also investigate how properties of $\mathcal{E}$ are reflected in $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$, for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ as a subcategory of $\operatorname{mod}(\mathcal{E})$ when $\mathcal{E}$ is a resolving subcategory of an abelian category. This includes the category of Gorenstein projective modules and the category of maximal Cohen-Macaulay modules as special cases. Finally, we use $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ to give a bijection between exact structures on an idempotent complete additive category $\mathcal{C}$ and certain resolving subcategories of $\operatorname{mod}(\mathcal{C})$.

Auslander's formula and correspondence for exact categories

Abstract

The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category of admissibly finitely presented functors and use it to give a version of Auslander correspondence for any exact category . An important ingredient in the proof is the localization theory of exact categories. We also investigate how properties of are reflected in , for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe as a subcategory of when is a resolving subcategory of an abelian category. This includes the category of Gorenstein projective modules and the category of maximal Cohen-Macaulay modules as special cases. Finally, we use to give a bijection between exact structures on an idempotent complete additive category and certain resolving subcategories of .

Paper Structure

This paper contains 23 sections, 64 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Localizations/Quotients of exact categories
  3. Exact categories
  4. Localization of categories
  5. Localizations via percolating subcategories
  6. Admissible morphisms in the quotient
  7. Torsion theory with percolating torsion class
  8. Auslander's formula for exact categories
  9. Basic definitions and results
  10. The quotient
  11. Auslander's formula for exact categories
  12. Torsion theory in admissibly presented functors
  13. Admissibly presented functors and the category of sheaves
  14. Auslander correspondence for exact categories
  15. The 2-category AEx
  16. ...and 8 more sections

Key Result

Theorem 1.2

The following holds:

Figures (2)

  • Figure 1: The Auslander-Reiten quivers of $\operatorname{mod}(\mathop{\mathrm{rep}}\nolimits_k(Q))$ (left) and $\operatorname{mod}(\mathop{\mathrm{rep}}\nolimits_k(Q)^{\operatorname{op}})$ (right) from \ref{['example:MainTheorem1']}.
  • Figure 2: The Auslander-Reiten quivers of $\operatorname{mod}({\mathcal{E}})$ and $\operatorname{mod}({\mathcal{E}}^{\operatorname{op}})$ from \ref{['example:MainTheorem2']}.

Theorems & Definitions (173)

  • Definition 1.1: \ref{['Definition:smodadAndeff']}
  • Theorem 1.2: \ref{['Proposition:EffAreTorsion']} and \ref{['Theorem:AuslandersFormulaForExactCategories']}
  • Definition 1.3: \ref{['AuslanderExactCategories']}
  • Theorem 1.4: \ref{['FirstAuslanderCorrespondence']}
  • Theorem 1.5: \ref{['Theorem:AuslanderSecond']}
  • Theorem 1.6: \ref{['Theorem:ExactStructuresResolving']}
  • Corollary 1.7: \ref{['Corollary:ExactStructuresResolving']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • ...and 163 more