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 $\mathrm{mo}{\mathrm{d}}_{\mathsf{adm}}(\mathcal{E})$ 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 $\mathrm{mo}{\mathrm{d}}_{\mathsf{adm}}(\mathcal{E})$, for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe $\mathrm{mo}{\mathrm{d}}_{\mathsf{adm}}(\mathcal{E})$ as a subcategory of $\mathrm{mod}(\mathcal{E})$ 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 $\mathrm{mo}{\mathrm{d}}_{\mathsf{adm}}(\mathcal{E})$ to give a bijection between exact structures on an idempotent complete additive category $\mathcal{C}$ and certain resolving subcategories of $\mathrm{mod}(\mathcal{C})$.

Accepted in Advances in Mathematics

References in corpus (13)

• Classifying exact categories via Wakamatsu tilting
• Classifications of exact structures and Cohen-Macaulay-finite algebras
• Auslander-Reiten theory in quasi-abelian and Krull-Schmidt categories
• Classifying substructures of extriangulated categories via Serre subcategories
• Localizations of (one-sided) exact categories
• The Yoneda isomorphism commutes with homology
• Exact structures and degeneration of Hall algebras
• On the lattices of exact and weakly exact structures
• Auslander Correspondence for Triangulated Categories
• Derived categories of (one-sided) exact categories and their localizations
• On the obscure axiom for one-sided exact categories
• Auslander's Formula: Variations and Applications
• Recollements for dualizing $k$-varieties and Auslander's formulas

Cited by in corpus (6)

• The left heart and exact hull of an additive regular category
• Groupoids and skeletal categories form a pretorsion theory in $\mathsf{Cat}$
• Localization of extriangulated categories
• A functorial approach to $n$-abelian categories
• The index in $d$-exact categories
• Higher Morita-Tachikawa correspondence