# Classifications of exact structures and Cohen-Macaulay-finite algebras

## TL;DR

This paper classifies exact structures on additive categories and explores their implications for Cohen-Macaulay-finite algebras, providing explicit classifications and developing Auslander-Reiten theory in this context.

## Contribution

It offers a comprehensive classification of exact structures and applies this to classify Cohen-Macaulay-finite algebras and cotilting modules, advancing the understanding of their structure.

## Key findings

- Grothendieck group relations generated by AR conflations
- Explicit classification of Cohen-Macaulay-finite algebras and cotilting modules
- Development of AR theory over noetherian complete local rings

## Abstract

We give a classification of all exact structures on a given idempotent complete additive category. Using this, we investigate the structure of an exact category with finitely many indecomposables. We show that the relation of the Grothendieck group of such a category is generated by AR conflations. Moreover, we obtain an explicit classification of (1) Gorenstein-projective-finite Iwanaga-Gorenstein algebras, (2) Cohen-Macaulay-finite orders, and more generally, (3) cotilting modules $U$ with $^\perp U$ of finite type. In the appendix, we develop the AR theory of exact categories over a noetherian complete local ring, and relate the existence of AR conflations to the AR duality and dualizing varieties.