Classifications of exact structures and Cohen–Macaulay-finite algebras

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 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.