Classifications of exact structures and Cohen-Macaulay-finite algebras
Published in Adv. Math., September 2018. Links: journal, arXiv:1705.02163
Citation: H. Enomoto, Classifications of exact structures and Cohen-Macaulay-finite algebras, Adv. Math. 335 (2018), 838--877.
Comment
I wanted an analogue of Auslander correspondence for exact categories, especially a kind of CM Auslander correspondence for Iwanaga-Gorenstein algebras.
Since non-Morita equivalent algebras may give equivalent CM categories (as additive categories), to classify CM-finite Iwanaga-Gorenstein algebras, one needs some extra info together with its CM category. That is (my favorite) exact structure, the structure of Quillen’s exact category. Then the original Iwanaga-Gorenstein algebra can be recovered by taking the endomorphism ring of progenerator in this exact category.
To do this, For a given category, I classify possible exact structures by using functor category. Also, Auslander-Reiten theory for exact categories are studied.