Auslander-type correspondences for exact categories and cotilting modules
Seminar Talk in South Osaka Algebra Seminar at Osaka Prefecture University, Japan
Based on papers:
- Classifying exact categories via Wakamatsu tilting
- Classifications of exact structures and Cohen-Macaulay-finite algebras
In this talk, I will give an Auslander-type correspondence for cotilting modules $U$ such that their Ext-perpendicular category $^\perp U$ have finitely many indecomposables. The typical examples of such categories $^\perp U$ are the category of Cohen-Macaulay modules over Iwanaga-Gorenstein algebras (and complete Cohen-Macaulay local ring), thus we can obtain a kind of classification of CM-finite algebras. Firstly, I will explain the Morita-type characterization of $^\perp U$ as an exact category. Then, I will talk about an Auslander-type correspondence for exact categories, more precisely, a classification of exact structures on a given category using its Auslander algebra. Finally, by combining these results, we will obtain the correspondence for $^\perp U$.