Exact categories, big Cohen-Macaulay modules and finite representation type
Abstract
One of the first remarkable results in the representation theory of artin algebras, due to Auslander and Ringel-Tachikawa, is the characterization of when an artin algebra is representation-finite. In this paper, we investigate aspects of representation-finiteness in the general context of exact categories in the sense of Quillen. In this framework, we introduce "big objects" and prove an Auslander-type "splitting-big-objects" theorem. Our approach generalises and unifies the known results from the literature. As a further application of our methods, we extend the theorems of Auslander and Ringel-Tachikawa to arbitrary dimension, i.e. we characterise when a Cohen-Macaulay order over a complete regular local ring is of finite representation type.
Cite
@article{arxiv.2001.04419,
title = {Exact categories, big Cohen-Macaulay modules and finite representation type},
author = {Chrysostomos Psaroudakis and Wolfgang Rump},
journal= {arXiv preprint arXiv:2001.04419},
year = {2021}
}
Comments
v2: 30 pages, some misprints corrected, references updated, comments are welcome