Relations for Grothendieck groups and representation-finiteness

TL;DR

This paper investigates the relation between Grothendieck groups and representation-finiteness in exact categories, establishing conditions under which the generation of the Grothendieck group by Auslander-Reiten conflations implies finiteness of indecomposable objects.

Contribution

It proves that under certain assumptions, the relation AR=Ex characterizes finite indecomposables in exact categories, extending to various module categories and Cohen-Macaulay modules.

Findings

AR=Ex is equivalent to finite indecomposables under certain conditions

Application to functorially finite torsion classes and Cohen-Macaulay modules

AR=Ex implies finiteness of syzygies in weaker assumptions

Abstract

For an exact category $\mathcal{E}$, we study the Butler's condition "AR=Ex": the relation of the Grothendieck group of $\mathcal{E}$ is generated by Auslander-Reiten conflations. Under some assumptions, we show that AR=Ex is equivalent to that $\mathcal{E}$ has finitely many indecomposables. This can be applied to functorially finite torsion(free) classes and contravariantly finite resolving subcategories of the module category of an artin algebra, and the category of Cohen-Macaulay modules over an order which is Gorenstein or has finite global dimension. Also we showed that under some weaker assumption, AR=Ex implies that the category of syzygies in $\mathcal{E}$ has finitely many indecomposables.