Relations for Grothendieck groups and representation-finiteness

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... Show more