Relations for Grothendieck groups and representation-finiteness

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.

16 pages

References in corpus (3)

• Classifying exact categories via Wakamatsu tilting
• Classifications of exact structures and Cohen-Macaulay-finite algebras
• Length categories of infinite height

Cited by in corpus (4)

• Auslander's defects over extriangulated categories: an application for the General Heart Construction
• Exact structures and degeneration of Hall algebras
• On the lattices of exact and weakly exact structures
• Exact subcategories, subfunctors of $\mathrm{Ext}$, and some applications