  We study the converse of a theorem of Butler and Auslander-Reiten. We show
that a Cohen-Macaulay local ring with an isolated singularity has only finitely
many isomorphism classes of indecomposable summands of syzygies of
Cohen-Macaulay modules if the Auslander-Reiten sequences generate the relation
of the Grothendieck group of finitely generated modules. This extends a recent
result of Hiramatsu, which gives an affirmative answer in the Gorenstein case
to a conjecture of Auslander.
