Cycles of modules and finite representation type

Abstract

We prove that an artin algebra A is of finite representation type if and only if almost all finitely generated indecomposable A‐modules are cycle‐finite. Moreover, we prove that every cycle‐finite module category of an artin algebra A of infinite representation type contains infinitely many left stable and infinitely many right stable directing modules. In particular, the main results of the paper provide solution of the open problem concerning infinity of directing modules over cycle‐finite artin algebras of infinite representation type.