Auslander-Reiten components containing modules with bounded Betti numbers

Let R be a connected selfinjective Artin algebra, and M an indecomposable nonprojective R-module with bounded Betti numbers lying in a regular component of the Auslander-Reiten quiver of R. We prove that the Auslander-Reiten sequence ending at M has at most two indecomposable sum-mands in the middle term. Furthermore we show that the component of the Auslander-Reiten quiver containing M is either a stable tube or of type ℤA_∞. We use these results to study modules with eventually constant Betti numbers, and modules with eventually periodic Betti numbers.