On lattices at the ends of connected components of the Auslander-Reiten quiver

Let R be a complete discrete valuation ring with radical Rπ and residue field R/πR of characteristic p dividing the order of a finite group G. We show that a virtually irreducible RG-lattice L with exponent π^a lies at the end of its Auslander Reiten component if and only if L/π^{a −1} L is indecomposable.