On exponents and Auslander-Reiten components of irreducible lattices

Abstract

Let G be a finite group, and let R be a complete discrete rank one valuation ring of characteristic zero with maximal ideal \(\max (R) = \pi R\), and residue class field \(R/\pi R\) of characteristic p > 0. The notion of the exponent of an RG-lattice L is due to J. F. Carlson and the first author [1]. In this note we use it to show that any non-projective absolutely irreducible RG-lattice L with indecomposable factor module \(\bar {L} = L/\pi L\) lies at the end of its connected component \(\Theta \) of the stable Auslander-Reiten quiver \(\Gamma _s(RG)\) of the group ring RG. Since such lattices L belong to p-blocks B with non-trivial defect groups \(\delta (B)\) we also study some relations between the order of \(\delta (B)\) and the exponent exp(L).