On tensor products and almost split sequences for Scott lattices over group rings

Abstract

Let G be a finite group and Q a p-subgroup of G. Let O be a complete discrete valuation ring of characteristic zero with algebraically closed residue class field of characteristic p. We show that for an indecomposable OG-lattice L with a Q-source T and the almost split sequence A(S(Q)) terminating in the Scott lattice S(Q) with vertex Q, the tensor sequence L⊗OA(S(Q)) is a direct sum of the almost split sequence A(L) terminating in L and a split sequence if and only if L is virtually irreducible and T is a splitting trace lattice.