Some cyclic virtually irreducible lattices. (Q645254)

Let \(R\) be a discrete valuation ring of characteristic \(0\) with residue field of positive characteristic \(p\) and with exponential valuation \(\nu\). Let \(G\) be a finite group. Following Knörr, an \(RG\)-lattice \(M\) is called virtually irreducible if for every endomorphism \(\alpha\) of \(M\) one has \(\nu(\text{rank}(M))\leq\nu(\text{trace}(\alpha))\), with equality precisely for automorphisms \(\alpha\). Knörr has shown that if every virtually irreducible lattice for Abelian \(p\)-groups \(D\) is necessarily of \(p'\)-rank, then one would get Brauer's height \(0\)-conjecture for these defect groups \(D\). Knörr also constructed explicitly a lattice of rank \(6\) for a dvr with \(p=2\) and \(D\) the elementary Abelian group of order \(16\). The paper under review shows that in this situation for this \(D\) all virtually irreducible lattices are cyclic. Moreover, for \(D\) elementary Abelian and \(p=2\) the author gives some character theoretic condition for a lattice being virtually irreducible.