On lattices at the ends of connected components of the Auslander-Reiten quiver (Q2782432)

The author proves a criterion for a virtually irreducible lattice \(L\) over a finite group \(G\) to be situated at the end of its stable Auslander-Reiten component. Let \(p\) be a rational prime dividing \(|G|\), and \(R\) a complete discrete valuation ring with radical \(R\pi\) such that \(Rp\subset R\pi\). The exponent \(\pi^a\) is defined as the smallest power of \(\pi\) such that \(\pi^a\colon L\to L\) factors through a projective \(RG\)-lattice. An absolutely indecomposable \(RG\)-lattice \(L\) is said to be virtually irreducible if \(\pi^{a-1}\) generates the socle of \(\underline{\text{End}}(L)\). The author proves that a virtually irreducible \(RG\)-lattice \(L\) with exponent \(\pi^a\), \(a\geq 2\), lies at the end of its component in the stable Auslander-Reiten quiver if and only if \(L/\pi^{a-1}L\) is indecomposable.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00038].