Polyhedrality of the Brookes–Groves invariant for the non-commutative torus

Abstract

In [3] Bieri and Strebel defined a geometric invariant Σ for finitely generated modules over the group algebras of finitely generated abelian groups. They used this to define a criterion for when metabelian groups are finitely presented. This invariant was further developed by Bieri, Strebel and Groves and has many interesting applications. In [2] Bieri and Groves showed that when the group algebra is defined over a Dedekind domain the complement of Σ must be a closed rational spherical polyhedral cone. In [4] and [5] Brookes and Groves defined a similar invariant ∆ for modules over the crossed product of a division ring by a free finitely generated abelian group. Such a crossed product is often known as the (coordinate ring of) the non-commutative torus since in the special case where it is commutative it is the coordinate ring of an algebraic torus. If in the commutative case we take the complement of ∆ and identify points that differ by a positive scalar multiple we obtain Σ . Brookes and Groves were unable to prove that their invariant must be a rational polyhedral cone, although using the methods of [2] they do prove a weaker version of the result; they show that for any finitely generated module M , ∆(M) must contain a rational polyhedral cone ∆∗(M) of dimension equal to the Gelfand–Kirillov dimension of M and moreover that the complement ∆(M) ∆∗(M) must be contained inside a rational polyhedral cone of strictly smaller dimension. In this paper we use Grobner basis methods to prove the following theorem: