A remark on the polyhedrality theorem for the Σ-invariants of modules over Abelian groups

Abstract

1·1. Background. Throughout this note Q stands for a finitely generated multiplicative Abelian group of torsion-free rank n, R for a commutative ring with 1, and M for a finitely generated RQ-module. The geometric invariant Σ_M of M was introduced in [BS1, BS2]. It can be viewed as a subset of the ℝ-vector space of all (additive) characters of Q, Q* = Hom(Q, ℝ) ≅ ℝⁿ, as follows: for every character χ: Q → ℝ one considers the submonoid Q_χ = {q ∈ Q [mid ] χ(q) [ges ] 0} of Q and puts

Note that 0 ∈ Σ_M. It is often convenient to work with the complement Σ^c_M of Σ_M in Q*.

The geometric invariant Σ_M has been investigated for two reasons. Firstly, if R is a Dedekind domain, then Σ_M turns out to be a polyhedral (i.e. a finite union of finite intersection of (open) vector half spaces) subset of Q*. This rather subtle fact was conjectured, for R a field, by Bergman [B] and established by Bieri and Groves in [BG2]; it opens the possibility for computations and imposes arithmetic restrictions on automorphisms of M. Secondly, for R = ℤ, Σ_M contains interesting information on the (metabelian) groups G which are extensions of M by Q (i.e. G fits into a short exact sequence M [rarrtl ] G [Rarr ] Q). In [BS1] it is proved that G has a finite presentation if and only if Σ_M ∪ −Σ_M = Q*. A number of attempts have been undertaken to extend this result to a characterization of the higher dimensional finiteness property that G is of typeA group of G is of type FP_m if the trivial G-module ℤ admits a free resolution F [Rarr ] ℤ with finitely generated m-skeleton. For metabelian groups G it is known, by [BS1], that FP₂ is equivalent to finite presentability.FP_m for m > 2, and they all revolve around the following:

FP_m-Conjecture: G is of type FP_mif and only if 0 ∈ Q* is not in the convex hull of m points of Σ^c_M.

The conjecture appeared in [BG1].