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 ΣM 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 Qχ
= {q ∈ Q [mid ] χ(q) [ges ] 0} of Q and puts
Note that 0 ∈ ΣM. It is often convenient to work with the complement
ΣcM of ΣM in Q*.
The geometric invariant ΣM has been investigated for two reasons. Firstly, if R
is a Dedekind domain, then ΣM 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, ΣM 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
ΣM ∪ −ΣM = 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 FPm 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.FPm for m > 2, and they all revolve around
the following:
FPm-Conjecture: G is of type FPmif
and only if 0 ∈ Q* is not in the convex hull of
m points of ΣcM.
The conjecture appeared in [BG1].
