ON LENGTHS OF HZ-LOCALIZATION TOWERS

arXiv:1605.08198v2 [] 27 Jun 2016

SERGEI O. IVANOV AND ROMAN MIKHAILOV
Abstract. In this paper, the HZ-length of different groups is studied. By definition,
this is the length of HZ-localization tower or the length of transfinite lower central series
of HZ-localization. It is proved that, for a free noncyclic group, its HZ-length is ≥ ω + 2.
For a large class of Z[C]-modules M, where C is an infinite cyclic group, it is proved
that the HZ-length of the semi-direct product M ⋊ C is ≤ ω + 1 and its HZ-localization
can be described as a central extension of its pro-nilpotent completion. In particular,
this class covers modules M , such that M ⋊ C is finitely presented and H2 (M ⋊ C) is
finite.

MSC2010: 55P60, 19C09, 20J06

1. Introduction
Let R be either a subgring of rationals or a cyclic ring. In his fundamental work [5], A.K.
Bousfield introduced the concept of HR-localization. This is a functor in the category of
groups, closely related to the functor of homological localization of spaces. In this paper
we will study the case R = Z, that is, HZ-localization for different groups.
An HZ-map between two groups is a homomorphism which induces an isomorphism
on H1 and an epimorphism on H2 . A group Γ is HZ-local if any HZ-map G → H induces
a bijection Mor(H, Γ) ≃ Mor(G, Γ). Recall that ([5], Theorem 3.10) the class of HZ-local
groups is the smallest class which contains the trivial group and closed under inverse
limits and central extensions. Given a group G, the HZ-localization
η ∶ G → EG
can be uniquely characterized by the following two properties: η is an HZ-map and the
group EG is HZ-local. These two properties are given as a definition of HZ-localization
in [5]. It is shown in [5] that, for any G, the HZ-localization EG exists, unique and
transfiniltely nilpotent.
For a group G, denote by {γτ (G)} the transfinite lower central series of G, defined
inductively as γτ +1 (G) ∶= [γτ (G), G] and γα = ⋂τ <α γτ (G) for a limit ordinal α. For a G,
we will call the length of transfinite lower series of EG, i.e. the least ordinal τ, such that
γτ (EG) = 1 by HZ-length of G and denote it as HZ-length(G).
Let C be an infinite cyclic group. A Z[C]-module M is tame if and only if M ⋊ C is
a finitely presented group [3]. If M is a tame C-module, then dimQ (M ⊗ Q) < ∞ and
there exist a generator t ∈ C such that the minimal polynomial of the linear map t ⊗ Q ∶
M ⊗ Q → M ⊗ Q is an integral monic polynomial, which is denoted by µM ∈ Z[x] (see [2,
Theorem C] and Lemma 4.7). We prove the following
1

ON LENGTHS OF HZ-LOCALIZATION TOWERS

2

Theorem. Let G be a metabelian group of the form G = M ⋊C, where M is a tame Z[C]module and µM = (x − λ1 )m1 . . . (x − λl )ml for some distinct complex numbers λ1 , . . . , λl
and mi ≥ 1.
(1) Assume that the equality λi λj = 1 holds only if λi = λj = 1. Then
HZ-length(G) ≤ ω.
(2) Assume that the equality λi λj = 1 holds only if either mi = mj = 1 or λi = λj = 1.
Then
HZ-length(G) ≤ ω + 1.
As a contrast, we give an example of a finitely presented metabelian group of the form
M ⋊C, where M is tame, whose HZ-length is greater than ω +1. In the following example,
the Z[C]-module M is tame but it does not satisfy the condition of Theorem 5.6. Let
G = ⟨a, b, t ∣ at = a−1 , bt = ab−1 , [a, b] = 1⟩ = Z2 ⋊ C,

(1.1)

( −10 −11 ).

It is shown in Theorem 5.5 that the HZ-length
where C acts on Z2 by the matrix
of G is ≥ ω + 2.
Let M be a tame Z[C]-module and µM = (x − 1)m f, for some m ≥ 0 and f ∈ Z[x] such
that f (1) ≠ 0. Assume that f = f1m1 . . . flml where f1 , . . . , fl ∈ Z[x] are distinct irreducible
monic polynomials. If f (1) ∈ {−1, 1}, then HZ-length(M ⋊ C) < ω. (Corollary 4.17).
Conjecture. If f (1) ∉ {−1, 1}, then HZ-length(M ⋊ C) ≤ ω + n, where

n = max({mi ∣ fi (0) ∈ {−1, 1} ∧ fi (1) ∉ {−1, 1}} ∪ {0}).

In particular, for any tame Z[C]-module M, HZ-length(M ⋊ C) < 2ω.
It is easy to check that the above theorem together with Corollary 4.17 and Proposition
4.8 imply the conjecture for n = 0, 1.
For a group G, denote by Ĝ its pro-nilpotent completion:
Ĝ ∶= lim G/γn (G).
←Ð n

For a finitely generated group G, there is a natural isomorphism (Prop. 3.14 [5])
EG/γω (EG) = Ĝ.
Therefore, for finitely generated groups, HZ-localization gives a natural extension of the
pro-nilpotent completion.
The pro-nilpotent completion of a finitely generated group G is always HZ-local and
the map G → Ĝ induces an isomorphism on H1 . Therefore, for such a group, the following
conditions are equivalent:
1) the natural epimorphism EG ↠ Ĝ is an isomorphism;
2) HZ-length of G ≤ ω;
3) The natural map H2 (G) → H2 (Ĝ) is an epimorphism.
A simple example of a group with HZ-length ω is the following. Let
G = ⟨a, t ∣ at = a3 ⟩ = Z[1/3] ⋊ C.
Here C acts on Z[1/3] as the multiplication by 3. Then the pro-nilpotent completion
has the structure Ĝ = Z2 ⋊ C, where the cyclic group C = ⟨t⟩ acts on 2-adic integers

ON LENGTHS OF HZ-LOCALIZATION TOWERS

3

as the multiplication by 3. Looking at the homology spectral sequence for an extension
1 → Z2 → Ĝ → C → 1, we obtain H2 (Ĝ) = Λ2 (Z2 ) ⊗ Z/9 = 0. Therefore, EG = Ĝ. Since the
group G is not pre-nilpotent, HZ-length(G) = ω.
The above example is an exception. In most cases, the description of HZ-localization
as well as the computation of HZ-length for a given group is a difficult problem. It is
shown in [5] that HZ-length of the Klein bottle group GKl ∶= ⟨a, t ∣ at = a−1 ⟩ = Z ⋊ C is
greater than ω. As a corollary, it is concluded in [5] that HZ-length of any non-cyclic free
group also is greater than ω. Our Theorem 5.6 implies that HZ-length(GKl ) = ω + 1 and
that the HZ-localization EGKl lives in the central extension
1 → Λ2 (Z2 ) → EGKl → Z2 ⋊ ⟨t⟩ → 1,
where the action of t on 2-adic integers is negation. Moreover, we give a more explicit
description of EGKl in Proposition 9.2. The HZ-length of a free non-cyclic group remains
a mystery for us, however, we prove the following
Theorem. Let F be a free group of rank ≥ 2. Then HZ-length(F ) ≥ ω + 2.
Briefly recall the scheme of the proof. Consider an extension of the group (1.1) given by
presentation
Γ ∶= ⟨a, b, t ∣ [[a, b], a] = [[a, b], b] = 1, at = a−1 , bt = ab−1 ⟩

(1.2)

We follow the Bousfield scheme of comparison of pro-nilpotent completions for a free
group and the group Γ. Consider a free simplicial resolution of Γ with F0 = F . Group Γ
has finite second homology, therefore, lim 1 of its Baer invariants is zero, therefore, π0 of
←Ð
the pro-nilpotent completion of the free simplicial resolution equals to the pro-nilpotent
completion of Γ. The group Γ has HZ-length greater than ω + 1 and the result about
HZ-length of F follows from natural properties of the HZ-localization tower. Observe
that the same method does not work for the group (1.1) (as well as for all groups of the
type M ⋊ C for abelian M), since lim 1 of Baer invariants of G is huge (this follows from
←Ð
proposition 5.8 and an analysis of the tower of Baer invariants for a metabelian group).
The paper is organized as follows. In section 2, we present the theory of relative central
extensions, which is a generalisation of the standard theory of central extensions. A nonlimit step in the construction of Bousfield’s tower can be viewed as the universal relative
extension. Section 2 is technical and introductory, it may be viewed just as a comment
to the section 3 of [5]. In section 3 we recall the exact sequences in homology from [9],
[10] for central stem-extensions. Observe that the universal relative extensions used for
the construction of HZ-tower are stem-extensions. Proposition 3.1 gives the main trick:
for a cyclic group C, Z[C]-module M, and a central stem-extension N ↪ G ↠ M ⋊ C, the
composite map H3 (M ⋊ C) → (M ⋊ C) ⊗ N ↠ N can be decomposed as H3 (M ⋊ C) →
H2 (M)C → H2 (M)C → N. This trick gives a possibility to analyze the homology of the
ω + 1-st term of the HZ-localization tower for groups of the type M ⋊ C.
Using the properties of tame modules we show in section 4 that the question about the
HZ-length of the group M ⋊ C with a tame Z[C]-module M, can be reduced to the same
question for the group (M/N) ⋊ C, where N is the largest nilpotent submodule of M. It
is shown in [14] that, for a finitely presented metabelian group G, the cokernel (denoted
by H2 (ηω )(G)) of the natural map H2 (G) → H2 (Ĝ) is divisible. Using this property we

ON LENGTHS OF HZ-LOCALIZATION TOWERS

4

conclude that, HZ-length of M ⋊ C is not greater than ω + 1 if and only if the composite
map
Λ2 (M̂ )C → Λ2 (M̂ )C → H2 (ηω )
is an epimorphism (see proposition 5.3). Theorem 5.6 is our main result of section 5. There
is a simple condition on a tame Z[C]-module M which implies that HZ-length(M ⋊ C) ≤
ω + 1. Theorem 5.6 provides a large class of groups for which one can describe HZlocalization explicitly. In particular, we show that, if the homology H2 (M ⋊ C) is finite,
then the module M satisfies the condition (ii) of Theorem 5.6 and therefore, for such M,
HZ-length(M ⋊ C) ≤ ω + 1.
In section 6 we recall the method of Bousfield from [5], which gives a possibility to
study the second homology of the pro-nilpotent completion of a free group. In section 7
we present our root examples (1.1) and (1.2) and prove that they have HZ-length greater
than ω + 1. Following the scheme described above, we get the same result for a free
non-cyclic group. In the last section of the paper we present an alternative approach
for proving that some groups have a long HZ-localization tower. Consider the wreath
product
i
Z ≀ Z = ⟨a, b ∣ [a, ab ] = 1, i ∈ Z⟩
Using functorial technique, we show in theorem 8.2 that HZ-length(Z ≀ Z) ≥ ω + 2. In the
last section, as an application of the theory developed in the paper, we give an explicit
construction of EGKl .
2. Relative central extensions and HZ-localization
Throughout this section G, H denote groups, f ∶ H → G a homomorphism and A an
abelian group.
2.1. (Co)homology of a homomorphism. Consider the continuous map between classifying spaces Bf ∶ BH → BG and its mapping cone Cone(Bf ). Following Bousfield [5,
2.14], we define homology and cohomology of f with coefficients in A as follows
Hn (f, A) = Hn (Cone(Bf ), A),

H n (f, A) = H n (Cone(Bf ), A).
Then there are long exact sequences
⋅ ⋅ ⋅ → H2 (H, A) → H2 (G, A) → H2 (f, A) → H1 (H, A) → H1 (G, A) → H1 (f, A) → 0,

0 → H 1 (f, A) → H 1 (G, A) → H 1 (H, A) → H 2 (f, A) → H 2 (G, A) → H 2 (H, A) → . . . .
In particular, H1 (f ) = Coker{Hab → Gab } and H1 (f, A) = H1 (f ) ⊗ A.
We denote by C̄ ● (G, A) the complex of normalized cochains of G with coefficients in A,
[18, 6.5.5] by ∂ n ∶ C̄ n (G, A) → C̄ n+1 (G, A) its differential and by Z̄ n (G, A) and B̄ n (G, A)
the groups of normalized cocycles and coboundaries. For a homomorphism f ∶ H → G
and an abelian group A we denote by Z̄ n (f, A) and B̄ n (f, A) the following subgroups of
Z̄ n (G, A) ⊕ C̄ n−1 (H, A)
Z̄ n (f, A) = {(c, α) ∣ f ∗ c = −∂α},

B̄ n (f, A) = {(−∂β, f ∗ β + ∂γ) ∣ β ∈ C̄ n−1 (G, A), γ ∈ C̄ n−2 (H, A)}.

Since the map ∂ ∶ C̄ 0 (H, A) → C̄ 1 (H, A) is trivial, we have

B̄ 2 (f, A) = {(−∂β, βf ) ∣ β ∈ C̄ 1 (G, A)}.

ON LENGTHS OF HZ-LOCALIZATION TOWERS

5

Lemma 2.1. For n ≥ 1 there is an isomorphism H n (f, A) ≅ Z̄ n (f, A)/B̄ n (f, A).

Proof. For a space X, we denote by C● (X) the complex of integral chains. Then
C● (X, A) = C● (X) ⊗ A and C ● (X, A) = Hom(C● (X), A). For a continuous map F ∶ X → Y
we denote by C● (F ) ∶ C● (X) → C● (Y ) the induced morphism of complexes. Then there
is a natural homotopy equivalence of complexes Cone(C● (F )) ≃ C● (Cone(F )). It follows
that there is a natural homotopy equivalence of complexes
C ● (Cone(F ), A) ≃ Cone(C ● (F, A))[−1].

(2.1)

Denote by C ● (G, A) the complex of (non-normalised) cochains of the group G. For
a homomorphism f ∶ H → G we denote by C ● (f, A) ∶ C ● (G, A) → C ● (H, A) the induced morphism of complexes. There is a natural homotopy equivalence C ● (G, A) ≃
C ● (BG, A). Moreover, there is a natural homotopy equivalence of complexes of normalised and non-normalised cochains C̄ ● (G, A) ≃ C ● (G, A) because that come from
two different functorial resolutions. It follows that there is a natural homotopy
equivalence Cone(C̄ ● (f, A)) ≃ Cone(C ● (Bf, A)). Combining this with (2.1) we get
∼
C ● (Cone(Bf ), A) Ð
→ Cone(C̄ ● (f, A))[−1]. The assertion follows.

2.2. Relative central extensions.
Definition 2.2. A relative central extension of G by A with respect to f is a couple
ι
π
ι
π
E = (A ↣ E ↠ G, f˜), where A ↣ E ↠ G is a central extension of G and f˜ ∶ H → E is a
homomorphism such that π f˜ = f.
H

0
//

A

ι

❅❅
❅❅ f
❅❅
f˜
❅❅

π //
// E

G
//

1

ι1
π1
ι2
π2
Two relative central extensions (A ↣ E1 ↠ G, f˜1 ) and (A ↣ E2 ↠ G, f˜2 ) are said to be
≅
equivalent if there exist an isomorphism θ ∶ E1 → E2 such that θι1 = ι2 , π2 θ = π1 and
θf˜1 = f˜2 .

Let (c, α) ∈ Z̄ 2 (f, A). Consider the central extension A ↣ Ec ↠ G corresponding to the
2-cocycle c. The underling set of Ec is equal to A × G and the product is given by
(a1 , g1 )(a2 , g2 ) = (a1 + a2 + c(g1 , g2 ), g1 g2 ).
Denote by f˜α ∶ H → Ec the map given by f˜α (h) = (α(h), f (h)). Note that the equality
f ∗ c = −∂α implies the equality
c(f (h1 ), f (h2 )) = −α(h1 ) + α(h1 h2 ) − α(h2 )
for all h1 , h2 ∈ H. It follows that f˜α is a homomorphism. Indeed
f˜α (h1 )f˜α (h2 ) = (α(h1 ), f (h1 ))(α(h2 ), f (h2 )) =

(α(h1 ) + α(h2 ) + c(f (h1 ), f (h2 )), f (h1 )f (h2 )) =

(α(h1 h2 ), f (h1 h2 )) = f˜α (h1 h2 ).

Then we obtain a relative central extension
E(c, α) = (A ↣ Ec ↠ G, f˜α ).

ON LENGTHS OF HZ-LOCALIZATION TOWERS

6

Proposition 2.3. The map (c, α) ↦ E(c, α) induces a bijection between elements of
H 2 (f, A) and equivalence classes of relative central extensions of G by A with respect
to f .
Proof. Any central extension is equivalent to the extension A ↣ Ec ↠ G for a normalised
2-cocycle c. Hence, it is sufficient to consider only them. Consider a relative central
extension (A ↣ Ec ↠ G, f˜). Define α ∶ H → A so that f˜(h) = (α(h), f (h)). Since f˜(1) =
(0, 1), α is a normalised 1-cochain. Since f˜ is a homomorphism, we get
(α(h1 ) + α(h2 ) + c(f (h1 ), f (h2 )), f (h1 )f (h2 )) = (α(h1 h2 ), f (h1 h2 )).

Thus f ∗ c = −∂α. It follows that any relative central extension is isomorphic to the relative
central extension E(c, α) for some (c, α) ∈ Z 2 (f, A).
Consider two elements (c, α), (c′ , α′ ) ∈ Z 2 (f, A) such that (c′ , α′ ) − (c, α) = (−∂β, βf )
for some β ∈ C̄ 1 (G, A). It follows that
c′ (g1 , g2 ) + β(g1 ) + β(g2 ) = c(g1 , g2 ) + β(g1 g2 )

for any g1 , g2 ∈ G and α′ (h) = α(h) + β(f (h)) for any h ∈ H. Denote by θβ ∶ Ec → Ec′ the
map given by θβ (a, g) = (a + β(g), g). Then θβ is a homomorphism. Indeed,
θβ (a1 , g1 )θβ (a2 , g2 ) = (a1 + β(g1 ), g1 )(a2 + β(g2 ), g2 ) =

(a1 + a2 + β(g1 ) + β(g2 ) + c′ (g1 , g2 ), g1 g2 ) =

(a1 + a2 + β(g1 g2 ) + c(g1 , g2 ), g1 g2 ) = θβ ((a1 , g1 )(a2 , g2 )).

Moreover, θβ is an isomorphism, because θ−β is its inverse, and it is easy to see that it is
an equivalence of the relative central extensions. It follows that the map (c, α) ↦ E(c, α)
induces a surjective map from elements of H 2 (f, A) to equivalence classes of relative
central extensions.
Consider two elements (c, α), (c′ , α′ ) ∈ Z 2 (f, A) such that the relative central extensions
E(c, α) and E(c′ , α′ ) are equivalent. Then there is an equivalence θ ∶ Ec → Ec′ . Since θ
respects the i njections from A, ϕ(a, 1) = (a, 1) for any a ∈ A. Since θ respects the
projections on G, there exist a unique normalised 1-cochain β ∶ G → A such that θ(0, g) =
(β(g), g) for any g ∈ G. Using that c is a normalised 2-cocycle, we obtain
θ(a, g) = θ((a, 1)(0, g)) = (a, 1)(β(g), g) = (a + β(g), g).

Then the fact that θ is a homomorphism implies that
c′ (g1 , g2 ) + β(g1 ) + β(g2 ) = c(g1 , g2 ) + β(g1 g2 ),
and hence c′ − c = −∂β, and the equality θf˜α = f˜α′ implies α′ − α = βf. The assertion
follows.

2.3. Universal relative central extensions. Let A1 and A2 be abelian groups. Recall
ι1
π1
ι2
π2
that c morphism from a central extension A1 ↣ E1 ↠ G to a central extension A2 ↣ E2 ↠
G is a couple (ϕ, θ), where ϕ ∶ A1 → A2 and θ ∶ E1 → E2 are homomorphisms such that
θι1 = ι2 τ, π2 θ = π1 .
Lemma 2.4 (cf. [18, Lemma 6.9.6]). Let (ϕ, θ) and (ϕ′ , θ′ ) be morphisms from a central
extension A1 ↣ E1 ↠ G to a central extension A2 ↣ E2 ↠ G. Then the restrictions on
the commutator subgroup coincide θ∣[E1 ,E1 ] = θ′ ∣[E1 ,E1 ] .

ON LENGTHS OF HZ-LOCALIZATION TOWERS

7

Proof. For the sake of simplicity we identify A1 with the subgroup of E1 and A2 with the
subgroup of E2 . Consider the map ρ ∶ E1 → A2 given by ρ(x) = θ(x)θ′ (x)−1 . Since A2 is
central, we get
ρ(x)ρ(y) = θ(x)θ′ (x)−1 ρ(y) = θ(x)ρ(y)θ′ (x)−1 =

θ(x)θ(y)θ′ (y)−1 θ′ (x)−1 = θ(xy)θ′ (xy)−1 = ρ(xy).

Hence ρ is a homomorphism to an abelian group. Thus ρ∣[E1 ,E1 ] = 1.



Lemma 2.5. Let (ϕ, θ) be a morphism from a central extension A1 ↣ E1 ↠ G to a central
ι2
π2
extension A2 ↣ E2 ↠ G. If ϕ = 0, then A2 ↣ E2 ↠ G splits.
ι1

π1

Proof. ϕ = 0 implies θι1 = 0. Since G is a cokernel of ι1 , there exists s ∶ G → E2 such that
sπ1 = θ. Then π2 sπ1 = π2 θ = π1 . It follows that π2 s = id.

Definition 2.6. A morphism from a relative central extension (A1 ↣ E1 ↠ G, f˜1 ) to
a relative central extension (A2 ↣ E2 ↠ G, f˜2 ) is a morphism (ϕ, θ) from the central
extension A1 ↣ E1 ↠ G to the central extension A2 ↣ E2 ↠ G such that θf˜1 = f˜2 . So
relative central extensions of G with respect to f form a category. The initial object of
this category is called the universal relative central extension of G with respect to f.
Example 2.7. For any homomorphism ϕ ∶ A1 → A2 and any (c, α) ∈ Z 2 (f, A1 ) there is a
morphism of relative central extensions
E(ϕ) = (ϕ, ϕ × id).

E(ϕ) ∶ E(c, α) Ð→ E(ϕc, ϕα),

(2.2)

Definition 2.8. A homomorphism f ∶ H → G is said to be perfect if fab ∶ Hab → Gab is an
epimorphism. In other words, f is perfect if and only if H1 (f ) = 0.
Lemma 2.9 (cf. [18, Lemma 6.9.6]). Let (ϕ, θ) and (ϕ′ , θ′ ) be morphisms from a relative
central extension (A1 ↣ E1 ↠ G, f˜1 ) to a relative central extension (A2 ↣ E2 ↠ G, f˜2 ). If
f˜1 is perfect, then (ϕ, θ) = (ϕ′ , θ′ ).
Proof. Since f˜1 is perfect, we obtain Im(f˜1 )[E1 , E1 ] = E1 . Since θf˜1 = f˜2 = θ′ f˜1 , we have
θ∣Im(f˜1 ) = θ′ ∣Im(f˜1 ) . Lemma 2.4 implies θ∣[E1 ,E1] = θ′ ∣[E1 ,E1 ] . The assertion follows.


Proposition 2.10. The universal relative central extension of G with respect to f exists
if and only if f is perfect. Moreover, in this case it is unique (up to isomorphism) and
given by a short exact sequence
H
u

0
//

H2 (f )


//

U

❄❄
❄❄ f
❄❄
❄❄

//

G
//

1

where u ∶ H → U is a perfect homomorphism.
Proof of Proposition 2.10. Assume that there is a universal relative central extension
π
(A ↣ U ↠ G, u) of G with respect to f and prove that f is perfect. Set B =
Coker(fab ∶ Hab → Gab ) and consider the epimorphism τ ∶ U ↠ B given by the composition U ↠ G ↠ Gab ↠ B. Then we have two morphisms (0, (πτ )) and (0, (π0 )) from

ON LENGTHS OF HZ-LOCALIZATION TOWERS

8

(A ↣ U ↠ G, u) to the split extension (B ↣ B × G ↠ G, (f0)). The universal property
implies that they are equal, and hence B = 0. Thus f is perfect.
Assume that f is perfect. Since H1 (Cone(Bf )) = H1 (f ) = 0, the universal coefficient formula for the space Cone(Bf ) implies that there is an isomorphism H 2 (f, A) ≅
Hom(H2 (f ), A) natural by A. Chose an element (cu , αu ) ∈ Z 2 (f, H2 (f )) that represents
the element of H 2 (f, H2 (f )) corresponding to the identity map in Hom(H2 (f ), H2 (f )).
ιu
πu
Set U ∶= Ecu , u = f˜αu and Eu = E(cu , αu ). Then Eu = (H2 (f ) ↣ U ↠ G, u). Take a
homomorphism ϕ ∶ H2 (f ) → A and consider the commutative diagram
Z 2 (f, H2 (f ))

// //

H 2 (f, H2 (f )) oo

ϕ∗

Hom(H2 (f ), H2 (f ))

ϕ∗



Z 2 (f, A)

≅

ϕ○−





// //

H 2 (f, A) oo

≅

Hom(H2 (f ), A).

It shows that the isomorphism Hom(H2 (f ), A) ≅ H 2 (f, A) sends ϕ to the class of
(ϕcu , ϕαu ). Combining this with Proposition 2.3 we obtain that any relative central extension is isomorphic to the extension E(ϕcu , ϕαu ) for some ϕ ∶ H2 (f ) → A. For any relative
central extension E(ϕcu , ϕαu ) there exists a morphism E(ϕ) ∶ Eu → E(ϕcu , ϕαu ) from Example 2.7. Then we found a morphism from Eu to any other relative central extension.
In order to prove that Eu is the universal relative central extension, we have to prove that
such a morphism is unique. By Lemma 2.9 it is enough to prove that u ∶ H → U is perfect.
Prove that u ∶ H → U is perfect. In other words we prove that Im(u)[U, U] = U. Set
˜
˜
˜
E ∶= Im(u)[U, U], A = ι−1
u (E) and f ∶ H → E given by f (h) = u(h). Note that f is perfect.
Since f is perfect, πu (E) = Im(f )[G, G] = G. Consider the restriction π = πu ∣E and the
π
relative central extension E = (A ↣ E ↠ G, f˜) with the obvious embedding E ↪ Eu .
Consider the projection ϕ′ ∶ H2 (f ) → H2 (f )/A and take the composition
E(ϕ′ )

E ↪ Eu ÐÐÐ→ E(ϕ′ cu , ϕ′ αu ).

The composition is equal to (0, (ϕ′ × 1)∣E ). By Lemma 2.5 E(ϕ′ cu , ϕ′ αu ) splits. Thus
(ϕ′ cu , ϕ′ αu ) represents 0 in H 2(f, A) ≅ Hom(H2 (f ), A), and hence ϕ′ = 0. It follows that
A = H2 (f ). Then the extension A ↣ E ↠ G is embedded into the extension A ↣ U ↠ G.
It follows that E = U.

Remark 2.11. If fab ∶ Hab → Gab is an isomorphism, then H2 (f ) = Coker{H2 (H) →
H2 (G)}.
Remark 2.12. In the proof of Proposition 2.10 we show that, if f is perfect, then the
universal relative central extension corresponds to the identity map Hom(H2 (f ), H2 (f ))
with respect to the isomorphism H 2 (f, H2 (f )) ≅ Hom(H2 (f ), H2 (f )) that comes from
the universal coefficient theorem.
2.4. HZ-localization tower via relative central extensions. Here we give an approach to the HZ-localization tower [5] via relative central extensions.
Let G be a group and η ∶ G → EG be its HZ-localization. For an ordinal α we define
αth term of the HZ-localization tower by Tα G ∶= EG/γα (EG), where γα (EG) is the αth
term of the transfinite lower central series (see [5, Theorem 3.11]). By ηα ∶ G → Tα G we
denote the composition of η and the canonical projection, and by tα ∶ Tα+1 G ↠ Tα G we

ON LENGTHS OF HZ-LOCALIZATION TOWERS

9

denote the canonical projection. The main point of [5] is that Tα G can be constructed
inductively and EG = Tα G for big enough α. We threat the construction of Tα G for a
non-limit ordinal α via universal relative central extensions.
Proposition 2.13. Let G be a group and α > 1 be an ordinal. Then (ηα )ab ∶ Gab →
(Tα G)ab is an isomorphism, the universal central extension of Tα G with respect to ηα is
given by
G

0
//

●●
●●
● ηα
ηα+1 ●●●
●●

##
tα
// Tα+1 G
//

H2 (ηα )

Tα G
//

1,

and H2 (ηα ) = Coker{H2 (G) → H2 (Tα G)}.
Proof. It follows from [5, 3.2], [5, 3.4], Proposition 2.10 and Remarks 2.11, 2.12.



3. Homology of stem-extensions
Consider a central extension of groups
1→N →G→Q→1

(3.1)

It is shown in [10] that, there is a natural long exact sequence
H3 (G) → H3 (Q) → (Gab ⊗ N)/U → H2 (G) → H2 (Q) → N → H1 (G) → H1 (Q) → 0 (3.2)
β

where U is the image of the natural map
H4 K(N, 2) → Gab ⊗ N.
Here H4 K(N, 2) is the forth homology of the Eilenberg-MacLane space K(N, 2) which
can be described as the Whitehead quadratic functor
H4 K(N, 2) = Γ2 N.
A central extension (3.1) is called a stem-extension if N ⊆ [G, G]. For a stem extension
(3.1), the exact sequence (3.2) has the form (see [9], [10])
H3 (G) → H3 (Q) → Gab ⊗ N → H2 (G) → H2 (Q) → N → 0
δ

β

(3.3)

The map δ is given as follows. We present (3.1) in the form
1 → S/R → F /R → F /S → 1,

for a free group F and normal subgroups R, S with R ⊂ S, [F, S] ⊆ R. Then the map δ is
induced by the natural epimorphism Sab → S/R:
H3 (Q) = H1 (F /S, Sab ) → H1 (F /S, S/R) = Qab ⊗ N = Gab ⊗ N.

The isomorphism H1 (F /S, S/R) = Qab ⊗ N follows from the triviality of F /S-action on
S/R.
In this section we consider the class of metabelian groups of the form Q = M ⋊ C, where
C is an infinite cyclic group and M a Z[C]-module. It follows immediately from the
homology spectral sequence that, for any n ≥ 2, there is a short exact sequence
0 → H0 (C, Hn (M)) → Hn (Q) → H1 (C, Hn−1 (M)) → 0

ON LENGTHS OF HZ-LOCALIZATION TOWERS

10

which can be presented in terms of (co)invariants as
0 → Hn (M)C → Hn (Q) → Hn (M)C → 0.

Composing the last epimorphism with Hn−1 (M)C → Hn−1 (M)C ↪ Hn−1 (Q), we get a
natural (in the category of Z[C]-modules) map
αn ∶ Hn (Q) → Hn−1 (Q).
In the next proposition we will construct a composite map
α3′ ∶ H3 (Q) → H1 (C, Λ2 (M)) ↪ Λ2 (M) ↠ Λ2 (M)C ↪ H2 (Q)

(3.4)

using group-theoretical tools, without spectral sequence. Probably, α3′ coincides with α3
up to isomorphism, but we will not use this comparison later.
Proposition 3.1. For a stem extension (3.1) of a group Q = M ⋊ C, there exists a map
α3′ ∶ H3 (Q) → H2 (Q), given as a composition (3.4), such that the following diagram is
commutative
H3 (Q)

α′3

H2 (Q)
//

β

δ




Gab ⊗ N
//

N

where the lower horizontal map is the projection
Gab ⊗ N → C ⊗ N = N.
Proof. We choose a free group F with normal subgroups R ⊂ S ⊂ T such that
F /T = C, F /S = Q, F /R = G.

In the above notation, we get M = T /S, N = S/R. The proof follows from the direct
analysis of the following diagram, which corners are exactly the roots of the diagram
given in proposition:
α′3

H3 (Q)
H1 (F /S, Sab )

H1 (F /T, S∩T
S′ )

//

′

H1 (F /T, Sab )

//

S∩T
H1 (F /T, [S,T
])
′

(3.5)







H2 (T /S)

H2 (Q)
//

β

δ



H1 (F /S, S/R)

(F /S)ab ⊗ S/R
//



S/R

All arrows of this diagram are natural. We will make comments only about two maps
from the diagram, other maps are obviously defined. The map
S ∩ T′
H1 (F /T,
) → H1 (F /T, Sab )
S′

ON LENGTHS OF HZ-LOCALIZATION TOWERS

11

is an isomorphism since
S
) ↪ H1 (F /T, Tab ) = H3 (F /T ) = 0.
S ∩ T′
S∩T ′
The vertical map in the diagram H1 (F /T, [S,T
] ) = H1 (F /T, H2 (S/T )) ↪ H2 (S/T ) follows
from the identification of H1 of a cyclic group with invariants.
The commutativity of (3.5) follows from the commutativity of the natural square
H1 (F /T,

H1 (F /S, Sab )

// //





H1 (F /S, S/R)

H1 (F /T, Sab )

// //

H1 (F /T, S/R)

and identification of H1 (F /T, −) with invariants of the F /T -action.



4. Tame modules and completions
Throughout the section C denotes an infinite cyclic group. If R is a commutative ring,
R[C] denotes the group algebra over R. We use only R = Z, Q, C. The augmentation ideal
is denoted by I. If t is one of two generators of C, R[C] = R[t, t−1 ] and I = (t − 1).
4.1. Finite dimensional K[C]-modules. Let K be a field (we use only K = Q, C), V
be a right K[C]-module such that dimK V < ∞. If we fix a generator t ∈ C we obtain a
linear map ⋅t ∶ V → V that defines the module structure. We denote the linear map by
aV ∈ GL(V ). The characteristic and minimal polynomials of aV are denoted by χV and
µV respectively. These polynomials depend of the choice of t ∈ C. Note that for any such
modules V and U we have
χV ⊕U = χV χU ,

µV ⊕U = lcm(µV , µU ).

(4.1)

Lemma 4.1. Let V be a right K[C]-module such that dimK V < ∞ and t ∈ C be a
generator. Then there exist distinct irreducible monic polynomials f1 , . . . , fl ∈ K[x] and
an isomorphism
V ≅ V1 ⊕ ⋅ ⋅ ⋅ ⊕ Vl ,
where
mi,l
m
Vi = K[C]/(fi i,1 (t)) ⊕ ⋅ ⋅ ⋅ ⊕ K[C]/(fi i (t)),
and mi,1 ≥ mi,2 ≥ ⋅ ⋅ ⋅ ≥ mi,li ≥ 1. Moreover, if we set mi = ∑j mi,j , then χV = f1m1 . . . flml
m
m
and µV = f1 1,1 . . . fl l,1 .
Proof. Note that K[C] = K[t, t−1 ] is the polynomial ring K[t] localised at the element
t. Then it is a principal ideal domain. Then the isomorphism follows from the structure
theorem for finitely generated modules over a principal ideal domain. The statement
about χV and µV follows from the fact that both characteristic and minimal polynomials
m
m

of K[C]/(fi i,j (t)) equal to fi i,j and the formulas (4.1).
Let R be a commutative ring and t be a generator of C. For an R[C]-module M and
a polynomial f ∈ R[x] we set
M f = {m ∈ M ∣ m ⋅ f (t) = 0}.

ON LENGTHS OF HZ-LOCALIZATION TOWERS

12

Note that M C = M x−1 . It is easy to see that for any f, g ∈ R[x] we have
(M/M f )g = M f g /M f .

(4.2)

Corollary 4.2. Let V be a right K[C]-module such that dimK V < ∞ and t ∈ C be a
generator. Assume that µV = f1m1 . . . flml , where f1 , . . . , fl ∈ K[x] are distinct irreducible
monic polynomials. Consider the filtration
0 = F0 V ⊂ F1 V ⊂ ⋅ ⋅ ⋅ ⊂ Fl V = V
given by Fi V = V

m
m
f1 1 ...fi i

. Then

V = ⊕ Fi V /Fi−1 V,
i

and Fi V /Fj−1 V = (V /Fj−1V )fj

mj

mi

...fi

m

µFi V /Fj−1 = fj j . . . fimi

.

Proof. In the notation of Lemma 4.1 we obtain Fi V = V1 ⊕⋅ ⋅ ⋅⊕Vi . The assertion follows. 
4.2. Tame Z[C]-modules. The rank of an abelian group A is dimQ (A ⊗ Q). The torsion
subgroup of A is denoted by tor(A). The following statement seems to be well known but
we can not find a reference, so we give it with a proof.
Lemma 4.3. Let A be a torsion free abelian group of finite rank and B be a finite abelian
group. Then A ⊗ B is finite.
Proof. It is sufficient to prove that A ⊗ Z/pk is finite for any prime p and k ≥ 1. Consider a
p-basic subgroup A′ of A (see [12, VI]). Then A′ ≅ Zr , where r is the rank of A and A/A′
is p-divisible. Thus (A/A′ ) ⊗ Z/pk = 0. Hence the map (Z/pk )r ≅ A′ ⊗ Z/pk ↠ A ⊗ Z/pk is
an epimorphism.

A finitely generated Z[C]-module M is said to be tame if the group M ⋊ C is finitely
presented.
Proposition 4.4 (Theorem C of [2]). Let M be a finitely generated Z[C]-module. Then
M is tame if and only if the following properties hold:
● tor(M) is finite;
● the rank of M is finite;
● there is a generator t of C such that χM ⊗Q is integral.
Lemma 4.5 (Lemma 3.4 of [2]). Let M and M ′ be tame Z[C]-modules. Then M ⊗ M ′
is a tame Z[C]-module (with the diagonal action).
Definition 4.6. Let M be a tame module. The generator t ∈ C such that such that χM ⊗Q
is integral is called an integral generator for M. When we consider a tame module, we
always denote by t an integral generator for M. We set aM ∶= t ⊗ Q ∶ M ⊗ Q → M ⊗ Q, and
denote by χM , µM the characteristic and the minimal polynomial of aM . In other words
χM = χM ⊗Q and µM = µM ⊗Q .
Lemma 4.7. µM is an integral monic polynomial for any tame Z[C]-module M.

Proof. Let χM = (x − λ1 )m1 . . . (x − λl )ml for some distinct λ1 , . . . , λl ∈ C and mi ≥ 1. Then
µM = (x − λ1 )k1 . . . (x − λl )kl , where 1 ≤ ki ≤ mi . Since χ is a monic integral polynomial,
λ1 , . . . , λl are algebraic integers. It follows that the coefficients of µM are algebraic integers
as well. Moreover, they are rational numbers, because aM is defined rationally. Using
that a rational number is an algebraic integer iff it is an integer number, we obtain
µM ∈ Z[x].


ON LENGTHS OF HZ-LOCALIZATION TOWERS

13

Proposition 4.8. Let M be a torsion free tame Z[C]-module and µM = f1m1 . . . flml where
f1 , . . . , fl are distinct irreducible integral monic polynomials and mi ≥ 1 for all i. Consider
the filtration
0 = F0 M ⊂ F1 M ⊂ ⋅ ⋅ ⋅ ⊂ Fl M = M
m1

mi

m

given by Fi M = M f1 ...fi . Then Fi M/Fj−1 M is torsion free and µFi M /Fj−1 M = fj j . . . fimi
for any i ≥ j. Moreover, the corresponding filtration on M ⊗ Q splits:
l

M ⊗ Q ≅ ⊕ (Fi M/Fi−1 M) ⊗ Q.
i=1

Proof. Prove that Fi M/Fj−1 M is torsion free. Let v + Fj−1 M ∈ Fi M/Fj−1 M and nv +
m
Fj−1 M = 0. Hence nv ⋅ f1m1 (t) . . . fj j (t) = 0 in M. Using that M is torsion free we get
m
v ⋅ f1m1 (t) . . . fj j (t) = 0, and hence v + Fj−1 M = 0. Thus Fi M/Fj−1 M is torsion free. Set
V = M ⊗Q, K = Q, apply Corollary 4.2 and note that Fi V /Fj−1 V = (Fi M/Fj−1 M)⊗Q. The
assertion follows. Here we use Gauss lemma about integral polynomials: an irreducible
polynomial in Z[x] is irreducible in Q[x].

Recall that a module N is said to be nilpotent if NI n = 0 for some n, where I is the
augmentation ideal. It is easy to see that a Z[C]-module N is nilpotent if and only if
n
N (x−1) = N for some n.
Definition 4.9. A Z[C]-module M is said to be invariant free if M C = 0.
Lemma 4.10. Let M be a torsion free tame Z[C]-module. Then the following equivalent.
(1)
(2)
(3)
(4)
(5)
(6)

M is invariant free;
M does not have non-trivial nilpotent submodules;
MC is finite;
aM − 1 is an automorphism;
χM (1) ≠ 0;
µM (1) ≠ 0.

Proof. (1) ⇔ (2) and (4) ⇔ (5) ⇔ (6) are obvious. The equality Ker(aM − 1) = M C ⊗ Q
implies (1) ⇔ (4). Since M is finitely generated Z[C]-module, MC is a finitely generated
abelian group. Then the equality Coker(aM − 1) = MC ⊗ Q implies (3) ⇔ (4).


Corollary 4.11. Let M be a torsion free tame Z[C]-module and µM = (x − 1)m f , where
f (1) ≠ 0. Then there exists the largest nilpotent submodule N ≤ M. Moreover, µN =
(x − 1)m , µM /N = f, M/N is torsion free and invariant free, and the short exact sequence
N ⊗ Q ↣ M ⊗ Q ↠ (M/N) ⊗ Q splits over Q[C].

Proof. If µM (1) ≠ 0, then M is already invariant free, N = 0 and there is nothing to
prove. If µM (1) = 0, then we can decompose µM = (x − 1)m1 f2m2 . . . flml into a product
of irreducible polynomials such that fi (1) ≠ 0 for i ≥ 2. Consider the filtration from
Proposition 4.8. Then N = F1 M.

Corollary 4.12. Let M be a tame Z[C]-module. Then there exists the largest nilpotent
submodule N ≤ M. Moreover, M/N is invariant free and (M/N)C is finite.
Recall that a module N is said to be prenilpotent if NI n = NI n+1 for n >> 1.

ON LENGTHS OF HZ-LOCALIZATION TOWERS

14

Corollary 4.13. Let M be a tame Z[C]-module and µM = (x − 1)m f , where f (1) ≠ 0.
Then there exists a prenilpotent submodule N ≤ M such that M/N is torsion free and
invariant free. Moreover, tor(N) = tor(M), µN = (x − 1)m , µM /N = f and the sequence
N ⊗ Q ↣ M ⊗ Q ↠ (M/N) ⊗ Q splits over Q[C].
Lemma 4.14. Let M be a tame torsion free Z[C]-module. If µM (0) ∈ {−1, 1}, then M is
finitely generated as an abelian group.

Proof. It follows from the fact that Z[t, t−1 ]/(µM (t)) is a finitely generated abelian group.

4.3. Completion of tame Z[C]-modules. If M is a finitely generated R[C]-module,
we set M̂ = lim M/MI i and we denote by
←Ð
ϕ = ϕM ∶ M Ð→ M̂

the natural map to the completion. Note that the functor M ↦ M̂ is exact [19, VIII] and
M̂ /M̂I i = M/MI i .
We set
Zn = lim Z/ni
←Ð
for any n ∈ Z. In particular, Zn = Z−n , Z0 = Z, Z1 = 0 and, if n ≥ 2, then Zn = ⊕ Zp , where
p runs over all prime divisors of n.
Lemma 4.15. Let M be a tame torsion free invariant free Z[C]-module and n = χM (1).
Then ni ⋅ M ⊆ MI i for any i ≥ 1 and there exists a unique epimorphism of Z[C]-modules
ϕ̂ ∶ M ⊗ Zn ↠ M̂ such that the diagrams
M ⊗ Zn

ϕ̂

// //



M ⊗ Z/ni

M̂


// //

M/MI i

are commutative.
Proof. We identify M with the subgroup of M ⊗ Q. Corollary 4.10 implies that n ≠ 0.
Set b = aM − 1. Then the characteristic polynomial of b is equal to χb (x) = χM (x + 1) and
if χb = ∑di=0 βi xi , then β0 = n. Thus nx = b(∑di=1 βi bi−1 (x)) for any x ∈ M. It follows that
nM ⊆ b(M). Hence ni M ⊆ bi (M) = MI i for any i ≥ 1 and we obtain homomorphisms
M ⊗ Z/ni → M/MI i . We define ϕ̂ as the composition M ⊗ Zn → lim(M ⊗ Z/ni ) → M̂.
←Ð
Since the rank M is finite, the abelian groups M ⊗ Z/ni are finite. Thus we get that
the homomorphism lim(M ⊗ Z/ni ) → M̂ is an epimorphism because lim1 of an inverse
←Ð
←Ð
sequence of finite groups is trivial. Then it is sufficient to prove that the homomorphism
M ⊗Zn → lim(M ⊗ Z/ni ) is an epimorphism. For this it is enough to prove that M ⊗Zp →
←Ð
lim(M ⊗ Z/pi ) is an epimorphism for any prime p. Consider a p-basic subgroup B of M
←Ð
(see [12, VI]). Since B ≅ Zl , we get B ⊗Zp = lim(B ⊗ Z/pi ). Using that B ⊗Z/pi ↠ M ⊗Z/pi
←Ð
are epimorphisms of finite groups, we obtain that lim(B ⊗ Z/pi ) → lim(M ⊗ Z/pi ) is an
←Ð
←Ð

ON LENGTHS OF HZ-LOCALIZATION TOWERS

15

epimorphism. Then analysing the diagram
B ⊗ Zp //
//

M ⊗ Zp

≅

lim(B
←Ð





⊗ Z/pi )

// //

lim(M ⊗ Z/pi )
←Ð

we obtain that the right vertical arrow is an epimorphism.



A Z[C]-module is said to be perfect if MI = M.
Corollary 4.16. Let M be a torsion free tame Z[C]-module. If µM (1) ∈ {−1, 1}, then M
is perfect.
Proof. By Lemma 4.10, M is invariant free. Then Lemma 4.15 implies M̂ = 0. Hence M
is perfect.

Corollary 4.17. Let M be a tame Z[C]-module. If µM = (x−1)m f, where f is an integral
polynomial such that f (1) ∈ {−1, 1}, then M is prenilpotent.
Proof. A finite module is always prenilpotent, so we can assume that M has no torsion.
Further, by Lemma 4.11, we can consider the largest nilpotent submodule N ≤ M such
that µN = (x − 1)m and µM /N = f. Corollary 4.16 implies that M/N is perfect. Then N
and M/N are prenilpotent, and hence, M is prenilpotent.

Proposition 4.18. Let M and M ′ be tame Z[C]-modules with the same integral generator
t ∈ C, λ1 , . . . , λl ∈ C are eigenvalues of aM and λ′1 , . . . , λ′l′ ∈ C are eigenvalues of aM ′ .
Assume that the equality λi λ′j = 1 holds only if λi = λ′j = 1. Then the homomorphism
(M ⊗ M ′ )C Ð→ (M̂ ⊗ M̂ ′ )C
is an epimorphism.
Proof. Note that if M1 ↣ M2 ↠ M3 is a short exact sequence of tame modules and
(M1 ⊗ M ′ )C → (M̂1 ⊗ M̂ ′ )C , (M3 ⊗ M ′ )C → (M̂3 ⊗ M̂ ′ )C are epimorphisms, then (M2 ⊗
M ′ )C → (M̂2 ⊗ M̂ ′ )C is an epimorphism. Indeed, since the functor of completion is exact,
we have the commutative diagram with exact rows
(M1 ⊗ M ′ )C


//

(M̂1 ⊗ M̂ ′ )C

(M2 ⊗ M ′ )C

// //




//

(M̂2 ⊗ M̂ ′ )C

(M3 ⊗ M ′ )C

// //

(M̂3 ⊗ M̂ ′ )C

that implies this. Then, using Corollary 4.13, we obtain that we can divide our prove
into two parts: (1) prove the statement for the case of torsion free invariant free modules
M, M ′ ; (2) prove the statement for the case of a prenilpotent module M and arbitrary
tame module M ′ . Throughout the proof we use that (M ⊗ M ′ )C ≅ M ⊗Z[C] Mσ′ , where
Mσ′ is the module with the same underling abelian group M ′ but with the twisted action
of C ∶ m ∗ t = mt−1 .

ON LENGTHS OF HZ-LOCALIZATION TOWERS

16

(1) Assume that M, M ′ are torsion free invariant free tame Z[C]-modules. Lemma
4.10 implies that λi ≠ 1 and λ′j ≠ 1 for all i, j. Then we have λi λ′j ≠ 1 for all i, j. Note
that the eigenvalues of aM ⊗ aM ′ equal to the products λi λj , and hence 1 is not an
eigenvalue of aM ⊗ aM ′ . It follows that det(aM ⊗ aM ′ − 1) ≠ 0. Consider the minimal
polynomial µ of the tensor square aM ⊗ aM ′ . Since the aM ⊗ aM ′ is defined over Q, the
coefficients of µ are rational (because they are invariant under the action of the absolute
Galois group). Moreover, µ = ∏(x − λi λ′j )ki,j for some ki,j , and hence, its coefficients are
algebraic integers. It follows that µ is a monic polynomial with integral coefficients. The
polynomial µ(x + 1) is the minimal polynomial for aM ⊗ aM ′ − 1. Let µ(x + 1) = ∑ki=0 ni xi .
Then n0 = det(aM ⊗aM ′ −1) ≠ 0 and n0 ⋅(M ⊗M ′ ) ⊆ (M ⊗M ′ )(t−1). Since the rank of M ⊗M ′
is finite, (M ⊗M ′ )/n0 (M ⊗M ′ ) is finite, and hence, (M ⊗M ′ )C = (M ⊗M ′ )/(M ⊗M ′ )(t−1)
is finite. By Lemma 4.15 we have epimorphisms M ⊗ Zn ↠ M̂ and M ′ ⊗ Zn′ ↠ M̂ ′ , where
n = det(aM − 1) and n′ = det(aM ′ − 1). It is easy to see that
((M ⊗ Zn ) ⊗ (M ′ ⊗ Zn′ ))C = (M ⊗ M ′ )C ⊗ (Zn ⊗ Zn′ ).

Since (M ⊗M ′ )C is finite, (M ⊗M ′ )C → (M ⊗M ′ )C ⊗(Zn ⊗Zn′ ) is an epimorphism. Then
(M ⊗ M ′ )C → (M̂ ⊗ M̂ ′ )C is an epimorphism.
(2) Assume that M is a prenilpotent Z[C]-module and M ′ is a tame Z[C]-module.
Then there exists i such that M̂ = M/MI i . Since (M̂ ⊗ M̂ ′ )C ≅ M̂ ⊗Z[C] M̂σ′ , we get
(M̂ ⊗ M̂ ′ )C ≅ (M/MI i ⊗ M̂ ′ )C ≅ (M/MI i ⊗ M̂ ′ /M̂ ′ I i )C ≅ (M/MI i ⊗ M ′ /M ′ I i )C .

It follows that (M ⊗ M ′ )C → (M̂ ⊗ M̂ ′ )C is an epimorphism.



Corollary 4.19. Let M be a tame Z[C]-module and µM = (x − 1)m f1m1 . . . flml for some
distinct monic irreducible polynomials f1 , . . . , fl ∈ Z[x] such that fi (1) ≠ 0 and fi (0) ∉
{1, −1} for all 1 ≤ i ≤ l. Then the the homomorphism
(M ⊗2 )C Ð→ (M̂ ⊗2 )C

is an epimorphism.
Proof. Let λ1 , . . . , λk be roots of µM . Assume that λi λj = 1. Then λi is an invertible
algebraic integer, and hence, the absolute term of its minimal polynomial equals to ±1.
Thus λi can not be a root of fm for 1 ≤ m ≤ l. It follows that it is a root of x − 1. Then
λi = λj = 1.

Proposition 4.20. Let M, M ′ be tame Z[C]-modules with the same integral generator
t ∈ C, µM = (x − λ1 )m1 . . . (x − λl )ml for some distinct λ1 , . . . , λl ∈ C and µM ′ = (x −
′
′
λ′1 )m1 . . . (x − λ′l′ )ml′ for some distinct λ′1 , . . . , λ′l′ ∈ C. Assume that the equality λi λ′j = 1
holds only if either mi = m′j = 1 or λi = λ′j = 1. Then the cokernel of the homomorphism
(M ⊗ M ′ )C ⊕ (M̂ ⊗ M̂ ′ )C Ð→ (M̂ ⊗ M̂ ′ )C
is finite.
Proof. Corollary 4.13 implies that the proof can be divided into proofs of the following
two statements: (1) the statement for torsion free invariant free modules M, M ′ ; (2)
if N ↣ M ↠ M0 is a short exact sequence of tame Z[C]-modules such that N ⊗ Q ↣
M ⊗Q ↠ M0 ⊗Q splits, N is prenilpotent, and the statement holds for the couple M0 , M ′ ,
then it holds for the couple M, M ′ .

ON LENGTHS OF HZ-LOCALIZATION TOWERS

17

(1) Here we prove that the cokernel of (M̂ ⊗ M̂ ′ )C Ð→ (M̂ ⊗ M̂ ′ )C is already finite. Set
n = χM (1) and n′ = χM ′ (1). Lemma 4.15 implies that there are epimorphisms M ⊗Zn ↠ M̂
and M ′ ⊗ Zn′ ↠ M̂ ′ . Using that − ⊗ (Zn ⊗ Zn′ ) is an exact functor, we obtain that there is
an epimorphism (M ⊗M ′ )C ⊗(Zn ⊗Zn′ ) ↠ (M̂ ⊗ M̂ ′ )C . Moreover, there is an epimorphism
Coker((M ⊗ M ′ )C → (M ⊗ M ′ )C ) ⊗ (Zn ⊗ Zn′ ) ↠ Coker((M̂ ⊗ M̂ ′ )C → (M̂ ⊗ M̂ ′ )C ).
It follows that it is enough to prove that Coker((M ⊗ M ′ )C → (M ⊗ M ′ )C ) is finite.
Lemma 4.5 implies that M ⊗ M ′ is finitely generated, and hence, (M ⊗ M ′ )C is a finitely
generated abelian group. It follows that it is enough to prove that (M ⊗ M ′ )C ⊗ C →
(M ⊗ M ′ )C ⊗ C is an epimorphism. Eigenvalues of aM ⊗ aM ′ are products λi λ′j . Assume
that λi λ′j = 1 for some i, j. Since M and M ′ are invariant free, λi ≠ 1 and λ′j ≠ 1. Then
mi = 1 = m′j . It follows that all Jordan blocks of aM ⊗ C corresponding to λi and all
Jordan blocks of aM ′ ⊗ C corresponding to λ′j are 1 × 1-matrices. It follows that all Jordan
blocks of aM ⊗ aM ′ ⊗ C corresponding to 1 are 1 × 1-matrices. Hence all Jordan blocks of
B ∶= aM ⊗ aM ′ ⊗ C − 1 corresponding to 0 are 1 × 1-matrices. It is easy to see that, if all
Jordan blocks of a complex linear map B ∶ V → V corresponding to 0 are 1 × 1-matrices,
then V = Ker(B)⊕Im(B). It follows that the map Ker(B) → Coker(B) is an isomorphism.
Then (M ⊗ M ′ )C ⊗ C → (M ⊗ M ′ )C ⊗ C is an isomorphism.
(2) Note that N̂ = N/NI i for some i >> 1. Since, (N̂ ⊗ M̂ ′ )C can be interpret as
N̂ ⊗Z[C] M̂σ′ , (tensor product over Z[C]), we obtain (N̂ ⊗ M̂ ′ )C = (N/NI i ⊗ M ′ /M ′ I i )C .
It follows that (N̂ ⊗ M̂ ′ )C is a finitely generated abelian group and the map (N ⊗ M ′ )C →
(N̂ ⊗ M̂ ′ )C is an epimorphism. Set
N ∶= N ⊗ M ′ , M ∶= M ⊗ M ′ ,
M0 ∶= M0 ⊗ M ′ , Ñ ∶= N̂ ⊗ M̂ ′ ,
M̃ ∶= M̂ ⊗ M̂ ′ ,

M̃0 ∶= M̂0 ⊗ M̂ ′ ,

L̃ ∶= Ker(M̃ ↠ M̃0 ).
Then ÑC is a finitely generated abelian group and the map NC → ÑC is an epimorphism.
Consider the exact sequence
0 → L̃C → M̃C → M̃C
0 → L̃C → M̃C → (M̃0 )C → 0.
Since M̃ ⊗ Q ↠ M̃0 ⊗ Q is a split epimorphism, the image of M̃C
0 → L̃C lies in the torsion
subgroup, which is finite because of the epimorphism ÑC ↠ L̃C . Then the cokernel of M̃C
→ M̃C
0 is finite. Set
Q = Coker(MC ⊕ M̃C → M̃C ),

Q0 = Coker((M0 )C ⊕ M̃C
0 → (M̃0 )C ).

ON LENGTHS OF HZ-LOCALIZATION TOWERS

18

Then we know that Q0 is finite and Coker(M̃C → M̃C
0 ) is finite, and we need to prove
that Q is finite. Consider the diagram with exact columns.
NC ⊕ Ñ C
//

MC ⊕ M̃C



ÑC

α

//

(M0 )C ⊕ M̃0



//

M̃C

C

//

(M̃0 )C



Q


//

Q0

Using the snake lemma, we obtain that Ker(Q → Q0 ) = Coker(α). Since Q0 is finite and
Coker(α) = Coker(M̃C → M̃C
0 ) is finite, we get that Q is finite.

Corollary 4.21. Let M be a tame Z[C]-module and µM = (x − 1)m f1m1 . . . flml for some
distinct monic irreducible polynomials f1 , . . . , fl ∈ Z[x] such that fi (1) ≠ 0. Assume that
for any 1 ≤ i ≤ l either fi (0) ∉ {−1, 1} or mi = 1. Then the cokernel of the homomorphism
(M ⊗2 )C ⊕ (M̂ ⊗2 )C Ð→ (M̂ ⊗2 )C

is finite.
Remark 4.22. We prove Propositions 4.18 and 4.20 for tensor products of some modules
and their completions. Further we need the same statements for exterior squares. Of
course, the statements for tensor products imply the statements for exterior squares, so
it is enough to prove for tensor products. Moreover, it is more convenient to prove such
statements for tensor products because they have two advantages.
The first obvious advantage is that we can change modules M and M ′ in the tensor
product M ⊗ M ′ independently doing some reductions to ‘simpler’ modules.
The second less obvious advantage is the following. Let A be an abelian group and
M, M ′ are Z[A]-modules. Then we can interpret coinvariants of the tensor product as
the tensor product over Z[A]
(M ⊗ M ′ )A = M ⊗Z[A] Mσ′ ,
where Mσ′ is the module M ′ with the twisted module structure m∗a = ma−1 . In particular,
there is an additional nontrivial structure of Z[A]-module on (M ⊗ M ′ )A . But there is no
such a structure on (Λ2 M)A . More precisely, the kernel of the epimorphism
(M ⊗ M)A ↠ (Λ2 M)A

is not always a Z[A]-submodule. For example, if A = C = ⟨t⟩, M = Z2 where t acts on M
via the matrix ( 10 11 ) , it is easy to check that the kernel is not a submodule.
In our article [14] there are two mistakes concerning this that can be fixed easily.
(1) On page 562 we define ∧2σ M as a quotient module of M ⊗Λ Mσ by the submodule
generated by the elements m ⊗ m. Then we prove Corollaries 3.4 and 3.5 for such
a module. In the proof of Proposition 7.2 we assume that ∧2σ M = (∧2 M)A which
is the first mistake. In order to fix this mistake we have define ∧2σ M as a quotient

ON LENGTHS OF HZ-LOCALIZATION TOWERS

19

of M ⊗Λ M by the abelian group generated by the elements m ⊗ m and prove
Corollaries 3.4 and 3.5 using this definition. The prove is the same. We just need
to change the meaning of the word ‘generated’ form ‘generated as a module’ to
’generated as an abelian group’.
(2) In the proof of Lemma 7.1 we assume that (∧2 M)A is an Z[A]-module. This is
the second mistake. In order to fix it, we have replace (∧2 M)A by (M ⊗ M)A in
the first sentence of the proof of Lemma 7.1.
5. HZ-localization of M ⋊ C
Now consider our group G = M ⋊ C and the maximal nilpotent submodule N ⊆ M, such
that (M/N)C is finite. We have a natural commutative diagram
N //
//

G

// //

G/N
ηω

ηω





N //
//

Ĝ

// //

̂
G/N

Observe that, for any Z[C]-submodule N ′ ⊆ N,
̂′ = Ĝ/N ′ .
G/N
Lemma 5.1. For any Z[C]-submodule N ′ of N, there is a natural isomorphism
H2 (ηω )(G) = H2 (ηω )(G/N ′ ).
Proof. We can present the submodule N ′ as a finite tower of central extensions. If we will
prove that
H2 (ηω )(G) = H2 (ηω )(G/N ′ )
for any N ′ ⊆ N such that N ′ (1−t) = 0, than we will be able to prove the general statement
by induction on class of nilpotence of N ′ .
The assumption that N ′ (1 − t) = 0 implies that the extensions
̂′ → 1
1 → N ′ → G → G/N ′ → 1, 1 → N ′ → Ĝ → G/N
are central. Consider the natural map between sequences (3.2) for these extensions:
(Gab ⊗ N ′ )/U
//

H2 (G)

H2 (G/N ′ )
//

//

N′

H1 (G)
//





(Ĝab ⊗ N ′ )/U

//

H2 (Ĝ)


//

H2 (ηω )(G)

̂′ )
H2 (G/N
//

N′
//

H1 (Ĝ)

//



H2 (ηω )(G/N ′ )

Elementary diagram chasing implies that the lower horizontal map is an isomorphism and
the needed statement follows.

Lemma 5.2. If E(G/N) = Tω+1 (G/N), then EG = Tω+1 (G).

ON LENGTHS OF HZ-LOCALIZATION TOWERS

20

Proof. First we observe that, for any N ′ ⊆ N, there is a natural isomorphism
Tω+1 (G/N ′ ) = Tω+1 (G)/N ′ .
Indeed, lemma 5.1 implies that there is a natural diagram

H2 (ηω )(G) //

N′

N′








Tω+1 (G)
//

// //





H2 (ηω )(G/N ′ ) //

Tω+1 (G/N ′ )
//

Ĝ

// //

̂′
G/N

Hence, we have a natural diagram
N ′ //
//

G

// //

G/N ′
ηω+1

ηω+1





N ′ //
//

Tω+1 (G)

// //

Tω+1 (G/N ′ )

Again, as in the proof of lemma 5.1, we will assume that N ′ is central and will prove that,
in this case, EG = Tω+1 (G) provided E(G/N ′ ) = Tω+1 (G/N ′ ).
This follows from comparison of sequences (3.2) applied to the above central extensions:
(Gab ⊗ N ′ )/U
(Ĝab

⊗ N ′ )/U

//

H2 (G)
//

//



H2 (Tω+1 (G))

//

N′
//



H2 (Tω+1 (G/N ′ ))
//



H2 (ηω+1 )(G)

H2 (G/N ′ )

//

N′
//

H1 (G)

H1 (Tω+1 (G))

//



H2 (ηω+1 )(G/N ′ )

Again, elementary diagram chasing shows that the lower horizontal map is an isomorphism
and the needed statement follows.

Proposition 5.3. For a tame Z[C]-module M, the following conditions are equivalent:
(i) HZ-length(M ⋊ C) ≤ ω + 1;
(ii) the composition
Λ2 (M̂ )C → Λ2 (M̂ )C → H2 (ηω )
is an epimorphism.

ON LENGTHS OF HZ-LOCALIZATION TOWERS

21

Proof. It follows from (3.3) and construction of Tω+1 (G) that we have a natural diagram:

H3 (Ĝ)

Gab ⊗ H2 (ηω )(G)
//

//

H2 (G)

H2 (G)





H2 (Tω+1 (G))

H2 (Ĝ)
//

// //

H2 (ηω )(G)



q
qqqq
q
q
qqq
qqq

H2 (ηω )(G)

(5.1)
This diagram implies that the condition (i) for G = M ⋊ C is equivalent to the surjectivity
of the map δ ∶ H3 (Ĝ) → Gab ⊗ H2 (ηω )(G). Proposition 3.1 implies the following natural
diagram
H3 (Ĝ)
//

Gab ⊗ H2 (ηω )(G)




Λ2 (M̂ )C

//

H2 (ηω )(G)

♣77
♣♣♣
♣
♣
♣♣
♣♣♣

❏❏
❏❏
❏❏
❏❏
❏%% %%

Λ2 (M̂ )C

and the implication (i) ⇒ (ii) follows.
Now assume that (ii) holds. Let N be the maximal nilpotent submodule of M such
that (M/N)C is finite. Denote H ∶= (M/N) ⋊ C. We have a natural diagram:
H3 (Ĝ)


//

H3 (Ĥ)
//

Λ2 (M̂ )C
Λ2 (̂
M/N )C
//

//

Λ2 (M̂ )C
Λ2 (̂
M/N )C

//

H2 (ηω )(G)

//



H2 (ηω )(H)

Lemma 5.1 implies that the right hand vertical map in this diagram is a natural isomorphism. Condition (ii) implies that the composition of three lower arrows in the last
diagram must be an epimorphism. Now observe that Hab ⊗ H2 (ηω )(H) = H2 (ηω )(H),
since the group H2 (ηω )(H) is divisible and (M/N)C is finite. Therefore, δ ∶ H3 (Ĥ) →
Hab ⊗ H2 (ηω )(H) is surjective. The diagram (5.1) with G replaced by H implies that
EH = Tω+1 (H). Now the statement (i) follows from lemma 5.2.

Now we will consider the key example of a tame Z[C]-module M, such that
HZ-length(M ⋊ C) > ω + 1. For the construction of such an example, recall first certain well-known properties of quadratic functors.
Let X1 , . . . , Xn , Y1 , . . . , Ym be abelian groups and X = ⊕ni=1 Xi , Y = ⊕m
j=1 Yj . An element
of a direct sum will be written as a column (x1 , . . . , xn )T ∈ X and a homomorphism

ON LENGTHS OF HZ-LOCALIZATION TOWERS

22

f ∶ X → Y will be written as a matrix f = (fji ), where fji ∶ Xi → Yj and
n

n

f ((x1 , . . . , xn ) ) = (∑ f1i (xi ), . . . , ∑ fmi (xi ))T .
T

i=1

i=1

For an abelian group X we denote by
and X ⊗2 its exterior, symmetric,
divided (the same as the Whitehead quadratic functor) and tensor squares respectively.
If X is torsion free, then there are short exact sequences
Λ2 X,

S2 X,

ι∧

πS

ιΓ

π∧

Γ2 X

0 Ð→ Λ2 X ÐÐ→ X ⊗2 ÐÐ→ S2 X Ð→ 0,
0 Ð→ Γ2 X ÐÐ→ X ⊗2 ÐÐ→ Λ2 X Ð→ 0,
where π∧ , πS are the canonical projections
ι∧ (x1 x2 ) = x1 ⊗ x2 − x1 ⊗ x2 ,

ιΓ (γ2 (x)) = x ⊗ x

(see [13, ch.1 Section 4.3]). We will identify Γ2 X with the kernel of π∧ for torsion free
groups.
Lemma 5.4. Let M = Z2 be the Z[C]-module with the action of C given by the matrix
1
2n = 4n M for any natural n and M̂ = (Z )2 with the action of
c = ( −1
2
0 −1 ) . Then M(t − 1)
C given by the same matrix. Moreover,
Λ2 M̂ = Λ2 Z2 ⊕ Λ2 Z2 ⊕ Z⊗2
2 ,
(Λ2 M̂ )C = Λ2 Z2 ⊕ 0 ⊕ Γ2 Z2 ,
(Λ2 M̂ )C = 0 ⊕ Λ2 Z2 ⊕ S2 Z2

and the cokernel of the natural map
(Λ2 M̂)C Ð→ (Λ2 M̂ )C
is isomorphic Λ2 Z2 .
1
n
n
n
2
Z2 /dn (Z2 ).
Proof. Set d ∶= c−1 = ( −2
0 −2 ) . Since MI = M(t−1) = d (Z ), we have M̂ = lim
←
Ð
Computations show that d2 = −4c, and hence d2n = (−4)n cn . Since c induces an automorphism on Z2 , we obtain d2n (Z2 ) = 4n Z2 . Thus the filtration dn (Z2 ) of Z2 is equivalent
to the filtration 2n Z2 . It follows that M̂ = (Z2 )2 with the action of C given by c. Then
Λ2 M̂ ≅ (Λ2 Z2 )2 ⊕ Z⊗2
2 . We identify the element

xe1 ∧ x′ e1 + ye2 ∧ y ′e2 + ze1 ∧ z ′ e2 ∈ ∧2 M̂
with the column

(x ∧ x′ , y ∧ y ′ , z ⊗ z ′ )T ∈ (∧2 Z2 )2 ⊕ (Z2 ⊗ Z2 ),
where e1 , e2 is the standard basis of (Z2 )2 over Z2 . Let us present the homomorphism
∧2 ĉ ∶ Λ2 M̂ → Λ2 M̂ that defines the action of C as a matrix. Since
∧2 ĉ(xe1 ∧ x′ e1 ) = xe1 ∧ x′ e1
∧2 ĉ(ye2 ∧ y ′e2 ) = ye1 ∧ y ′e1 + ye2 ∧ y ′ e2 − (ye1 ∧ y ′ e2 − y ′ e1 ∧ ye2 )
∧2 ĉ(ze1 ∧ z ′ e2 ) = −ze1 ∧ z ′ e1 + ze1 ∧ z ′ e2 ,

we obtain

1 1 −π∧
1
0 ),
0 −ι∧ 1

∧2 ĉ = ( 0

0 1 −π∧
0
0 ).
0 −ι∧ 0

∧2 ĉ − 1 = ( 0

ON LENGTHS OF HZ-LOCALIZATION TOWERS

23

Now it is easy to compute that
(Λ2 M̂ )C = Ker(∧2 ĉ − 1) = Λ2 Z2 ⊕ 0 ⊕ Γ2 Z2
and

Im(∧2 ĉ − 1) = Λ2 Z2 ⊕ 0 ⊕ ι∧ (Λ2 Z2 ).

It follows that (Λ2 M̂ )C = Coker(∧2 ĉ − 1) = 0 ⊕ Λ2 Z2 ⊕ S2 Z2 . In order to prove that
Coker((Λ2 M̂)C → (Λ2 M̂ )C ) = Λ2 Z2 ,

2
we have to prove that Coker(Γ2 Z2 → Z⊗2
2 → S Z2 ) = 0. In other words we have to prove
⊗2
that Z2 is generated by elements x ⊗ x and x ⊗ y − y ⊗ x for x, y ∈ Z2 . Fist note that any
2-divisible element is generated by elements

2x ⊗ y = (x ⊗ y − y ⊗ x) + (x + y) ⊗ (x + y) − x ⊗ x − y ⊗ y.
Since any element of Z2 is equal to 2x or 1+2x for some x ∈ Z2 , all other elements of Z2 ⊗Z2
can be presented as sums of elements (1 +2x)⊗(1 +2y) = 1 ⊗1 +2(x⊗1 +1 ⊗y +2x⊗y). 
Theorem 5.5. Let M be the Z[C]-module from lemma 5.4 The HZ-length of the group
G ∶= M ⋊ C = ⟨a, b, t ∣ [a, b] = 1, at = a−1 , bt = ab−1 ⟩
is greater than ω + 1.
Proof. Consider the central sequence
1 → H2 (ηω )(G) → Tω+1 (G) → Ĝ → 1

We will show that the cokernel of the map δ ∶ H3 (Ĝ) → Gab ⊗ H2 (ηω )(G) from (3.1)
contains Λ2 (Z2 ). The theorem will immediately follow.
By proposition 3.1, the map
δ ∶ H3 (Ĝ) → Gab ⊗ H2 (ηω )(G) = H2 (ηω )(G)
factors as

H3 (Ĝ) → (Λ2 M̂ )C → (Λ2 M̂ )C ↠ H2 (ηω )(G)

The direct summand Λ2 Z2 from (Λ2 M̂)C maps isomorphically to a direct summand of
H2 (ηω )(G) = Λ2 Z2 ⊕ (S2 Z2 )/Z. By lemma 5.4, the summand Λ2 Z2 lies in the cokernel of
the map (Λ2 M̂ )C → (Λ2 M̂ )C , therefore, it lies also in the cokernel of the composite map
(Λ2 M̂ )C → H2 (ηω )(G) as well as of the map δ.

Theorem 5.6. Let G be a metabelian group of the form G = M ⋊ C, where M is a tame
C-module and µM = (x − λ1 )m1 . . . (x − λl )ml for some distinct complex numbers λ1 , . . . , λl
and mi ≥ 1.
(1) Assume that the equality λi λj = 1 holds only if λi = λj = 1. Then
HZ-length(G) ≤ ω.
(2) Assume that the equality λi λj = 1 holds only if either mi = mj = 1 or λi = λj = 1.
Then
HZ-length(G) ≤ ω + 1.
Proof. Follows from propositions 5.3, 4.18 and 4.20.



ON LENGTHS OF HZ-LOCALIZATION TOWERS

24

Lemma 5.7. Let M be a tame Z[C]-module such that H2 (M ⋊ C) is finite and
µM = (x − λ1 )m1 . . . (x − λl )ml ,
where λ1 , . . . , λl are distinct complex numbers and mi ≥ 1. Then λi λj = 1 implies mi =
mj = 1.
Proof. Set V ∶= M ⊗C. Then by Lemma 4.1 we get V = V1 ⊕⋅ ⋅ ⋅⊕Vl such that µVi = (x−λi )mi .
The short exact sequence
(Λ2 M)C ↣ H2 (M ⋊ C) ↠ M C
implies that (∧2 M)C is finite. Then (Λ2C V )C = (Λ2 M)C ⊗ C = 0. Assume the contrary,
that there exist i, j such that λi λj = 1 and mi ≥ 2. Consider two cases i = j and i ≠ j.
(1) Assume that i = j. Then λi = λj = −1. Since mi ≥ 2, at least one of Jordan blocks of
aM ⊗ C corresponding to −1 has size bigger than 1. It follows that there is a epimorphism
V ↠ U, where U = C2 and C acts on U by the matrix ( −10 −11 ) . The epimorphism induces a
epimorphism 0 = (Λ2C V )C ↠ (∧2C U)C . From the other hand, a simple computation shows
that (Λ2C U)C ≅ C. So we get a contradiction.
(2) Assume that i ≠ j. Then λi ≠ λj . Because of the isomorphism of C[C]-modules
Λ2C V = (⊕ Λ2C Vk ) ⊕ ( ⊕ Vk ⊗C Vk′ ),
k

k<k ′

we have an epimorphism of Z[C]-modules Λ2C V ↠ Vi ⊗C Vj . Since mi ≥ 2, there is an
epimorphism Vi ↠ Ui , where Ui = C2 and C acts on Ui by the matrix ( λ0i λ1i ) . Moreover,
there is an epimorphism Vj ↠ Uj , where Uj = C and C acts on Uj by the multiplication
on λj . It follows that C acts on Ui ⊗C Uj by the matrix ( 10 λ1j ) . Thus (Ui ⊗C Uj )C ≅ C.
From the other hand we have an epimorphism 0 = (Λ2C V )C ↠ (Ui ⊗ Uj )C . So we get a
contradiction.

Proposition 5.8. Let G be a metabelian finitely presented group of the form G = M ⋊ C
for some Z[C]-module M and H2 (G) is finite. Then
HZ-length(G) ≤ ω + 1.
Proof. It follows from Lemma 5.7 and Theorem 5.6.



6. Bousfield’s method
Let G be a finitely presented group given by presentation
⟨x1 , . . . , xm ∣ r1 , . . . , rk ⟩
Consider the free group F = F (x1 , . . . , xn ) and an epimorphism F → G with the kernel
normally generated by k elements ker F → G = ⟨r1 , . . . , rk ⟩F . Here we will study the
induced map
h ∶ H2 (F̂ ) → H2 (Ĝ).

(6.1)

ON LENGTHS OF HZ-LOCALIZATION TOWERS

25

We follow the scheme due to Bousfield from [5]. Let
Ð→

Ð→

d0 ,d1

Ð→

F∗ ∶

. . . Ð→ F1 Ð→ F0 (= F ) → G
←Ð
←Ð

←Ð

be a free simplicial resolution of G, where F1 a free group with m + k generators. The
structure of F1 is F (y1 , . . . , yk ) ∗ F , and the maps d0 , d1 are given as
id

d0 ∶ yi ↦ 1, F ↦ F
id

d 1 ∶ y i ↦ ri , F ↦ F
The following short exact sequence follows from Lemma 5.4 and Proposition 3.13 in [5]:
̂∗ ) → Ĝ → 1
1 → lim 1 π1 (F∗ /γk (F∗ )) → π0 (F
←Ð k

(6.2)

The first homotopy group π1 (F∗ /γk (F∗ )) is isomorphic to the kth Baer invariant known
also as k-nilpotent multiplicator of G (see [7], [11]). If G = F /R for a normal subgroup
R ⊲ F , the Baer invariant can be presented as the quotient
π1 (F∗ /γ∗(F∗ )) ≃

R ∩ γk (F )
.
[R, F, . . . , F ]
´¹¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹¶
k−1

1

Now assume that, for a group G, the lim -term vanishes in (6.2), that is, there is a natural
←Ð
isomorphism
̂∗ ) ≃ Ĝ
π0 (F
There is the first quadrant spectral sequence (see [4], page 108)
1
̂∗ ))
= Hq (F̂p ) ⇒ Hp+q (W (F
Ep,q
r
r
→ Ep−r,q+r−1
dr ∶ Ep,q

As a result of convergence of this spectral sequence, we have the following diagram:
1
E0,1
= H2 (F̂ )

// //

∞
E0,1

(6.3)




̂∗ ))
H2 (W (F

// //

H2 (Ĝ)


∞
E1,0
∞ is a quotient of π ((F ) ) =
Since F1 is finitely generated, H1 (F̂1 ) ≃ H1 (F1 ), therefore, E1,0
1
∗ ab
H2 (G). Now the diagram (6.3) implies the following

ON LENGTHS OF HZ-LOCALIZATION TOWERS

26

Lemma 6.1. Assuming that lim 1 -term in (6.2) vanishes, the cokernel of the map (6.1)
←Ð
is isomorphic to a quotient of the homology group H2 (G). For a group G with H2 (G) = 0,
the map h ∶ H2 (F̂ ) → H2 (Ĝ) is an epimorphism.
7. Main construction
Lemma 7.1. Let K = ⟨a, b ∣ [[a, b], a] = [[a, b], b] = 1⟩ be the free 2-generated nilpotent
group of class 2 and C = ⟨t⟩ acts on K by the following automorphism
a ↦ a−1
b ↦ ab−1 .
Set K̃ ∶= lim K/γn (K ⋊ C). Then the following holds
←Ð
(1) the pronilpotent completion of K ⋊ C is equal to K̃ ⋊ C;
(2) Kab is isomorphic to the Z[C]-module from Lemma 5.4;
≅ ̂
(3) the obvious map (K̃)ab Ð
→ (K
ab ) is an isomorphism of Z[C]-modules;
(4) either [K̃, K̃] ≅ Z2 or [K̃, K̃] ≅ Z/2m for some m.

Proof. For the sake of simplicity we set Kn ∶= γn (K ⋊ C). (1) follows from the equality
(K ⋊ C)/Kn = (K/Kn ) ⋊ C

for n ≥ 2. (2) is obvious. Prove (3) and (4).
It is obvious that γ2 (K) = Z(K) = {[a, b]k ∣ k ∈ Z} ≅ Z and [ai , bj ] = [a, b]ij for
all i, j ∈ Z. Moreover, any element of K can be uniquely presented as ai bj [a, b]k for
i, j, k ∈ Z. Set M = Kab . We consider M with the additive notation as a module over
C. Then γn+1 (M ⋊ C) = M(t − 1)n . Lemma 5.4 implies that γ2n+1 (M ⋊ C) = 4n M. Since
the map K2n+1 ↠ γ2n+1 (K ⋊ C) is an epimorphism, there exist k, l ∈ Z (that depend of
n
n
2n
n
n
n) such that a4 [a, b]k , b4 [a, b]l ∈ K2n+1 . Since, [a, b]4 = [a4 [a, b]k , b4 [a, b]l ], we obtain
2n
m(2n+1)
[a, b]4 ∈ K2n+1 . Then K2n+1 ∩ γ2 (K) = ⟨[a, b]2
⟩ for some natural number m(2n + 1)
m(n)
2n
because 4 is divisible only on powers of 2. Hence Kn ∩ γ2 (K) = ⟨[a, b]2
⟩ for some
m(n). Then the short exact sequence γ2 (K) ↣ K ↠ M induces the short exact sequence
ιn

0 Ð→ Z/2m(n) ÐÐÐ→ K/Kn Ð→ M/M(t − 1)n Ð→ 1,
where ιn (1) = [a, b]. Since Z/2m(n) and M/M(t − 1)n are finite 2-groups, the order of
′
K/Kn is equal to 2m (n) for some m′ (n). We obtain a short exact sequence
0 Ð→ lim Z/2m(n) ÐÐ→ K̃ Ð→ M̂ Ð→ 1.
←Ð
If m(n) → ∞, then lim Z/2m(n) = Z2 , else m(n) stabilizes and lim Z/2m(n) = Z/2m .
←Ð
←Ð
Now it is sufficient to prove that [K̃, K̃] = Im(ι̂). Since M̂ is abelian, [K̃, K̃] ⊆ Im(ι̂).
Prove that [K̃, K̃] ⊇ Im(ι̂). Any element of K̃ can be presented as a sequence (xn )∞
n=1 ,
m(n)
where xn ∈ K/Kn such that xn ≡ xn+1 mod Kn . Any element of lim Z/2
can be
←Ð
∞
k
presented as an image of a 2-adic integer ∑k=0 αk 2 , where αk ∈ {0, 1}. Then an element
of Im(ι̂) can be presented as a sequence
ι̂

([a, b]∑k=0

m(n)

αk 2k ∞
)n=1

= ([a∑k=0

m(n)

αk 2k

, b])∞
n=1 .

ON LENGTHS OF HZ-LOCALIZATION TOWERS

27

2
Note that the element (a∑k=0 αk 2 )∞
n=1 is a well defined element of K̃ because a
It follows that
m(n)
m(n)
k
∑k=0 αk 2k ∞
)n=1 , (b)∞
([a, b]∑k=0 αk 2 )∞
n=1 ],
n=1 = [(a
m′ (n)

m′ (n)

k

and hence, [K̃, K̃] ⊇ Im(ι̂).

∈ Kn .



Theorem 7.2. Let F be a free group of rank ≥ 2. Then HZ-length(F ) ≥ ω + 2.

Proof. Consider the following group:
Γ ∶= ⟨a, b, t ∣ [[a, b], a] = [[b, a], b] = 1, at = a−1 , bt = ab−1 ⟩
Observe that Γ is the semidirect product K ⋊ C from the previous lemma.
Consider the natural diagram induced by the projection K → Kab = M, i.e. by Γ =
K ⋊ C ↠ G = M ⋊ C:
H3 (Γ̂)

δ



H3 (Ĝ)

δ

//

H1 (Γ) ⊗ H2 (ηω )(Γ)

(7.1)

◗ ◗
◗ ◗
◗ ◗
◗((

// H1 (G) ⊗ H2 (ηω )(G)
// // Coker(δ)(G)

It is shown in the proof of theorem 5.5 that Coker(δ)(G) contains Λ2 Z2 . This term
is an epimorphic image of one of the terms Λ2 Z2 in

Λ2 Z2

H2 (M̂ ) = Λ2 M̂ = Λ2 Z2 ⊕ Λ2 Z2 ⊕ Z2 ⊗ Z2
(see lemma 5.4). By lemma 7.1, there is a short exact sequence
H2 (K̃) → H2 (M̂ ) ↠ [K̃, K̃]

Now, by lemma 7.1, [K̃, K̃] is either Z2 or a finite group, therefore, the image of both
terms Λ2 Z2 in [K̃, K̃] are zero (2-adic integers do not contain divisible subgroups). In
particular, the term Λ2 Z2 which maps isomorphically onto a subgroup of Coker(δ)(G)
lies in the image of H2 (K̃). The natural square
H2 (K̃)

// //



H2 (M̂ )

H2 (ηω )(Γ)


// //

H2 (ηω )(G)

is commutative and we conclude that the diagonal arrow in (7.1) maps onto the subgroup
Λ2 Z2 of Coker(δ)(G). Hence, Coker(δ)(Γ) maps epimorphically onto Λ2 Z2 .
Now lets prove that the second homology H2 (Γ) is finite. Since the group Γ is the
semi-direct product K ⋊ C, its homology is given as
0 → H2 (K)C → H2 (Γ) → H1 (C, Kab ) → 0

The right hand term is zero: H1 (C, Kab ) = H1 (C, M) = M C = 0. It follows immediately
that H2 (K)C = (γ3 (F (a, b))/γ4 (F (a, b)))C = Z/4.

ON LENGTHS OF HZ-LOCALIZATION TOWERS

28

Observe that the group Γ can be defined with two generators only. Let F be a free
group of rank ≥ 2. Consider a free simplicial resolution of Γ with F0 = F : F∗ → Γ. Since
H2 (Γ) is finite, all Baer invariants of Γ are finite (see, for example, [11])
lim 1 π1 (F∗ /γk (F∗ )) = 0
←Ð

and, by lemma 6.1, the cokernel of the natural map H2 (F̂ ) → H2 (Γ̂) is finite.
We have a natural commutative diagram:
H3 (F̂ )

δ

H1 (F ) ⊗ H2 (ηω )(F )
//



H3 (Γ̂)

// //

Coker(δ)(F )



δ

//

H1 (Γ) ⊗ H2 (ηω )(Γ)

(7.2)


// //

Coker(δ)(Γ)

with H2 (ηω )(F ) = H2 (F̂ ). Since the cokernel of H2 (F̂ ) → H2 (Γ̂) is finite,
Coker{Coker(δ)(F ) → Coker(δ)(Γ)}
also is finite. However, Coker(δ)(Γ) maps onto Λ2 Z2 , as we saw before, hence Coker(δ)(F )
is uncountable. Therefore, by (3.3), H2 (Tω+1 (F )) ≠ 0 and the statement is proved.


8. Alternative approaches
In general, given a group, the description of its pro-nilpotent completion is a difficult
problem. If a group is not pre-nilpotent, its pro-nilpotent is uncountable and may contain
complicated subgroups. In this paper, as well as in [14] we essentially used the explicit
structure of pro-nilpotent completion for metabelian groups. Now we observe that, there
is a trick which gives a way to show that some groups have HZ-length greater than ω
without explicit description of their pro-nilpotent completion. In a sense, the Bousfield
scheme described above also gives such a method, however in that way one must compare
the considered group with a group with clear completion. The trick given bellow is
different.
We put Φi H2 (G) = Ker{H2 (G) → H2 (G/γi+1 (G))}. Then Φi H2 (G) is the Dwyer filtration on H2 (G) (see [8]).
Proposition 8.1. Let G be a group with the following properties:
(i) γω (G) ≠ γω+1 (G)
(ii) ∩i Φi H2 (G/γω (G)) = 0.
Then the HZ-length of G is greater than ω.

ON LENGTHS OF HZ-LOCALIZATION TOWERS

29

Proof. All ingredients of the proof are in the following diagram with exact horizontal and
vertical sequences:
H2 (G)

▼▼▼
▼▼▼
▼▼▼
▼▼&&

//
// H2 (G/γω (G))

ker //



∩i Φi H2 (G/γω (G))

H2 (Ĝ)



γω (G)/γω+1 (G)

The fact that the kernel lies in ∩i Φi H2 (G/γω (G)) follows from the standard property of
Dwyer’s filtration: the map G/γω (G) → Ĝ induces isomorphisms on H2 /Φi for all i (see
[8]). The vertical exact sequence is the part of 5-term sequence in homology. Now assume
that the map H2 (G) → H2 (Ĝ) is an epimorphism. Then H2 (G/γω (G)) → H2 (Ĝ) is an
epimorphism as well. However, condition (ii) implies that the last map is a monomorphism
as well. Hence, it is an isomorphism and surjectivity of H2 (G) → H2 (Ĝ) contradicts the
property (i).

An example of a group with satisfies both conditions (i) and (ii) from proposition, is
the group
⟨a, b ∣ [a, b3 ] = [[a, b], a]2 = 1⟩

(see [16], examples 1.70 and 1.85). However, in this example, G/γω (G) is metabelian and
one can use explicit description of its pro-nilpotent completion to show the same result.
The above proposition can be used for more complicated groups.
Now we consider another example of finitely generated metabelian group of the type
M ⋊ C with HZ-length greater than ω + 1. In our example, M = Z[C] with a regular
action of C as multiplication. This is not a tame Z[C]-module and the group is not
finitely presented. The proof of the bellow result uses functorial arguments.
Theorem 8.2. Let
G = Z[C] ⋊ C = Z ≀ Z = ⟨a, b ∣ [a, ab ] = 1, i ∈ Z⟩,
i

then HZ-length(G) ≥ ω + 2.
Proof. We define a functor from the category fAb of finitely generated free abelian groups
to the category of groups as follows:
F ∶ A ↦ (A ⊗ M) ⋊ C, A ∈ fAb,
where the action of C on A is trivial and M = Z[C]. Now our main example can be
written as G = F (Z). Since the action of C on A is trivial, the pro-nilpotent completion
of F (A) can be described as follows:
̂
F
(A) = (A ⊗ M̂) ⋊ C.

One can easily see that

H1 (C, Hi (A ⊗ M̂ )) = 0, i ≥ 1.

ON LENGTHS OF HZ-LOCALIZATION TOWERS

30

Since the abelian group A ⊗ M̂ is torsion-free,
̂
Hi (F
(A)) = Λi (A ⊗ M̂ )C , i ≥ 1.
Now we have
H2 (ηω )(F (A)) = Coker{Λ2 (A ⊗ M)C → Λ2 (A ⊗ M̂ )C }.
Now consider the functor from the category of finitely generated free abelian groups to
all abelian groups
G ∶ A ↦ H2 (ηω )(F (A))
It follows immediately that G is a quadratic functor. Indeed, it is a proper quotient of
the functor A ↦ A ⊗ A ⊗ B for a fixed abelian group B.
The exact sequence (3.3) applied to the stem-extension
̂
1 → H2 (ηω )(F (A)) → Tω+1 (F (A)) → F
(A) → 1
can be viewed as an exact sequence of functors in the category sAb. Consider the map δ
̂
as a natural transformation in sAb. The functor H3 (F
(A)) = Λ3 (A ⊗ M̂)C is a quotient
of the cubic functor
A⊗A⊗A⊗D
for some fixed abelian group D (which equals to the tensor cube of M̂ ). Recall that any
natural transformation of the form
A ⊗ A ⊗ A ⊗ D → quadratic functor
is zero. This follows from the simple observation that A⊗A⊗A⊗D is a natural epimorphic
image of its triple cross-effect. See, for example, [17] for generalizations and detailed
discussion of this observation.
Therefore, for any non-zero A, the map δ in (3.3) is zero and H2 (Tω+1 (F (A))) contains a subgroup H2 (ηω )(F (A)) which is uncountable. In particular, H2 (Tω+1 (G)) =
H2 (Tω+1 (F (Z)) is uncountable and hence H2 (G) → H2 (Tω+1 (G)) is not an epimorphism.

Remark 8.3. If we change the group Z ≀ Z by lamplighter group by adding one more
relation, we will obtain another wreath product (for n ≥ 2)
Z/n ≀ Z = ⟨a, b ∣ an = 1, [a, ab ] = 1, i ∈ Z⟩
i

One can use the scheme of this paper to prove that HZ-length(Z/n≀Z) > ω+1. Essentially it
follows from the triviality of Λ2 (M̂ )C as in Theorem 8.2. As it was shown by G. Baumslag
[1], there are different ways to embed this group into a finitely-presented metabelian group.
For example, Z/n ≀ Z is a subgroup generated by a, b in
G[n] ∶= ⟨a, b, c ∣ an = [a, ab ] = [a, ab ] = [b, c] = 1, ac = aab a−b ⟩.
2

2

Now we have a decomposition G[n] = (Z/n)⊕∞ ⋊ (C × C) and it follows immediately from
[14] that HZ-length(G[n] ) = ω, since H2 (ηω )(G[n] ) is divisible.

ON LENGTHS OF HZ-LOCALIZATION TOWERS

31

9. Concluding remarks
Lemma 9.1. For any prime p and n ≥ 2 the embedding Zp ↪ Qp induces an isomorphism
Λn Zp ≅ Λn Qp .
In particular, Λn Zp is a Q-vector space of countable dimension.
⊗n
Proof. Since Zp and Qp are torsion free, the map Z⊗n
p → Qp is a monomorphism. Since
for torsion free groups the exterior power is embedded into the tensor power, we obtain
that the map Λn Zp → Λn Qp is a monomorphism. So we can identify Λn Zp with its image
in Λn Qp . Since Qp = ⋃ p1i ⋅ Zp , we obtain Λn Qp = ⋃ p1i ⋅ Λn Zp . Then it is enough to prove
that Λn Zp is p-divisible. For any a ∈ Zp we consider the decomposition a = a(0) + a(1) ,
where a(0) ∈ {0, . . . , p − 1} and a(1) = p ⋅ b for some b ∈ Zp . Then for any a1 , . . . , an ∈ Zp
(i )
(i )
we have a1 ∧ ⋅ ⋅ ⋅ ∧ an = ∑ a1 1 ∧ ⋅ ⋅ ⋅ ∧ an n , where (i1 , . . . , in ) runs over {0, 1}n . For any
(i )
(i )
sequence (i1 , . . . , in ) except (0, . . . , 0) the element a1 1 ∧ ⋅ ⋅ ⋅ ∧ an n is p divisible because
(1)
(0)
(0)
ai is p-divisible. Since Λn Z = 0, we get a1 ∧ ⋅ ⋅ ⋅ ∧ an = 0. It follows that a1 ∧ ⋅ ⋅ ⋅ ∧ an is
p-divisible. Thus Λn Zp is p-divisible.


Remind that the Klein bottle group is given by GKl = Z ⋊ C, where C acts on Z by the
multiplication on −1. Its pronilpotent completion is equal to Z2 ⋊ C. Consider the map
w ∶ Z2 × Z2 → Λ2 Z2 given by w(a, b) = 21 a ∧ b. Here we use that Λ2 Zp is a Q-vector space.
It is easy to see that w is a 2-cocycle and we get the corresponding central extension
Λ2 Z2 ↣ Nw ↠ Z2 ,
whose underlying set is (Λ2 Z2 ) × Z2 and the product is given by
(α, a)(β, b) = (α + β +

1
⋅ a ∧ b, a + b) .
2

We define the action of C on Nw by the map (α, a) ↦ (α, −a).
Proposition 9.2. There is an isomorphism
EGKl = Nw ⋊ C.
In other words, EGKl can be described as the set (Λ2 Z2 ) × Z2 × C with the multiplication
given by
(α, a, ti )(β, b, tj ) = (α + β +

(−1)i
⋅ a ∧ b, a + (−1)i b, ti+j ) .
2

Proof. Set G ∶= GKl . Then Ĝ = Z2 ⋊ C. The Lyndon-Hochschild-Serre spectral sequences
≅
imply that H2 (G) = 0 and the map Λ2 Z2 = H2 (Z2 ) Ð
→ H2 (Ĝ) is an isomorphism. Theorem 5.6 implies that Tω+1 G = EG. Then EG is the universal relative central extension
(H2 (ηω ) ↣ EG ↠ Ĝ, ηω+1 ). Since H2 (ηω ) = Coker{H2 (G) → H2 (Ĝ)} and H2 (G) = 0,
≅
≅
we obtain that H2 (Z2 ) Ð
→ H2 (Ĝ) Ð
→ H2 (ηω ) are isomorphisms. The continious maps

ON LENGTHS OF HZ-LOCALIZATION TOWERS

32

BZ2 → B Ĝ → Cone(Bηω ) give the commutative diagram for any abelian group A ∶
H 2 (ηω , A)

Hom(H2 (ηω ), A)

≅ //

(9.1)

≅




H 2 (Ĝ, A)
//

Hom(H2 (Ĝ), A)
≅



H 2 (Z2 , A)
//



Hom(H2 (Z2 ), A)

If A is a divisible abelian group, then Ext(H1 (Ĝ), A) = 0 = Ext(Z2 , A), and all morphisms
in the diagram (9.1) are isomorphisms. In particular, if A = H2 (ηω ) = H2 (Ĝ) = Λ2 Z2 , then
the morphisms induce isomorphisms
H 2 (ηω , Λ2 Z2 ) ≅ H 2 (Ĝ, Λ2 Z2 ) ≅ H 2 (Z2 , Λ2 Z2 ) ≅ Hom(Λ2 Z2 , Λ2 Z2 )

and the extension Λ2 Z2 ↣ EG ↠ Ĝ corresponds to the identity map in Hom(Λ2 Z2 , Λ2 Z2 ).
Therefore, it is sufficient to prove that the extension Λ2 Z2 ↣ Nw ⋊ C ↠ Ĝ goes to the
identity map via the composition
H 2 (Ĝ, Λ2 Z2 ) → H 2 (Z2 , Λ2 Z2 ) → Hom(Λ2 Z2 , Λ2 Z2 ).

The map H 2 (Ĝ, Λ2 Z2 ) → H 2 (Z2 , Λ2 Z2 ) on the level of extensions is given by the pullback.
It follows that the extension Λ2 Z2 ↣ Nw ⋊ C ↠ Ĝ goes to the extension Λ2 Z2 ↣ Nw ↠ Z2 .
Then we need to prove that w goes to the identity map under the map H 2 (Z2 , Λ2 Z2 ) →
Hom(Λ2 Z2 , Λ2 Z2 ).
For any group H and an abelian group A the map H 2 (H, A) → Hom(H2 (H), A) on
the level of cocycles is induced by the evaluation map C 2 (H, A) → Hom(C2 (H), A) given
by u ↦ ((h, h′ ) ↦ u(h, h′ )). For an abelian group H the map Λ2 H → H2 (H) given by
h ∧ h′ ↦ (h, h′ ) − (h′ , h) + B2 (H) is an isomorphism Λ2 H ≅ H2 (H) (see [6, Ch.V §5-6]).
Then the map H 2 (H, A) → Hom(Λ2 H, A) is given by u ↦ (h ∧ h′ ↦ u(h, h′ ) − u(h′ , h)).
Since w(a, b) − w(b, a) = 21 (a ∧ b − b ∧ a) = a ∧ b, w goes to the identity map.


Remark 9.3. If we identify EGKl with the set (Λ2 Z2 ) × Z2 × C, it is easy to check that
γ2 (EGKl ) = (Λ2 Z2 ) × 2Z2 × 1 and [γ2 (EGKl ), γ2 (EGKl )] = (Λ2 Z2 ) × 0 × 1. It follows that
EGKl is a solvable group of class 3 but it is not metabelian. In particular, the class of
metabelian groups is not closed under the HZ-localization.
Finishing this paper, we mention some possibilities of generalization of the obtained
results. We conjecture that, the tame Z[C]-module M defined by the k × k-matrix (k ≥ 2)
⎞
⎛
⎜−1 1 0 . . . 0 ⎟
⎟
⎜
⎟
⎜
⎜ 0 −1 1 0 . . .⎟
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜. . .
⎟
⎜
⎟
⎜
⎟
⎜
⎝ 0 . . . 0 0 −1⎠

ON LENGTHS OF HZ-LOCALIZATION TOWERS

33

defines the group M ⋊C with HZ-length ω +l(k), where l(k) ∈ N and l(k) → ∞ for k → ∞.
With the help of this group, one can try to use the same scheme as in this paper to prove
that HZ-length of a free non-cyclic group is ≥ 2ω.
Acknowledgement
The research is supported by the Russian Science Foundation grant N 16-11-10073.
References
[1] G. Baumslag: Subgroups of finitely presented metabelian groups, J. Austral. Math. Soc. 16 (1973),
98–110, Collection of articles dedicated to the memory of Hanna Neumann.
[2] R. Bieri and K. Strebel: Almost finitely presented soluble groups, Comm. Math. Helv. 53 (1978),
258–278.
[3] R. Bieri and R. Strebel: A geometric invariant for modules over an abelian group, J. Reine Angew.
Math. 322 (1981), 170–189.
[4] A. K. Bousfield and D. Kan: Homotopy limits, completions and localizations, Lecture Notes in
Mathematics 304, (1972).
[5] A. K. Bousfield: Homological localization towers for groups and π-modules, Mem. Amer. Math. Soc,
no. 186, 1977.
[6] K. Brown, Cohomology of Groups, Springer-Verlag GTM 87, 1982.
[7] J. Burns and G. Ellis: On the nilpotent multipliers of a group, Math. Z. 226 (1997), 405–428.
[8] W. Dwyer: Homology, Massey products and maps between groups, J. Pure Appl. Algebra, 6 (1975),
177–190.
[9] B. Eckmann, P. Hilton and U. Stammbach: On the homology theory of central group extensions: I
- the commutator map and stem extensions, Comm. Math. Helv. 47, (1972), 102–122.
[10] B. Eckmann, P. Hilton and U. Stammbach: On the homology theory of central group extensions: II
- the exact sequence in the general case, Comm. Math. Helv. 47, (1972), 171–178.
[11] G. Ellis: A Magnus-Witt type isomorphism for non-free groups, Georgian Math. J. 9 (2002), 703–708.
[12] L. Fuchs: Infinite abelian groups, Academic Press, New York and London.
[13] L.Illusie: Complexe Cotangent et Déformation I, Lecture Notes in Mathematics, vol. 239, Springer,
Berlin, 1971.
[14] S. O. Ivanov and R. Mikhailov: On a problem of Bousfield for metabelian groups: Advances in Math.
290 (2016), 552–589.
[15] F. Kasch: Modules and Rings, Acd. Press, London New York, 1982.
[16] R. Mikhailov and I. B. S. Passi: Lower Central and Dimension Series of Groups, LNM Vol. 1952,
Springer 2009.
[17] R. Mikhailov: Polynomial functors and homotopy theory, Progress in Math, 311 (2016), arXiv:
1202.0586.
[18] C. Weibel: An Introduction to Homological Algebra, Cambridge Univ. Press, 1994.
[19] O. Zariski and P. Samuel, Commutative Algebra, Vol. 2, Van Nostrand, Princeton 1960.
Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178 Russia
E-mail address: ivanov.s.o.1986@gmail.com
Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178 Russia and St. Petersburg Department of Steklov Mathematical Institute
E-mail address: rmikhailov@mail.ru

