Extensions of locally compact abelian, torsion-free groups by compact torsion abelian groups

0.09 MB·

by  Hossain Sahleh

Extensions of locally compact abelian, torsion-free groups by
compact torsion abelian groups
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1 HosseinSAHLEHandAliAkbarALIJANI
0
2

r
Department of Pure Mathematics, Faculty of Mathematical Sciences,
a
M University of Guilan, Rasht, Iran
[email protected]

8

] Abstract. Let X be a compact torsion abelian group. In this paper, we
R showthatanextensionofFpbyX splitswhereFp isthep-adicnumbergroup
G and p a prime number. Also, we show that an extension of a torsion-free,
non-divisibleLCAgroupbyX isnotsplit.
.
h
t Introduction
a
m
Throughout, all groups are Hausdorff abelian topological groups and will be
[
written additively. Let £ denote the category of locally compact abelian (LCA)

2 groups with continuous homomorphisms as morphisms. The Pontrjagin dual of a
v group G is denoted by Gˆ. A morphism is called proper if it is open onto its image
3 φ ψ
and a short exact sequence 0 → A → B → C → 0 in £ is said to be proper exact
8
0 if φ and ψ are proper morphisms. In this case the sequence is called an extension
7 of A by C ( in £). Following [5], we let Ext(C,A) denote the (discrete) group of
1. extensionsofAbyC. ThesplittingprobleminLCAgroupsisfindingconditionson
0 A and C under which Ext(C,A) = 0. In [3, 4, 6] the splitting problem is studied.
4 We havestudied the splitting problemin the categoryofdivisible, LCA groups[9].
1 By using the splitting problem, we determined the LCA groups G such that the
:
v maximaltorsionsubgroupofGisclosed[10]. LetX beacompacttorsiongroup. In
i [4,Theorem1],itisprovedthatifGisadivisibleLCAgroup,thenExt(X,G)=0.
X
However,thesuggestedproofin[4]appearstobeincompleteasitusestheincorrect
r
a Proposition 8 of [3]. In [9], we proved that if G is a divisible, σ−compact group,
then Ext(X,G)=0. Let P be the set of all prime numbers, J , the p-adic integer
p
group and F , the p-adic number group which is the minimal divisible extension
p
of J for every p ∈ P [7]. By [7, 25.33], a divisible, torsion-free LCA group G
p
has the form G ∼= Rn A M E, where A is a discrete, torsion-free, divisible
group, M a compactLconnLectedLtorsion-free group and E, the minimal divisible
extension of Jnp where n is a cardinal number for every p∈P. In general,
p∈P p p
E 6= FQnp. Let I be a finite subset of P such that
p∈P p
Q
0 if p6∈I
n =
p (cid:26) 1 if p∈I
2010 Mathematics Subject Classification. 20K35,22B05.
Keywordsandphrases. locallycompactabeliangroup,extension,divisiblegroup,torsion-free
group.
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2 H.SAHLEHandA.A.ALIJANI
Then the minimal divisible extension of Jnp is Fnp. In this paper,
p∈I p p∈I p
weshowthatifG∼=Rn A M p∈IFQpnp,thenExQt(X,G)=0(seeTheorem
1.6). L L LQ
The additive topological group of real numbers is denoted by R, Q is the group
ofrationaleswith discrete topologyandZ is the groupofintegers. The topological
isomorphismwill be denote by ” ∼= ”. For more on locally compact abelian groups
see [7].
1. Extensions of a torsion-free LCA group by a compact torsion
abelian group
Lemma 1.1. Let X ∈£ and p a prime number. Then nExt(X,F )=Ext(X,F )
p p
for every positive integer n.
Proof. Let n be a positive integer and f : F → F , f(x) = nx for all x ∈ F . By
p p p
[1, Lemma 2], f is open. So f is a proper morphism. Consider the exact sequence
f
0→Kerf →F −→F →0. By Corollary2.10 of [5], we have the exact sequence
p p
(1.1) →Ext(X,Kerf)→Ext(X,F )−f→∗ Ext(X,F )→0
p p
Sincef∗(Ext(X,Fp))=nExt(X,Fp),itfollowsfromsequence(1.1)thatnExt(X,Fp)=
Ext(X,F ). (cid:3)
p
Lemma 1.2. Let X be a compact torsion group. Then Ext(X,F )=0.
p
Proof. F isatotally disconnectedgroup. So,by Theorem24.30of[7], F contains
p p
a compact open subgroup K. Now we have the following exact sequence
(1.2) ...→Ext(X,K)→Ext(X,F )→Ext(X,F /K)→0
p p
Since F is divisible, so Ext(X,F /K) = 0 (see [5, Theorem 3.4]). Since X is
p p
compact and torsion, so by [7, Theorem 25.9], nX = 0 for some positive inte-
ger n. Hence, nExt(X,K) = 0 (see [8, Lemma 2.5]). Since (1.2) is exact, so
nExt(X,F )=0. Hence by Lemma 1.1, Ext(X,F )=0. (cid:3)
p p
Remark 1.3. Let X be a group and f : X → X,f(x) = nx for all x ∈ X. If
f is a topological isomorphism for every positive integer n, then X is a divisible,
torsion-free group.
Theorem1.4. LetX beacompact groupandpaprimenumber. Then Ext(X,F )
p
is a divisible, torsion-free group.
×n
Proof. Let n be a positive integer. Then the exact sequence 0 → X → X →
X/nX →0 induces the following exact sequence
×n
Ext(X/nX,F )→Ext(X,F ) → Ext(X,F )→0
p p p
×n
ByLemma 1.2, Ext(X/nX,F )=0. SoExt(X,F ) → Ext(X,F )is a topological
p p p
isomorphism. Hence by Remark 1.3, Ext(X,F ) is a divisible, torsion-free group.
p
(cid:3)
Corollary 1.5. Let X ∈£. Then Ext(X,F ) is a divisible, torsion-free group.
p
Extensions of locally compact abelian, torsion-free groups by compact torsion abelian groups 3
Proof. Let X ∈ £. By [7, Theorem 24.30], X = Rn H where H contains a
compact open subgroup K. Consider the exact sequenceL
Ext(H/K,F )→Ext(H,F )→Ext(K,F )→0
p p p
Since H/K is a discrete group and F a divisible group, so Ext(H/K,F ) = 0.
p p
∼
Hence Ext(H,Fp) = Ext(K,Fp). By Theorem 1.4, Ext(K,Fp) is a divisible,
torsion-free group. So Ext(X,F ) is a divisible, torsion-free group. (cid:3)
p
Theorem1.6. LetX beacompacttorsiongroupandG∼=Rn A M p∈IFpnp
where I is a finite subset of P defined as follows: L L LQ
0 if p6∈I
n =
p (cid:26) 1 if p∈I
Then Ext(X,G)=0.
Proof. First recall that by [5, Theorem 2.13],
Ext(X,G)∼=Ext(X,A) Ext(X,M) Ext(X,Fp)
M MpY∈I
Since X is a totally disconnected group, so by [5, Theorem 3.4], Ext(X,A) = 0.
Also Ext(X,M) ∼= Ext(Mˆ,Xˆ). Since Xˆ is a discrete bounded group and Mˆ a
discrete torsion-free group, so by [2, Theorem 27.5],Ext(Mˆ,Xˆ) = 0. By Lemma
1.2, Ext(X,F )=0. Hence Ext(X,G)=0. (cid:3)
p
Lemma 1.7. Let X be a compact torsion group. Then Hom(X,Q/Z)∼=Xˆ.
Proof. Theexactsequence0→Z→Q→Q/Zinducesthefollowingexactsequence
Hom(X,Q)→Hom(X,Q/Z)→Ext(X,Z)→Ext(X,Q)
Since X is torsion and Q is torsion-free, so Hom(X,Q) = 0. Also by [5, Theorem
3.4], Ext(X,Q)=0. Hence Hom(X,Q/Z)∼=Ext(X,Z). By [5, Theorem 2.12 and
Proposition 2.17], Ext(X,Z)∼=Ext(Zˆ,Xˆ)∼=Xˆ. So Hom(X,Q/Z)∼=Xˆ. (cid:3)
Theorem1.8. LetX beacompacttorsiongroupandGatorsion-free,non-divisible
group. Then Ext(X,G)6=0.
Proof. LetG∗ be the minimaldivisible extensionofG. By [7, A.13], G∗ is a divisi-
ble,torsion-freegroup. SinceX istorsionandG∗ torsion-free,soHom(X,G∗)=0.
By [5, Corollary 2.10], we have the following exact sequence
0=Hom(X,G∗)→Hom(X,G∗/G)→Ext(X,G)
Since G∗/G is a discrete, torsion divisible group, so Hom(X,G∗/G) containing a
copy of Hom(X,Q/Z). Hence by Lemma 1.7, Ext(X,G)6=0. (cid:3)
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