Abstract: Let be the class of all torsion-free, locally compact abelian (LCA) groups. In this paper, we determine the LCA groups such that every pure extension splits for each in .
keywords: Pure extension, split, torsion-free,locally compact abelian group.
MSC 2000: Primary 22B05, Secondary 22A05
Throughout, all groups are Hausdorff abelian topological groups and will be written additively. Let £ denote the category of locally compact abelian (LCA) groups with continuous homomorphisms as morphisms. The Pontrjagin dual of a group is denoted by . A morphism is called proper if it is open onto its image, and a short exact sequence in £ is said to be proper exact if and are proper morphisms. In this case, the sequence is called an extension of by . A subgroup of a group is called pure if for all positive integers . An extension is called a pure extension if is pure in . Following Fulp and Griffith , we let denote the (discrete) group of extensions of by . A subgroup of a group is called pure if for all positive integers . The elements represented by pure extensions of by form a subgroup of which is deenoted by . Assume that is any subcategory of £. This paper is part of an investigation which answers the following question:
under what conditions on , or for all Some answers are given in , in the category of locally compact abelian groups. Let be the category of all torsion-free, locally compact abelian groups. In , Fulp has determined the LCA groups such that for all . In this paper, we determine the discrete or compact groups satisfying for all .
The additive topological group of real numbers is denoted by , is the group of rationals , is the group of integers and is the cyclic group of order . By we mean the group with discrete topology and is the torsion part of . The topological isomorphism will be denoted by . For more on locally compact abelian groups, see .
2.Splitting in the category of torsion-free, locally compact abelian groups
In this section, we determine the structure of a discrete or a compact group such that for all torsion-free groups £.
Recall that a discrete group A is said to be cotorsion if for any discrete torsion-free group B , Ext(B,A)=0.
Lemma 2.1 for all torsion-free groups £.
Proof. Let be a discrete torsion-free group. By [1, Theorem 52.3], where is a divisible hull of . By [1, Theorem 46.1 and Theorem 54.6], and so is a reduced,algeraically compact and cotorsion group. It follows from [1, Prposition 54.2] that
Now suppose that is a torsion-free group in £.
Theorem 2.2 Let be a discrete group. Then for all torsion-free groups £ if and only if where is a discrete torsion group.
Proof. Let be a torsion-free group. By [2, Proposition 4], we have the following exact sequence: Since is torsion and a torsion-free group, so Now assume that . By (1), . It follows from [3, Proposition 2] that . By [1, Theorem 28.2], .
Conversely, suppose that where
is discrete torsion . If is torsion-free, then
It is enough to show that . There
exists a pure exact sequence
where is a direct sum of finite cyclic groups and, , a divisible torsion groupS. Consider the exact sequence
Since , so by lemma 2.1, . Clearly
. So .□
Lemma 2.3 Let and be two topological spaces, and an open map. If , then is open.
Proof. It is clear.□
Theorem 2.4 Let be a pure, closed and torsion-free subgroup of . Then for all .
Proof. Let be the natural map. For a fixed , we show that is a topological isomorphism. Assume that . Then . Since is pure, for some . So for some . Therefore, and ,that is is surjective. Now suppose that for some . Then and . Since is torsion-free, . So is injective. Since is open and is injective, so by lemma 2.3, is open. Hence .□
Definition 2.5 Let be an extension in £. Then is said to be pure if the sequence
is proper exact for all . Following Loth , we let denote the group of pure extensions of by .
Lemma 2.6 Let £. Then
for all torsion-free groups £.
Proof. Let be a pure extension of torsion-free group by . We may assume that is the inclusion homomorphism. So is a pure, closed and torsion-free subgroup of . Now by Theorem 2.4, for all . Since so is proper exact for all .
Theorem 2.7 Let be a compact group. Then for all torsion-free groups if and only if is torsion.
Proof. Assume that is a compact group such that for each torsion-free group . By [7, Lemma 2.13], . So . Since the sequence
Is exact, . Now consider the exact sequence
So is isomorphic to and therefore . It follows that is torsion. Since , so . Hence is cotorsion. By [1, Corollary 54.4], where is a bounded group and a divisible group. So . Clearly, for each torsion-free group . Consequently by [3, Theorem 2], or .
Conversely, let be a compact torsion group. Then by [6, Theorem 4.2] and Lemma 2.6, for all torsion-free groups .
L. Fuchs, Infinite abelian groups, Vol. I, Academic Press, New York, 1970.
R.O. Fulp, Homological study of purity in locally compact groups, Proc.London Math. Soc.21, pp 501-512(1970).
E. Hewitt and K. Ross, Abstract Harmonic Analysis, Vol I, Second Edition, Springer Verlag, Berlin, 1979.
R.O Fulp and P. Griffith, Extensions of locally compact abelian groups I, Trans. Amer. Math.Soc.154,pp 341-356(1971).
P. Loth, Topologically pure extensions abelian groups, Rings and modules, Proceedings of the AGRAM 2000 Conference in Perth, Western Australia, July 9-15,2000, Contemporary Mathematics 273, American Mathematical Society, pp 191-201(2001).
P. Loth, Pure extensions of locally compact abelian groups, Rend. Sem. Mat. Univ. Padova. 116, pp 31-40 (2006).