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Pure extensions of locally compact abelian groups. (English) Zbl 1165.22301

Rend. Semin. Mat. Univ. Padova 116, 31-40 (2006); erratum ibid. 118, 267 (2007).
Summary: We study the group Pext\((C,A)\) for locally compact abelian (LCA) groups \(A\) and \(C\). Sufficient conditions are established for Pext\((C,A)\) to coincide with the first Ulm subgroup of Ext\((C,A)\). Some structural information on pure injectives in the category of LCA groups is obtained. Letting \(\mathfrak C\) denote the class of LCA groups which can be written as the topological direct sum of a compactly generated group and a discrete group, we determine the groups \(G\) in \(\mathfrak C\) which are pure injective in the category of LCA groups. Finally we describe those groups \(G\) in \(\mathfrak C\) such that every pure extension of \(G\) by a group in \(K\) splits and obtain a corresponding dual result.

MSC:

22B05 General properties and structure of LCA groups
20K35 Extensions of abelian groups
20K25 Direct sums, direct products, etc. for abelian groups
20K40 Homological and categorical methods for abelian groups
20K45 Topological methods for abelian groups
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References:

[1] L. FUCHS, Infinite Abelian Groups, Vol. I, Academic Press, New York, 1970. (*) Indirizzo dell’A.: Department of Mathematics, Sacred Heart University, 5151 Zbl0209.05503 MR255673 · Zbl 0209.05503
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