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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Extensions of locally compact abelian groups. II


Authors: Ronald O. Fulp and Phillip A. Griffith
Journal: Trans. Amer. Math. Soc. 154 (1971), 357-363
MSC: Primary 18.20; Secondary 22.00
Part I: Trans. Amer. Math. Soc. (1971), 341-356
MathSciNet review: 0272870
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Abstract: It is shown that the extension functor defined on the category $ \mathcal{L}$ of locally compact abelian groups is right-exact. Actually $ {\text{Ext}^n}$ is shown to be zero for all $ n \geqq 2$. Various applications are obtained which deal with the general problem as to when a locally compact abelian group is the direct product of a connected group and a totally disconnected group. One such result is that a locally compact abelian group G has the property that every extension of G by a connected group in $ \mathcal{L}$ splits iff $ G = {(R/Z)^\sigma } \oplus {R^n}$ for some cardinal $ \sigma $ and positive integer n.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0272870-8
PII: S 0002-9947(1971)0272870-8
Keywords: Locally compact abelian groups, topological group extensions, right exactness of Ext, $ {\text{Ext}^n} = 0$, split exact sequences, connected by totally disconnected, totally disconnected by connected
Article copyright: © Copyright 1971 American Mathematical Society