Extensions of locally compact abelian groups. I

Fulp, Ronald O.; Griffith, Phillip A.

doi:10.1090/S0002-9947-1971-99931-0

Extensions of locally compact abelian groups. I
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by Ronald O. Fulp and Phillip A. Griffith

Trans. Amer. Math. Soc. 154 (1971), 341-356

DOI: https://doi.org/10.1090/S0002-9947-1971-99931-0

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Part II: Trans. Amer. Math. Soc. (1971), 357-363

Abstract:

This paper is concerned with the development of a (discrete) group-valued functor Ext defined on $\mathcal {L} \times \mathcal {L}$ where $\mathcal {L}$ is the category of locally compact abelian groups such that, for A and B groups in $\mathcal {L}$, Ext (A, B) is the group of all extensions of B by A. Topological versions of homological lemmas are proven to facilitate the proof of the existence of such a functor. Various properties of Ext are obtained which include the usual long exact sequence which connects Hom to Ext. Along the way some applications are obtained one of which yields a slight improvement of one of the Noether isomorphism theorems. Also the injectives and projectives of the category of locally compact abelian totally disconnected groups are obtained. They are found to be necessarily discrete and hence are the same as the injectives and projectives of the category of discrete abelian groups. Finally we obtain the structure of those connected groups C of $\mathcal {L}$ which are direct summands of every G in $\mathcal {L}$ which contains C as a component.

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