Extensions of locally compact abelian groups. II

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.