A version of purity on local abelian groups

Abstract

In [6], generalizations of the standard notion of purity on $p$-local abelian groups were defined using functorial methods to create injective resolutions. For example, if $\lambda$ is a limit ordinal, then for a group $G$ the completion functor $L_{\lambda }G$ determines the notion of $L_{\lambda }$-purity. Another way of constructing a type of purity, called $p_{\mathrm{w}}^{<\lambda }$-purity, is defined using the functor ${\prod }_{\alpha <\lambda }(G/p^{\alpha }G)$. Properties of this second type of purity are studied; for example, it is shown to be hereditary if and only if $\lambda$ has countable cofinality. In addition, $L_{\lambda }$ and $p_{\mathrm{w}}^{<\lambda }$-purity are compared in a variety of contexts, for example, in the category of Warfield groups.