### Patrick W. Keef

Whitman College, Walla Walla, USA

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## 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 $p^{ < \lambda}_{\mathrm w}$-*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 $p^{< \lambda}_{\mathrm w}$-purity are compared in a variety of contexts, for example, in the category of Warfield groups.

## Cite this article

Patrick W. Keef, A version of purity on local abelian groups. Rend. Sem. Mat. Univ. Padova 144 (2020), pp. 159–176

DOI 10.4171/RSMUP/63