\centerline{Comparing Completions of a Space at a Prime}
\centerline{Paul G. Goerss}
\bigskip\bigskip
There are two practical ways to complete an abelian group $A$ at a
prime $p$. One could complete $A$ with respect to the neighborhood base of
zero given by the subgroups $p^nA$. This completion, called $p$-completion,
is $\lim A/p^nA$. Or one could complete $A$ with respect to the neighborhood
base of zero given by subgroups $B\subseteq A$ of finite $p$-power index.
This completion, called $p$-pro-finite completion, is $\lim A_\alpha$ where
$A_\alpha$ runs over the finite $p$-group quotients of $A$. They agree for
finitely generated groups, but not in general: if $V$ is an $F_p$ vector
space it is $p$-complete, but its $p$-pro-finite completion is the double dual
$V^{\ast\ast}$. Also, the former, $p$-completion, is easier to define, but
it is neither left nor right exact and the category of $p$-complete groups is
not abelian -- it is not closed under cokernels. The latter, $p$-pro-finite
completion, is initially less tractable, but it is right exact and the
category of $p$-pro-finite abelian groups is an abelian category.
There are two corresponding completions for topological spaces. The analog
of $p$-completion is Bousfield-Kan completion [3] and the analog of
$p$-pro-finite completion is related to Sullivan's pro-finite completion and
has recently been given a homotopical definition by Morel [13]. The purpose
of this note is to compare these two completions; in addition, we seek to
give Morel's completion the same sort of computational footing that the
Bousfield-Kan completion enjoys.
\end