My question has to do with that post: https://mathoverflow.net/questions/215467/p-local-space-vs-p-completion. Here the guy who posted the answer asserts that when $\pi_{*}X$ is finite (or more generally finitely generated), then the $p$-localization agrees with the $p$-completion. How could someone prove this?
Because it might be easier for someone to answer if they now what's my background roughly, here is a quick summary:
Since $\pi_{*}X$ is always abelian ($* \geq 2$), the $p$-localization for me is just the morphism $\pi_{*}X \to L_K(\pi_{*}X)$, where $K=\oplus_{p \in J} \mathbb{Z}/p$, $J=P-\{p\}$ and $P$ the set of all primes, while the $p$-completion of the abelian group $\pi_{*}X$ is just the inverse limit, $\varprojlim_{r \geq 1} \pi_{*}X/p^r \pi_{*}X$.
As far as I know regardless if an abelian group $A$ is finitely generated or not, the $\oplus_{p \in J} \mathbb{Z}/p$-localization over an arbitrary set of primes $J$ always exists and is the morphism $$ A \to A \otimes \mathbb{Z}[J^{-1}].$$ On the other hand regarding the completion, the finitely generated assumption seems more crucial, since when it is fulfilled the $p$-completion coincides with the localization $L_{\mathbb{Z}[1/p]}A$. Therefore here is where I start becoming confused. How these two things are related and why?