I am trying to understand the paper named " Localization with respect to Certain Periodic Homology Theories"
Here is the part of it I am trying to understand the proof of proposition 1.5 in it.
But I do not know how to prove proposition 1.5(i) and 1.5(ii). could someone show me the proofs please?
I think there is an omission in Ravenel's paper: the definition of a localization functor should also include the requirement that $L_E X$ is $E_*$-local. Once you include this, it's not that hard to prove the statements. First prove:
Lemma: if $W \to Z$ is an $E_*$-equivalence between $E_*$-local spectra, then it is an equivalence.
Given this, if we have two candidates for $E_*$-localizations of $X$, say $L_E X$ and $L_E'X$, then by condition (ii) of the definition, we will get maps between $L_E X$ and $L_E' X$, and those maps will be $E_*$-equivalences of $E_*$-local spectra, and hence equivalences. That's 1.5(i).
To get 1.5(ii), we have $\eta_{L_E X}: L_E X \to L_E L_E X$, and this is an $E_*$-equivalence between $E_*$-local objects.
Finally, to prove the lemma, let $A$ be the fiber of $W \to Z$; then $A$ is $E_*$-acyclic, so the map $A \to W$ is zero (by definition of $E_*$-local). So the fiber sequence splits and $Z = W \vee A$, but since $Z$ is also local, there are no nontrivial maps $A \to Z$. Hence $A=0$ and $W = Z$.