Is there a clean proof that the $L\_n$, localization at $E(n)$, is simply rationalization (i.e. $L\_0$) on Eilenberg-MacLane spectra? Eric Peterson asked this [here](http://chat.stackexchange.com/transcript/message/19941162#19941162), but I haven't seen an answer.



Localization at the Johnson-Wilson spectrum and rationalization
First, recall that any rational spectrum is a wedge of suspended copies of $H\mathbb{Q}$. It follows that $H\mathbb{Q}$ is a retract of $E(n)\mathbb{Q}$ and so is $E(n)$-local. From this we see that the canonical map $H\to H\mathbb{Q}$ factors uniquely through $L\_nH$.


Over $\pi\_\*(E(n)\wedge H/p)$ we have a formal group law $F$ coming from $E(n)$, and a formal group law $G$ coming from $H$ (which is just $x+\_Gy=x+y$). These are isomorphic, by the standard yoga of complex oriented cohomology theories. We have $[p]\_G(x)=0$, and it follows easily that $[p]\_F(y)=0$. However, we also have $[p]\_F(y)=\sum^F\_{0\leq k\leq n}v\_ky^{p^k}$, so $v\_k=0$ in $\pi\_\*(E(n)\wedge H/p)$ for all $k$. But $v\_n$ is invertible, so $E(n)\wedge H/p=0$. It follows that $L\_n(H/p)=0$. But $L\_n(H/p)$ is the cofibre of $p$ times the identity on $L\_nH$, so we see that $p.1\_{L\_nH}$ is an equivalence, so $L\_nH$ is rational. It follows that the canonical map $H\to L\_nH$ factors through a map $H\mathbb{Q}\to L\_nH$. This is inverse to the map $L\_nH\to H\mathbb{Q}$ that we exhibited earlier.


