Localisation with respect to $H \mathbb{Q}$ I am trying to understand what happens when we localise with respect to the cohomology theory with $\mathbb{Q}$ coefficients $H\mathbb{Q}$. In the notes on Morava K theories and localisation by M Hovey, he says that this is smashing that is $L_{H\mathbb{Q}}X=X \wedge L_{H\mathbb{Q}}S^0$. Can somebody give any hint as to how to prove this?
Also is $L_{H\mathbb{Q}}S^0$ the rational sphere? By a rational sphere I mean the sphere whose cohomology with $\mathbb{Z}$ coefficients is $\mathbb{Q}$. I am not sure how one would prove this either. Any hints are appreciated. Thanks
 A: Isn't it just smashing with $H\mathbb Q$ ?
Indeed, let me first prove that $H\mathbb Q$-local equivalences are equivalences after smashing with $H\mathbb Q$ : this is pretty easy, indeed $map(X,H\mathbb Q)\simeq map_{H\mathbb Q}(H\mathbb Q\wedge X, H\mathbb Q)$
But now if $A,B$ are $H\mathbb Q$-modules (equivalently, $\mathbb Q$-chain complexes, equivalently, $\mathbb Z$-graded $\mathbb Q$-vector spaces) and $f:B\to A$ is a map such that $map(A,H\mathbb Q)\to map(B,H\mathbb Q)$ is an equivalence, so is $f$ (this follows from the last part of the previous parenthesis : we're looking at graded vector spaces; and a map of vector spaces is an isomorphism if and only if its dual is)
NB: note that I'm not claiming that two vector spaces are isomorphic if and only if their duals are, just that a map is an isomorphism if and only if its dual map is.
If $X\to Y$ is an $H\mathbb Q$-local equivalence, we can then apply this to $H\mathbb Q\wedge X\to H\mathbb Q\wedge Y$ : it follows that this is an equivalence.
Conversely, it's easy to prove, by the same argument that if $H\mathbb Q\wedge X\to H\mathbb Q\wedge Y$ is an equivalence, then $X\to Y$ was an $H\mathbb Q$-local equivalence.
So equivalences are just smashing equivalences.
Now there are two points to check : that $H\mathbb Q\wedge E$ is $H\mathbb Q$-local and $E\to H\mathbb Q\wedge E$ is an $H\mathbb Q$-local equivalence. Both ones are easily checked once you know that $H\mathbb Q \simeq \mathrm{colim}(S^0\to S^0 \to \dots)$ where the maps are $2,3,5$ and so on.
Once you have this, it follows that $\mathrm{Map}(H\mathbb Q, H\mathbb Q)\simeq H\mathbb Q$ (where $\mathrm{Map}$ is the mapping spectrum), and similarly,  $\mathrm{Map}(H\mathbb Q, H\mathbb Q\wedge E)\simeq H\mathbb Q\wedge E$
Both results follow easily from there and from what we did above. These two points show that $H\mathbb Q\wedge -$ is a localization functor, which is what we wanted.
