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