If $X$ is a spectrum with trivial (integer-valued) homology groups, does it have to be weakly-equivalent to a point?
This is easy to prove for connective spectrum, as a Hurewitz-type argument is then possible, but what about the general case?
Furthermore, if this is not the case, how should I think of the functor $L_{H\mathbb{Z}}$ (Bousfield-localization at the spectrum EM spectrum $H\mathbb{Z}$)?
If $K(n)$ is the $n$-th Morava K-theory for $n>0$, then $K(n)\otimes H\mathbb{Z}=0$ because, via the 2 complex orientations of $K(n)\otimes H\mathbb{Z}$, there are two formal groups over the ring $\pi_*(K(n)\otimes H\mathbb{Z})$ and an isomorphism between them. But one has height 0 (additive formal group from $H\mathbb{Z}$) and the other has height $n>0$ (coming from $K(n)$). This is impossible unless $\pi_*(K(n)\otimes H\mathbb{Z})=0$. But of course $K(n)\neq0$.
