There are maps of spaces which are not null-homotopic, but when localized at any prime become null. I don't know explicit constructions of any, but an example is given in Section 6 of Chapter 25 of the Handbook of Algebraic Topology (this chapter was written by C.A. McGibbon), of a phantom map $\Omega^2S^5\to \mathbb{H}P^\infty$ which has this property. 


My question is really simple to state: are there maps of spectra which are globally non-trivial, but become null when localized at any prime? Or, can anyone give a proof that this is impossible? 


It's well known that there's an "arithmetic square" for reconstructing spectra from their rationalizations and their $p$-completions (cf. [these notes](http://www-math.mit.edu/~lurie/917notes/Lecture32.pdf)), and I believe this might have some bearing on this situation? 



Essential maps of spectra which are null when localized at any prime
Such maps exist, even when the target is the sphere $S^{0}$. 


A map $X \rightarrow S^{0}$ is $p$-locally trivial for each $p$ if it vanishes when composed with each of the $p$-localizations $S^{0} \rightarrow S^{0}\_{(p)}$. This is the same as being in the kernel of the map $[X, S^{0}] \rightarrow [X, \prod S^{0}\_{(p)}]$ induced by the diagonal $S^{0} \rightarrow \prod S^{0}\_{(p)}$. 


Assume, by contradiction, that all such maps are zero. This would mean that $S^{0} \rightarrow \prod S^{0}\_{(p)}$ is a monomorphism in the stable homotopy category. Since the latter is triangulated, a monic necessarily splits, but $S^{0} \rightarrow S^{0}\_{(p)}$ is not split, as that would in particular mean that we have a splitting on $\pi\_{0}$, even though $\mathbb{Z} \hookrightarrow \prod \mathbb{Z}\_{(p)}$ is not split. We deduce that $S^{0} \rightarrow \prod S^{0}\_{(p)}$ is not split and hence not a monomorphism. 


The proof of the fact that in a triangulated category any monic splits (see the answer to [this question](https://mathoverflow.net/questions/15658/how-do-i-know-the-derived-category-is-not-abelian)) allows one to construct an explicit example of a map that vanishes $p$-locally at each prime. Namely, the inclusion of the fibre $F \rightarrow S^{0}$ of the map $S^{0} \rightarrow \prod S^{0}\_{(p)}$ is non-zero, as otherwise the latter would be split, although the composition $F \rightarrow S^{0} \rightarrow \prod S^{0}\_{(p)}$ does necessarily vanish. 


