Suppose $X$ and $Y$ are spectra (or homotopy classes thereof) such that $X$ is p-local and $Y$ is q-local, for primes $p\neq q$. Is it indeed true then, and if so how would one show that $[X,Y]\_\ast=0$?


Thanks!



Absence of Maps Between p-local and q-local spectra
The rational Eilenberg-Mac Lane spectrum $H\mathbb{Q}$ is $p$-local for every prime $p$, but certainly $[H\mathbb{Q}, H\mathbb{Q}]\_\*\neq 0$. 


If you replace "$p$-local" and "$q$-local" with "$p$-complete" and "$q$-complete", then your conclusion does hold. 


