Let $X\_k$ be a family of spectra equipped with maps $f\_k: X\_k \to X\_{k+1}$. If $Y$ is a compact object (such as a sphere), then I can compute homotopy classes of maps from $Y$ into the homotopy colimit $X\_\infty := hocolim \; X\_k$ of the sequence by


$$[Y, X\_\infty] = \lim\_{k \to \infty} [Y, X\_k]$$


Now if I assume that the $X\_k$ are local for Morava K-theory $K(n)$, then it need not be the case that the homotopy colimit is, too. In order to force this to be true, I can simply re-localise $X\_\infty$.


**My question is**: is there a similar formula for $[Y, L\_{K(n)} X\_\infty]$? Does it help that the $X\_k$ were local to begin with? I'm happy to assume that the $X\_k$ have finite $K(n)\_\*$, if that makes life any easier.



Homotopy of localisations of colimits
There are two natural finiteness conditions that you might impose on $Y$. The stronger one says that the Morava $E$-theory of $Y$ is finite in each degree, or equivalently that $Y$ is a retract of $L\_{K(n)}Y\_0$ for some finite spectrum $Y\_0$ with $E(n-1)\_\*Y\_0=0$. If this holds, then $[Y,L\_{K(n)}X\_\infty]$ is the colimit of the terms $[Y,X\_k]$, the key point being that $DY\wedge X\_\infty$ is already $K(n)$-local (even though $X\_\infty$ itself is not).


If $Y$ does not have the above smallness property then it will rarely be true that $[Y,L\_{K(n)}X\_\infty]$ is the colimit of $[Y,X\_k]$. For the canonical counterexample, let $\{M\_k\}$ be a tower of generalised Moore spectra of the usual kind, so 
$$ BP\_\*(M\_k)=BP\_\*/(v\_0^{a\_0},\dotsc,v\_{n-1}^{a\_{n-1}}) $$
with the exponents $a\_i$ tending to infinity as $k$ increases. Put $Y=L\_{K(n)}S^0$ and $X\_k=F(M\_k,L\_{K(n)}S^0)$. In this case it is known that $X\_\infty=L\_{K(n)}S^0$ so $[Y,X\_\infty]$ contains $\mathbb{Z}\_p$, but each group $[Y,X\_k]$ is finite so the colimit is torsion.


