TITLE: $S$-dual of filtered spectra
QUESTION [7 upvotes]: I hope this is research level. Suppose $E$ is the direct limit of finite spectra, say $E=\mathrm{colim }\ E_i$, which itself is not finite. I wonder how much and under which conditions the inverse limit $\mathrm{lim}\ D(E_i)$ is a good candidate for playing role of $D(E)$? Here, I write $D$ for the $S$-duality functor. I feel there is problem with the possibility of existence of phantom maps here. I will be very grateful for any advise on this, or possibly some references on the topic.
As to what one might expect. For instance, if $f:X\to Y$ is a map between finite spectra then $f_*=0$ if and only if $D(f)_*=0$. Or if we know about dimension of bottom cell of $E$ then that would tell about the dimension of the top cell of the dual spectrum. Here, $f_*$ is the map induced in ordinary homology and I consider CW spectra. And if $D(E)$ can always be identified with the function spectrum $F(E,S^0)$?!
I also would like to know what features of duality for finite spectra fail to hold in general. For the moment, homological behavior is very interesting for me. I also wonder if my worry about phantom maps has any point?!?

REPLY [4 votes]: Yet more fun examples ...  
If $A$ is an abelian torsion group and $n \ge 2$, the suspension spectrum of $K(A,n)$ has trivial dual: this is a result of Chun-Nip Lee [Amer.J.Math 114 (1992)]. (This implies the result of T.Y. Lin that Greg mentioned.)
An example that I observed in [Progress in Math 215 (2003) (arXiv)] goes as follows: if $K$ and $X$ are finite CW complexes, and the connectivity of $X$ is greater than the dimension of $K$, then the suspension spectrum of the space of continuous maps from $K$ to $X$ is an $S$-dual. (For example, $\Sigma^\infty \Omega^{2n-1}\mathbb CP^{m}/\mathbb CP^n$ is an $S$-dual, for all $m>n$.)  In fact, it is an $S$-dual of a filtered spectrum as a functor of $X$, and the dual of this filtration yields the Goodwillie tower.
Regarding intuition behind the Segal conjecture, it can be rephrased as saying that a map from a  fixed point spectrum to an associated homotopy fixed point spectrum is an appropriate sort of completion.  So, for example, there is an easily defined sensible map $\mathbb RP^{\infty} \vee S^0 \rightarrow D(\mathbb RP^{\infty})$ that turns out to be an equivalence after completing the sphere at 2.