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?!?



$S$-dual of filtered spectra
One key property of duality is that the canonical map from $E$ to its double dual $D(D(E))$ is a homotopy equivalence when $E$ is a finite spectrum. This fails (in general) for infinite spectra. Moreover, the map $E\to D^2(E)$ is not always a monomorphism in the homotopy category, or even injective on homotopy groups. For example, it is known that the SW-dual of the Eilenberg-Mac Lane spectrum $H{\mathbb Z/p}$ is contractible ([T.Y. Lin, Duality and Eilenberg-Mac Lane spectra](https://www.jstor.org/stable/2041622?seq=1#page_scan_tab_contents)). So it is not true that if $D(f)$ is null then so is $f$.


Regarding phantom maps: suppose $E$ is the homotopy direct limit of finite spectra $E\_i$. Then the homotopy groups of $D(E)$ fit into a lim$^1$ short exact sequence


$$0\to {\lim}^1 \pi\_{\*+1}(DE\_i)\to \pi\_\*(DE) \to \lim \pi\_\*(DE\_i) \to 0$$


The image of ${\lim}^1 \pi\_1(D(E\_i))$ in $\pi\_0(DE)$ represents precisely the phantom maps from $E$ to $S^0$.


Another key property of duality: in good cases the map $D(E) \wedge X \to F(E, X)$ is a homotopy equivalence. Here $F(-, -)$ stands for function spectrum. This holds if either $E$ or $X$ is a finite spectrum, but not in general.


