TITLE: Detecting the Brown-Comenetz dualizing spectrum
QUESTION [18 upvotes]: The Brown-Comenetz dualizing spectrum $I_{\mathbf{Q/Z}}$ is not detected by very many spectra: it is $BP, \mathbf{Z}, \mathbf{F}_2, X(n)$ for $n\geq 2$, and even $I_{\mathbf{Q/Z}}$-acyclic. However, if $X$ is any nontrivial finite spectrum, then $X\wedge I_{\mathbf{Q/Z}}$ is not contractible. This motivates a natural question: let $E$ be any spectrum such that $E\wedge I_{\mathbf{Q/Z}}$ is not contractible. Then, is it true that $\langle E\rangle \geq \langle X\rangle$ for some nontrivial finite spectrum $X$?

REPLY [12 votes]: Your question appears to be equivalent to the 'dichotomy conjecture' of Hovey, which I believe is still open.  
First, note that any finite spectrum has a type, and all finite spectrum of type $n$ have the same Bousfield class, usually denoted $F(n)$. In Hovey and Strickland's memoir (Appendix B) they conjecture that if $E \wedge I \ne 0$, then $\langle E \rangle \ge \langle F(n) \rangle$ for some $n$. This is precisely your question. 
In his paper on the chromatic splitting conjecture, Hovey conjectured that every spectrum has either a finite acyclic or a finite local (the dichotomy conjecture). With Palmieri (The structure of the Bousfield lattice) he proved that the following conjectures are equivalent:


(1) If $E \wedge I \ne 0$, then $\langle E \rangle \ge \langle F(n) \rangle$ for some $n$.
(2) The dichotomy conjecture.
(3) If $E$ has no finite acyclics, then $\langle E \rangle \ge \langle I \rangle$. 


The analog of the dichotomy conjecture is known to hold in other categories with a good theory of support satisfying the tensor product property - see 'The Bousfield lattice of a triangulated category and stratification' by Iyengar and Krause.