How can I see that $H\mathbb{Z}$ doesn't admit a Bousfield complement?

Question

From Ravenel's article "Localization and Periodicity in Homotopy Theory":

Two spectra $E$ and $F$ are said to be Bousfield equivalent when they give the same localization functor, or equivalently when $E_\ast (X)=0$ iff $F_\ast (X)=0$. The equivalence class of $E$ is denoted by $\langle E \rangle$. There is a partial ordering on the set of Bousfield classes. We say that $\langle E \rangle \geq \langle F \rangle$ if $E_\ast (X)=0$ implies that $F_\ast (X)=0$. Thus $\langle S^0 \rangle$ is the biggest class and $\langle pt \rangle $ is the smallest. Smash products and wedges are well defined on Bousfield classes. A class $\langle F \rangle$ is the complement of $\langle E \rangle$ if $\langle E \rangle \vee \langle F \rangle = \langle S^0 \rangle$ and $\langle E \rangle \wedge \langle F \rangle = \langle pt \rangle$. A class may or may not have a complement. It is easy to find examples of classes (e.g., that of an integer Eilenberg-Mac Lane spectrum) that do not.

I was trying to figure out why this last statement is true, and at first I wanted to apply cohomotopy to a hypothetical equivalence $H\mathbb{Z} \vee F \simeq S^0$, but then I realized that of course there's no reason that we should have such an equivalence. Is there some other easy approach?

Answer 1

Suppose we have $F$ such that $H\wedge F=0$. We need to show that $H\vee F$ has Bousfield class smaller than that of $S$, or in other words, that there exists $X\neq 0$ with $H\wedge X=0$ and $F\wedge X=0$. I claim that we can take $X=I$ (the Brown-Comenetz dual of the sphere, which is the standard counterexample for everything in this theory). Indeed, we have $\pi_k(I)=0$ for $k>0$, so we can write $I$ as the colimit of its Postnikov pieces $I[-n,0]$. Here $I[0,0]$ is an Eilenberg-MacLane spectrum, as are the fibres of the maps $I[-n,0]\to I[-n-1,0]$, so $F\wedge I[-n,0]=0$ for all $n$ by induction, so $F\wedge I=0$. It is also true that $H\wedge I=0$. This is Corollary B.12 in the memoir 'Morava K-theories and localisation' by Mark Hovey and myself; the proof is a fairly straightforward deduction from the fact that $[BP/p,S]_*=0$, which is contained in Ravenel's Lemma 3.2.