8
$\begingroup$

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?

$\endgroup$
1

1 Answer 1

10
$\begingroup$

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.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.