# Fracture Squares of Bousfield Localizations of Spectra

Suppose I have a spectrum $X$ and two homology theories $E$ and $F$. If I look at the Bousfield localizations, $L_E$, $L_F$, $L_{E\vee F}$ and $L_{E\wedge F}$, do I have a homotopy pullback square whose top row is $L_{E\vee F}(X)\to L_E(X)$, and whose lower row is $L_F(X)\to L_{E\wedge F}(X)$? If not, is it known what conditions I need to place on $E$ and $F$ to make this all work out? Does anyone know if I can iterate this process over some set of homology theories?

I went ahead and made this a reference request, because I imagine it could a rather significant answer.

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I might add that there is the well known case where we do this with completion at primes and rationalization. I think... –  Jon Beardsley Mar 12 '12 at 21:20
As well as situations with the Morava $K$ and $E$ theories. –  Jon Beardsley Mar 12 '12 at 22:04
You seem to have your arrows backwards. And it's possible that the well-known case you are thinking of involves a composition $L_E\circ L_F$ rather than $L_{E\wedge F}$ or $L_{E\vee F}$. –  Tom Goodwillie Mar 12 '12 at 22:04
And yes... you're right about the composition, to build the $E(n)$ localizations. I guess... hmm, what am I saying. I guess it should be something like that. In that case, it should be like wedging right? Since that's how we build our $E(n)$'s? –  Jon Beardsley Mar 12 '12 at 22:26
On the other hand, you have that $K(n) \wedge K(m)$ is contractible for $n \neq m$, and the same identity holds for their Bousfield classes. The situation you're describing actually relies on something special - namely, that for $n > m$ anything $K(m)$-local is $K(n)$-acyclic. –  Tyler Lawson Mar 12 '12 at 22:40
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I think the best available statement is as follows. Suppose that $E$ and $F$ have the property that whenever $F\wedge X=0$ we also have $F\wedge L_EX=0$. (This holds if $L_E$ is smashing, for example when $E$ is the Johnson-Wilson spectrum $E(n)$.) Then there is a natural homotopy pullback square $$\begin{array}{ccc} L_{E\vee F}X & \rightarrow & L_EX \\\\ \downarrow & & \downarrow \\\\ L_FX & \rightarrow & L_EL_FX \end{array}$$ Note that $L_{E\wedge F}X$ does not occur here. Probably the most important example is where $E=E(n-1)$ and $F=K(n)$ so $E\vee F$ is Bousfield equivalent to $E(n)$ but $E\wedge F=0$ and also $L_FL_E=0$ (but $L_EL_F\neq 0$).

For another important example, we can take $E=S\mathbb{Q}$ and $F=S/p$ so $E\vee F$ is Bousfield equivalent to $S_{(p)}$. In this case $L_{E\vee F}X=X_{(p)}$ and $L_EX=X\mathbb{Q}$ and $L_FX=X^\wedge_p$ and $L_EL_FX=(X^\wedge_p)\mathbb{Q}$. This gives the $p$-local arithmetic fracture square. For the global arithmetic fracture square, take $F=S(\mathbb{Q}/\mathbb{Z})$ (which is Bousfield equivalent to $\bigvee_pS/p$) instead.

I think that these ideas are all due to Mike Hopkins, but I don't remember what is the best place to read about them. I think there is a good paper by Mark Hovey.

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Ah yes Neil thankyou. I see the point now, where smash is a problem. I guess I'm trying to have some kind of descent property, so what you say may indeed work anyway. Thanks! –  Jon Beardsley Mar 13 '12 at 14:47
I think the relevant paper here might be Hovey's paper on the chromatic splitting conjecture? –  Jon Beardsley Mar 13 '12 at 22:38