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.



Fracture Squares of Bousfield Localizations of Spectra
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.


