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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
up vote 10 down vote accepted

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
A good place to read about these statements is Tilman Bauer's notes… . There is now also an nLab entry – Urs Schreiber Jul 25 '14 at 14:58

I am not really a MathOverflow reader, but I just came across this discussion. I first saw the fracture square that Neil describes (in the classic case of interest as above) in a (handwritten) letter to me from Pete Bousfield dated January 22, 1987. It is in the midst of a paragraph that begins with " ... I'll make some little comments which may be well known to you.", and describes how to (easily) construct distinct nice spectra X and Y whose K(n)-localizations agree for all n. (His letter was part of a correspondence we had around then about how one could generalize his telescopic functor for n=1 to all n.)

Very possibly Pete knew the fracture square result in the late 1970's, when he was thinking about the Boolean algebra of localization functors and such. But it doesn't have a lot of meat until one has some naturally arising smashing localizations, which needed developments in the 1980's.

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Thanks @Nick! I too am thinking about the Boolean algebra of localization functors! But probably if Bousfield didn't do much more with it, neither will I. – Jon Beardsley Oct 29 '12 at 18:19

To add to Neil Strickland's excellent answer just some more pointers to further resources:

One place where the general fracture theorem in stable homotopy theory is nicely spelled out with proofs is the note

  • Tilman Bauer, Bousfield localization and the Hasse square, 2011 (pdf)

The statement in the form of Neil Strickland's answer is proposition 2.2. there.

There is now also an $n$Lab entry reviewing some of this

This also includes some remarks on and pointers to

  1. the relation to arithmetic geometry, where this fracturing is essentially the adelic construction and motivates for instance the passage from number-theoretic Langlands to geometric Langlands;

  2. to fracturing in cohesive homotopy theory in the guise of the differential cohomology hexagon.

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