Fracture Squares of Bousfield Localizations of Spectra

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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

• $n$Lab, fracture theorem.

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.