Fracture Squares of Bousfield Localizations of Spectra - MathOverflow

Fracture Squares of Bousfield Localizations of Spectra

2012-03-12T21:16:48Z

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 by Neil Strickland for Fracture Squares of Bousfield Localizations of Spectra

2012-03-13T06:39:46Z

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 by Nick Kuhn for Fracture Squares of Bousfield Localizations of Spectra

2012-10-29T15:40:47Z

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.