Fracture Squares of Bousfield Localizations of Spectra - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T19:47:49Z http://mathoverflow.net/feeds/question/91021 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91021/fracture-squares-of-bousfield-localizations-of-spectra Fracture Squares of Bousfield Localizations of Spectra Jon Beardsley 2012-03-12T21:16:48Z 2012-10-29T15:40:47Z <p>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? </p> <p>I went ahead and made this a reference request, because I imagine it could a rather significant answer.</p> http://mathoverflow.net/questions/91021/fracture-squares-of-bousfield-localizations-of-spectra/91057#91057 Answer by Neil Strickland for Fracture Squares of Bousfield Localizations of Spectra Neil Strickland 2012-03-13T06:39:46Z 2012-03-13T06:39:46Z <p>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 $$\begin{array}{ccc} L_{E\vee F}X &amp; \rightarrow &amp; L_EX \\ \downarrow &amp; &amp; \downarrow \\ L_FX &amp; \rightarrow &amp; L_EL_FX \end{array}$$ 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 $L_EL_F\neq 0$). </p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/91021/fracture-squares-of-bousfield-localizations-of-spectra/111008#111008 Answer by Nick Kuhn for Fracture Squares of Bousfield Localizations of Spectra Nick Kuhn 2012-10-29T15:40:47Z 2012-10-29T15:40:47Z <p>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.)</p> <p>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.</p>