K(r)-localization and monochromatic layers in the chromatic spectral sequence - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T22:33:13Z http://mathoverflow.net/feeds/question/109023 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109023/kr-localization-and-monochromatic-layers-in-the-chromatic-spectral-sequence K(r)-localization and monochromatic layers in the chromatic spectral sequence Eric Peterson 2012-10-06T20:29:58Z 2013-04-17T01:57:12Z <p>While preparing some lecture notes, I had a basic point of confusion come up that I haven't been able to settle.</p> <p>The $BP$-Adams spectral sequence (or $p$-local Adams-Novikov spectral sequence) for the sphere begins with $E_2$-page <code>$$E_2^{*, *} = \operatorname{Ext}^{*, *}_{BP_* BP}(BP_*, BP_*)$$</code> and converges to $\pi_* \mathbb{S} \otimes_{\mathbb{Z}} \mathbb{Z}_{(p)}$.</p> <p>There are a variety of cool periodicities visible in this $E_2$-page, which we can organize via the following secondary spectral sequence. There is an ascending chain of $(BP_* BP)$-invariant ideals for $BP_*$ given by $I_r = (p, v_1, \ldots, v_{r-1})$, connected to one another by the short exact sequences $$0 \to BP_* / I_r^\infty \to v_r^{-1} BP_* / I_r^\infty \to BP_* / I_{r+1}^\infty \to 0.$$ The quotient $BP_* / I_r^\infty$ is thought of as the closed substack of the moduli of formal groups detected by the ideal sheaf corresponding to $I_r$ together with its formal neighborhood inside the parent stack. Applying $\operatorname{Ext}$ and stringing the resulting long exact sequences together, one arrives at the (trigraded) chromatic spectral sequence (CSS): <code>$$E_1^{r, *, *} = \operatorname{Ext}^{*, *}_{BP_* BP}(BP_*, v_r^{-1} BP_* / I_r^\infty) \Rightarrow \operatorname{Ext}^{*, *}_{BP_* BP}(BP_*, BP_*).$$</code> Much of the fun in chromatic homotopy theory after this point comes from identifying the groups in this $E_1$-page as other sorts of things, like certain group cohomologies.</p> <p>Shifting gears somewhat, Bousfield localization at the Johnson-Wilson $E(r)$-theories and the Morava $K(r)$-theories is meant to perform the same organization at the level of homotopy types. The spectra $E(\infty)$ and $E(0)$ correspond to $BP$ and to $H\mathbb{Q}$ respectively, so the sequence of localization functors $L_{E(r)}$ are meant to interpolate between rational homotopy theory and the sort of homotopy theory visible to the $p$-local Adams-Novikov spectral sequence.</p> <p>There are two ways to study these functors as $r$ increases. First, there there is a natural map $L_{E(r)} X \to L_{E(r-1)} X$. Its homotopy fiber detects the difference between these two spectra, denoted $M_r X$ and called the $r$th monochromatic layer of $X$. Second, there is a pullback square, dubbed chromatic fracture: <code>$$\begin{array}{ccc} L_{E(r)} X &amp; \to &amp; L_{K(r)} X \\ \downarrow &amp; &amp; \downarrow \\ L_{E(r-1)} X &amp; \to &amp; L_{E(r-1)} L_{K(r)} X. \end{array}$$</code> In both of these situations, you can hope to inductively study the filtering spectra $L_{E(r)} X$ by studying the "filtration layers", which are either $M_r X$ or $L_{K(r)} X$ depending upon your approach.</p> <blockquote> <p><strong>My question is:</strong> How exactly do these two approaches connect to the chromatic spectral sequence?</p> </blockquote> <p>I suspect that the CSS for $L_{E(R)} X$ looks like the CSS for $X$, after quotienting out the information in $r$-degrees $r > R$. I also suspect that the CSS for one of the two of $M_R X$ and $L_{K(R)} X$ looks like that for $L_{E(R)} X$, after additionally quotienting out the information in $r$-degrees $r &lt; R$. However, I can't seem to make the pieces line up. For instance, Prop. 7.4 of Hopkins, Mahowald, and Sadofsky's <em>Constructions of elements in Picard groups</em> suggests that this description holds for $L_{K(R)} \mathbb{S}$, as that statement matches their Adams-Novikov spectral sequence converging to $\pi_* L_{K(R)} \mathbb{S}$ --- just as one would expect from a collapsing chromatic spectral sequence. On the other hand, the bottom corner of the fracture square is of the form $L_{E(R-1)} L_{K(R)} X$, and this description seems to say that its CSS is empty, which doesn't sound right.</p> <p>I'd appreciate someone setting me straight about this. Thanks!</p> http://mathoverflow.net/questions/109023/kr-localization-and-monochromatic-layers-in-the-chromatic-spectral-sequence/124668#124668 Answer by Eric Peterson for K(r)-localization and monochromatic layers in the chromatic spectral sequence Eric Peterson 2013-03-16T02:44:40Z 2013-04-17T01:57:12Z <p>I have at least a partial answer to my question. It's fairly complicated, and pieces of it are written down in a variety of places, so I'm going to do what I can to be thorough. Before we do anything involving spectral sequences at all, it will turn out to be useful to have a certain pair of families of $BP_* BP$-comodules at our disposal, defined by the formulas $$N_r^s = BP_* / \langle p, \ldots, v_{r-1}, v_r^\infty, \ldots, v_{r+s-1}^\infty \rangle,$$ $$M_r^s = v_{r+s}^{-1} BP_* / \langle p, \ldots, v_{r-1}, v_r^\infty, \ldots, v_{r+s-1}^\infty\rangle = v_{r+s}^{-1} N_r^s.$$ In fact, these formulas even make sense on the level of spectra, since $N_0^0$ can be taken to be $BP$, $M_r^s$ appears as a mapping telescope built out of $N_r^s$, and there are cofiber sequences (/ short exact sequences) $$N_r^s \xrightarrow{\cdot v_{r+s}^\infty} M_r^s \to N_r^{s+1},$$ $$N_r^r \xrightarrow{\cdot v_r} N_r^r \to N_{r+1}^{r+1}.$$ The $BP_* BP$-comodules are recovered by taking homotopy groups.</p> <p>The most fundamental of all the spectral sequences in play was brought up by Drew in the comments above. The chromatic tower is a tower of fibrations $$\cdots \to L_{E(n+1)} \mathbb{S}^0 \to L_{E(n)} \mathbb{S}^0 \to \cdots \to L_{E(0)} \mathbb{S}^0,$$ and the fibers of these maps define the monochromatic layers. Applying <code>$\pi_*$</code> to the diagram produces a spectral sequence of signature <code>$$\pi_* M_r \mathbb{S}^0 \Rightarrow \pi_* \mathbb{S}^0_{(p)}.$$</code> To study this spectral sequence, there are two reasonable-sounding things to do involving the homology theory $BP_*$:</p> <ol> <li>Apply $BP_*$ to the chromatic tower diagram and study the resulting spectral sequence.</li> <li>Use the $BP$-Adams spectral sequence to compute $\pi_* M_n \mathbb{S}^0$ from $BP_* M_n \mathbb{S}^0$.</li> </ol> <p>These both turn out to be relevant, and they both rest upon a certain input, computed by Ravenel. Namely, he shows how to compute $BP_* L_{E(r)} \mathbb{S}^0$ and the surrounding pieces:</p> <blockquote> <p>Theorem 6.2 (Ravenel, <em>Localizations with respect to certain periodic cohomology theories</em>): The going-around maps $N_0^{s+1} \to \Sigma N_0^s$ compose to give a map $\Sigma^{-s-1} N_0^{s+1} \to N_0^0 = BP$. The cofiber of this map can be identified as $$\Sigma^{-s-1} N_0^{s+1} \to BP \to L_{E(s)} BP.$$ Moreover, the rotated triangle $BP_* \to \pi_* L_{E(s)} BP \to \pi_* \Sigma^{-s} N_0^{s+1}$ is split short exact. (There's an exception in the bottom case, where $BP_* L_{E(0)} \mathbb{S}^0 = BP_* \otimes \mathbb{Q}$.)</p> </blockquote> <p>Applying the octahedral axiom to the pair $\Sigma^{-s-1} N_0^{s+1} \to \Sigma^{-s} N_0^s \to BP$ and then applying <code>$BP_*$</code>-homology gives the calculation $$BP_* M_s \mathbb{S}^0 = \Sigma^{-s} M_0^s.$$</p> <p>Now we can address 1. and 2.:</p> <ol> <li>If we delete the boring <code>$BP_*$</code> summands in <code>$BP_* L_{E(n)} \mathbb{S}^0$</code>, then the exact couple coming from applying <code>$BP_*$</code>-homology to the chromatic tower just falls apart into a string of short exact sequences of $BP_* BP$-comodules. Now, we know that $H^{*, *} N_0^0$ is the input to the $BP$-Adams spectral sequence computing $\pi_* \mathbb{S}^0$, and applying <code>$H^{*, *}$</code> to this diagram of short exact sequences of <code>$BP_* BP$</code>-modules yields a spectral sequence of signature <code>$$E_1^{r, *, *} = H^{*, *} M_0^r \Rightarrow H^{*, *} N_0^0.$$</code> This is the usual chromatic spectral sequence, as stemming from algebraic considerations.</li> <li>Applying the $BP$-Adams spectral sequence to compute $\pi_* M_r \mathbb{S}^0$ begins with the computation of <code>$\operatorname{Ext}^{*, *}_{BP_* BP}(BP_*, BP_* M_r \mathbb{S}^0)$</code>, which we now know to be isomorphic to <code>$H^{*, *} M_0^r$</code>. (In fact, this spectral sequence is supposed to collapse for $p \gg n$, e.g., $p \ge 5$ for $n = 2$.)</li> </ol> <p>Another part of this whole story is how the $K(r)$-local sphere plays into this picture. Now, there is also an Adams-type spectral sequence computing <code>$\pi_* L_{K(r)} \mathbb{S}^0$</code>, and it has signature <code>$$\operatorname{Ext}^{*, *}_{\Gamma}(K(n)_*, K(n)_*) \Rightarrow \pi_* L_{K(n)} \mathbb{S}^0,$$</code> where <code>$\Gamma = K(n)_* \otimes_{BP_*} BP_* BP \otimes_{BP_*} K(n)_*$</code>. Morava's change of rings theorem states that the map <code>$M_r^0 \to K(n)_*$</code> induces an isomorphism between the sheaf cohomology groups <code>$H^{*, *} M_r^0$</code> and the $\operatorname{Ext}$-groups in the Adams-type spectral sequence.</p> <p>The difference, then, between the monochromatic sphere and the $K(r)$-local sphere is recorded in the index swap $M_0^r$ and $M_r^0$ --- i.e., whether the generators below $v_r$ are taken to be zero or to be torsion. The difference in these two situations is of course itself recorded as a spectral sequence: the inclusions $$v_{r+s}^{-1} BP_* / \langle p, \ldots, v_{r-1}, v_r^j, v_{r+1}^\infty, \ldots, v_{r+s-1}^\infty \rangle \xrightarrow{\cdot v_r} v_{r+s}^{-1} BP_* / \langle p, \ldots, v_{r-1}, v_r^{j+1}, v_{r+1}^\infty, \ldots, v_{r+s-1}^\infty \rangle$$ all have cofiber $M_{r+1}^{s-1}$, regardless of choice of index $j$. Applying $H^{*, *}$ to the string of inclusions (extended to cofiber sequences) yields the $v_r$-Bockstein spectral sequence, of signature $$H^{*, *} M_{r+1}^{s-1} \otimes \mathbb{F}_p[v_r] / v_r^\infty \Rightarrow H^{*, *} M_{r}^{s}.$$ So, there is a length $r$ string of <code>$v_*$</code>-Bockstein spectral sequences beginning with $H^{*, *} M_r^0$ and concluding with $H^{*, *} M_0^r$.</p> <p>Some things not included in this answer are: </p> <ul> <li>What happens when analyzing the chromatic tower of spaces other than the sphere spectrum?</li> <li>What is the relevance of the corner space $L_{E(r-1)} L_{K(r)} \mathbb{S}^0$ in terms of the chromatic spectral sequence?</li> <li>What parts of this story can be made sense of <em>mutatis mutandis</em> when replacing $BP$ with other spectra in the same family, like $E(r)$? My expectation (as the comments reveal) is that there should be an analogue of the algebraic chromatic spectral sequence for $E(R)$-homology, which is the truncation of the usual one for $r \le R$. (In October I even thought I knew how to prove this, but I've since forgotten. This is the least interesting question of the bunch.)</li> </ul>