While preparing some lecture notes, I had a basic point of confusion come up that I haven't been able to settle.

The $BP$-Adams spectral sequence (or $p$-local Adams-Novikov spectral sequence) for the sphere begins with $E_2$-page $$E_2^{*, *} = \operatorname{Ext}^{*, *}_{BP_* BP}(BP_*, BP_*)$$ and converges to $\pi_* \mathbb{S} \otimes_{\mathbb{Z}} \mathbb{Z}_{(p)}$.

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): $$E_1^{r, *, *} = \operatorname{Ext}^{*, *}_{BP_* BP}(BP_*, v_r^{-1} BP_* / I_r^\infty) \Rightarrow \operatorname{Ext}^{*, *}_{BP_* BP}(BP_*, BP_*).$$ 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.

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.

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: $$\begin{array}{ccc} L_{E(r)} X & \to & L_{K(r)} X \\ \downarrow & & \downarrow \\ L_{E(r-1)} X & \to & L_{E(r-1)} L_{K(r)} X. \end{array}$$ 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.

My question is: How exactly do these two approaches connect to the chromatic spectral sequence?

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 $r < R$. However, I can't seem to make the pieces line up. For instance, Prop. 7.4 of Hopkins, Mahowald, and Sadofsky's Constructions of elements in Picard groups 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.

I'd appreciate someone setting me straight about this. Thanks!

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    $\begingroup$ This deserves a real answer. But in the meantime, let me ask a question: What does "CSS for Y" mean (where Y is some random spectrum)? $\endgroup$ – Charles Rezk Oct 7 '12 at 3:46
  • $\begingroup$ Good question, something I've been wondering about. Have you read Behren's "The homotopy groups of the $E(2)$-local sphere at $p > 3$, revisited"? There is a lot of good stuff in there! You might be interested in section 7 where he uses the calculations of $H^*(M_1^1)$ and $H^*(M_0^2)$ to calculate the homotopy of the $K(2)$ local sphere, the $E(2)$ local sphere and the $K(2)/E(2)$ local mod $p$ Moore spectrum. $\endgroup$ – Drew Heard Oct 7 '12 at 4:57
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    $\begingroup$ In particular, I hadn't realised that there was a spectral sequence $E_1^{n,k} = \bigoplus_{n=0}^2 \pi_k M_n(S) \Rightarrow \pi_k L_{E(2)} S$ for example $\endgroup$ – Drew Heard Oct 7 '12 at 5:00
  • $\begingroup$ @Drew: I've opened it many times, but I still haven't properly read it; it's a very dense paper! That spectral sequence probably should have been mentioned above; it would also be nice to know what happens when using $BP_*$ rather than $\pi_*$ on the tower of fibrations $L_{E(r)} \mathbb{S} \to L_{E(r−1)} \mathbb{S}$ and whether the resulting spectral sequence is connected to the CSS. $\endgroup$ – Eric Peterson Oct 7 '12 at 20:58
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    $\begingroup$ @Charles: Ahh, OK, I see some of your point. You can build an $E(R)$-CSS converging to the $E(R)$-Adams SS's $E_2$-page via the same sexseqs as above. This produces the $(r < R)$-truncated CSS, avoids my confusion by keeping $Y = S$, and points out that a crucial component is flatness/Landweber-ness of $E(R)$. You can also get the $(r = R)$-truncated CSS from the $E_R^\vee$-CSS. That's enough to know what I'll say in the notes, but I'm still curious about how or whether $M_R$ belongs in this picture. A $BP$-CSS for $M_r S$ would require some flatness of $BP_* M_r S$ that I doubt we have...? $\endgroup$ – Eric Peterson Oct 7 '12 at 21:47

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.

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 $\pi_*$ to the diagram produces a spectral sequence of signature $$\pi_* M_r \mathbb{S}^0 \Rightarrow \pi_* \mathbb{S}^0_{(p)}.$$ To study this spectral sequence, there are two reasonable-sounding things to do involving the homology theory $BP_*$:

  1. Apply $BP_*$ to the chromatic tower diagram and study the resulting spectral sequence.
  2. Use the $BP$-Adams spectral sequence to compute $\pi_* M_n \mathbb{S}^0$ from $BP_* M_n \mathbb{S}^0$.

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:

Theorem 6.2 (Ravenel, Localizations with respect to certain periodic cohomology theories): 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 $BP_* L_{E(0)} \mathbb{S}^0 = BP_* \otimes \mathbb{Q}$.)

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 $BP_*$-homology gives the calculation $$BP_* M_s \mathbb{S}^0 = \Sigma^{-s} M_0^s.$$

Now we can address 1. and 2.:

  1. If we delete the boring $BP_*$ summands in $BP_* L_{E(n)} \mathbb{S}^0$, then the exact couple coming from applying $BP_*$-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 $H^{*, *}$ to this diagram of short exact sequences of $BP_* BP$-modules yields a spectral sequence of signature $$E_1^{r, *, *} = H^{*, *} M_0^r \Rightarrow H^{*, *} N_0^0.$$ This is the usual chromatic spectral sequence, as stemming from algebraic considerations.
  2. Applying the $BP$-Adams spectral sequence to compute $\pi_* M_r \mathbb{S}^0$ begins with the computation of $\operatorname{Ext}^{*, *}_{BP_* BP}(BP_*, BP_* M_r \mathbb{S}^0)$, which we now know to be isomorphic to $H^{*, *} M_0^r$. (In fact, this spectral sequence is supposed to collapse for $p \gg n$, e.g., $p \ge 5$ for $n = 2$.)

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 $\pi_* L_{K(r)} \mathbb{S}^0$, and it has signature $$\operatorname{Ext}^{*, *}_{\Gamma}(K(n)_*, K(n)_*) \Rightarrow \pi_* L_{K(n)} \mathbb{S}^0,$$ where $\Gamma = K(n)_* \otimes_{BP_*} BP_* BP \otimes_{BP_*} K(n)_*$. Morava's change of rings theorem states that the map $M_r^0 \to K(n)_*$ induces an isomorphism between the sheaf cohomology groups $H^{*, *} M_r^0$ and the $\operatorname{Ext}$-groups in the Adams-type spectral sequence.

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 $v_*$-Bockstein spectral sequences beginning with $H^{*, *} M_r^0$ and concluding with $H^{*, *} M_0^r$.

Some things not included in this answer are:

  • What happens when analyzing the chromatic tower of spaces other than the sphere spectrum?
  • 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?
  • What parts of this story can be made sense of mutatis mutandis 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.)

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