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!
Best Answer
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 $BP_*$:
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:
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.:
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: