Return to Revisions

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.)