Devinatz--Hopkins--Smith II

Recalling the notation from last time, in this post we address the

First half: If $X(n+1)_* \alpha$ is nilpotent, then $F_{p^k-1} \wedge \alpha^{-1} R$ is contractible for sufficiently large k.

To show that something’s contractible, it’s enough to show that its homotopy groups vanish, which will be our strategy. But first – what is that thing? Recalling that R is a ring and $\alpha$ is a class in $\pi_n R$, we actually get a self-map

which we call “multiplication by $\alpha$.” These maps can be stacked together into a sequence

and the colimit of this tower we denote $\alpha^{-1} R$, in reference to the identical process of localization in commutative algebra.

A standard tool in homotopy theory, especially in the stack-filled world that chromatic homotopy inhabits, the Adams spectral sequence is an invaluable tool for computing the homotopy groups of a spectrum. The Adams spectral sequence comes out of selecting a nice ring E, localizing the category of spectra at E, and trying to do some crafty homological algebra. What pops out, after a lot of thought, is a spectral sequence with $E_2$ term given by $E_2^{s, t} = \mathrm{Ext}_{E_* E}^{s, t}(E_*, E_* X)$ converging conditionally to something like $E_2 \Rightarrow \pi_* L_E X$.

The stacky perspective on this whole thing is worth touching on, since the theory of stacks is how we arrive at and understand the above $E_2$ term. A stack is, up to some trouble with functoriality, what you’d get if you took a sheaf and replaced the word ‘set’ everywhere with ‘groupoid’. The pair $(E_*, E_* E) = (A, \Gamma)$ in fact represents a stack, in the sense that the pair of functors $(\mathrm{Hom}(E_*, -), \mathrm{Hom}(E_* E, -))$ form a sheaf of groupoids on the category of commutative rings. In turn, we can consider the over-category of all presheaves of groupoids with a selected map down to $(A, \Gamma)$, and we can inherit the topology from below to consider sheaves on this new category, which we call sheaves on the stack $\mathrm{spec}\;(A, \Gamma)$. In this setting, quasicoherent sheaves correspond to $(A, \Gamma)$-comodules, by which we mean A-modules M with a coaction $M \to \Gamma \otimes_A M$. The modules $E_* X$ are in fact $(A, \Gamma)$-comodules, and hence correspond to quasicoherent sheaves on the stack associated to E, and the Adams spectral sequences relates their sheaf cohomology to the homotopy groups $\pi_* X$ (in the case that X is E-local, with some correction to be done in the case that it isn’t).

Anyway, in order to make use of this spectral sequence for $E = X(n+1)$, we need to show that is satisfies the niceness conditions alluded to above, as well as compute A and $\Gamma$ so that we can actually do the homological algebra required. To accomplish this, we can effectively re-run the standard structure analysis for MU, which at its core is powered by the canonical complex orientation $\mathbb{C}P^\infty \to MU$. Our spaces X(n) aren’t big enough to receive an orientation, but they do receive part of one; namely, the composites

are equal. Noting that the Thom spectrum of the unreduced tautological line bundle over $\mathbb{C}P^{n-1}$ is $\mathbb{C}P^n$, we get “partial” orientations $\mathbb{C}P^n \to X(n)$. This buys us

Calculation 1: For $k \le n$, we get: $X(n)_* \mathbb{C}P^k = X(n)_* \{\beta_1, \ldots, \beta_k\}$, and $X(n)_* X(k) = X(n)_*[b_0, \ldots, b_{k-1}] / \langle b_0 - 1 \rangle$ .  Moreover, $X(n+1)_* X(n+1)$ is flat over $X(n+1)_*$ and so represents a stack – in fact, it splits as: $X(n+1)_* \widehat{\otimes} \mathbb{Z}[b_0, \ldots, b_n] / \langle b_0 - 1\rangle$.

Given the standard calculation of the homology of SU(n) and related spaces, we can employ the Eilenberg-Moore spectral sequence to compute the homology of $F'_k$, and the Thom isomorphism and Atiyah-Hirzebruch spectral sequences finish the job to give

Calculation 2: $X(n+1)_* F_k$ is a subcomodule of $X(n+1)_* X(n+1)$ given by $X(n+1)_* X(n) \{1, b_n, \ldots, b_n^k\}$.

All this is enough to tell us that the Adams spectral sequence

is indeed meaningful.

Now we make a sequence of reductions, with the goal of establishing a vanishing line for this spectral sequence (i.e., a line above which all the Ext groups are zero). Our first step is to use flatness and the Künneth map to rewrite this as

Next, remembering the splitting noted above, we can use a fact due to Haynes Miller about the interaction between the coefficient ring $X(n+1)_*$ and the rest of the spectral sequence to discard $X(n+1)_*$ entirely in favor of

Now comes something more interesting. Given a pair of rings $(A, \Gamma)$ representing a stack and a faithfully flat map $f: A \to B$ of rings, we can build a new stack represented by $\Gamma_B = B \otimes_A \Gamma \otimes_A B$, and the stack $\mathrm{spec}\;(B, \Gamma_B)$ is equivalent to the original stack $\mathrm{spec}\;(A, \Gamma)$. Using this technique, the map $\mathbb{Z}_{(p)}[b_1, \ldots, b_n] \to \mathbb{Z}_{(p)}[b_n]$ is a faithfully flat map that induces an isomorphism between the above Ext groups and

significantly reducing our workload.

Now, we don’t particularly care about the exact information in this Ext calculation; all we want to get our hands on is the vanishing line. If we filter the underlying ring by powers of the ideal $\langle p \rangle$, then we arrive at a May spectral sequence (i.e., a Bockstein spectral sequence for Exts) which converges to the above Ext, up to some tensoring with $\mathbb{Z}_{(p)}$ – but this is a forgivable sin if we’re just looking for a vanishing line. The $E_2$ term of the May spectral sequence is

We can throw away more unnecessary information by establishing a vanishing line for the “underlying” spectral sequence (so to speak) by throwing away the tensor product, which is a sound thing to do because $X(n+1)_* R$ is a connective comodule. This reduces the situation to

What’s left now is just standard stuff; we’re done with all the creative Ext-jockeying. We can further reduce to

which is explicitly computable using the cobar resolution, where we can simply observe a vanishing line of slope $(2np^k - 1)^{-1}$. This is great, because it means if we pick k large enough, we can make the slope as small as we please!

Given this vanishing line, we can turn to the real prize. First, $\pi_* R$ acts on $\pi_* F_{p^k-1} \wedge R$ on the right, and this is reflected by a pairing in the relevant Adams spectral sequences. To achieve our original goal of proving $F_{p^k-1} \wedge \alpha^{-1} R$ contractible, we need to show that for every $\beta \in \pi_* F_{p^k-1} \wedge R$ we can find an m with $\beta \alpha^m = 0$. Since we’re already assuming that $X(n+1)_* \alpha$ is nilpotent, let’s replace $\alpha$ with one of its powers so that $X(n+1)_* \alpha = 0$. With this condition, we can find $\alpha$ somewhere in the spectral sequence – say, in

Now, if we choose k so that the associated vanishing line has slope less than $\mid s(t-s)^{-1} \mid$, for any

we can pick an m large enough so that

has coordinates lying above the vanishing line, and hence is zero. By localizing at $\alpha$, we conclude that $F_{p^k-1} \wedge \alpha^{-1} R$ has vanishing homotopy groups and so is contractible, proving the first half.