Chromatic Homotopy Theory – II

Let’s pick up from where we were before. Just to recall, in the previous post, we showed that every complex-oriented cohomology theory E gives rise to a formal group law over \pi_\ast E. In addition, formal group laws are classified by the Lazard ring L, that is isomorphic as a graded ring to MU^\ast, so every complex orientation of E gives rise to a formal group law over \pi_\ast E. The question we posed at the end of the previous post was the following. Given an L-module R we can form the functor E^\ast(X) = MU^\ast(X)\otimes_L R, but this isn’t necessarily a cohomology theory. Our goal is to find the conditions on R for which E is a cohomology theory. To do this, we must use the theory of stacks. This post is consequently uses a fair bit of algebraic geometry.

A map L\to R gives a map of schemes \mathrm{Spec}(R)\to\mathrm{Spec}(L); the latter is infinite-dimensional affine space. Define \mathcal{M}_\mathrm{FGL} to be \mathrm{Spec}(L). However, if we are considering formal groups, i.e., formal groups modulo the group of coordinate changes. This group G is the group of power series t+a_1t^2+a_2t^3+\cdots. The composite map \mathrm{Spec}(R)\to\mathrm{Spec}(L)\to\mathrm{Spec}(L)/G gives the formal group underlying the formal group law \mathrm{Spec}(R)\to\mathrm{Spec}(L). Thus, define \mathrm{Spec}(L)/G = \mathcal{M}_\mathrm{FG}, where G acts on \mathrm{Spec}(L) by sending (g(t) = b_0t + b_1t^2 + \cdots,F(x,y))\mapsto g(F(g^{-1}(x),g^{-1}(y))). Returning back to our question of whether E^\ast(X) = MU^\ast(X)\otimes_L R is a cohomology theory, it turns out that it suffices to show that the composite \mathrm{Spec}(R)\to\mathcal{M}_\mathrm{FGL}\to\mathcal{M}_\mathrm{FG} is flat. This motivates:

Definition: A formal group over R is Landweber exact if the map \mathrm{Spec}(R)\to\mathcal{M}_\mathrm{FG} is flat. A formal group law is Landweber exact if its underlying formal group is.

The Landweber exact functor theorem gives a necessary and sufficient condition for a formal group law to be Landweber exact. First, we need a definition. Let R be a commutative ring and M and R-module. A sequence of elements x_0,x_1\cdots\in R is said to be regular for M if x_0 is not a zero divisor on M, x_1 is not a zero divisor on M/x_0M, x_2 is not a zero divisor on M/(x_0M + x_1M), etc (so that x_n is not a zero divisor for M/\left(\sum^{n-1}_{k=0}x_kM\right)). Then the Landweber exact functor theorem is:

Theorem: A L-module M is flat over \mathcal{M}_\mathrm{FG} if and only if for every prime p the elements p,v_1,v_2,\cdots\in L is a regular sequence for M, where the elements v_i are defined as the coefficient of t^{p^n} in the p-series of the universal formal group law over L.

Now we can address the question of forming a cohomology theory from (certain kinds of) formal groups, as a kind of inverse to the functor from complex-oriented cohomology theories to formal group laws (and hence to formal groups). We first need a definition.

Definition: Let E be a ring spectrum. Say that it is weakly even periodic if \pi_i E vanishes for odd i and the map \pi_2\otimes_{\pi_0 E}\pi_{-2}E\to \pi_0 E is an isomorphism. (Therefore, \pi_{2k}E \cong(\pi_2 E)^{\otimes k}.) E is even periodic if \pi_i E vanishes for odd i and there is an invertible \alpha\in \pi_{-2}E such that multiplication by \alpha gives isomorphisms \pi_k E\cong \pi_{k-2}E.

If E is weakly even periodic, then it is complex-orientable. To see this, we first claim (without proof; see Proposition 7 of Lecture 4 of Lurie’s chromatic homotopy theory notes) that if E is such that the map E^2(\mathbf{CP}^\infty)\to E^2(S^2) is surjective, then the Atiyah-Hirzebruch spectral sequence degenerates (at least for \mathbf{CP}^n, so that we can compute E^\ast(\mathbf{CP}^n)). Then consider the image x of t\in \widetilde{E}^2(\mathbf{CP}^\infty) in \widetilde{E}^2(S^2). This is equivalent to a pointed map S^2\to\Omega^\infty E, and a complex orientation of E is an extension to \mathbf{CP}^\infty:

complex orientation

To solve this extension problem, we can use obstruction theory by filtering \mathbf{CP}^\infty as \mathbf{CP}^n\subset\mathbf{CP}^{n+1}. The obstruction to extending the map \mathbf{CP}^n\to\Omega^\infty E to \mathbf{CP}^{n+1} lies in \pi_{2n+1}\Omega^\infty E = \pi_{2n+1}E. Therefore, if \pi_{2n+1}E = \pi_{2m+1}E for all m,n>1, then E is complex-orientable, and hence if E is weakly even periodic, then it is complex-orientable. In addition, if E is weakly even periodic, then the formal group law of E reduces to a formal group law over \pi_0 E, in the sense that \mathrm{Spec}(\pi_\ast E)\to\mathcal{M}_\mathrm{FG} splits as \mathrm{Spec}(\pi_\ast E)\to \mathrm{Spec}(\pi_0 E)\to \mathcal{M}_\mathrm{FG}. This is because E^\ast(\mathbf{CP}^\infty)\cong E^0(\mathbf{CP}^\infty)\otimes_{\pi_0E}\pi_\ast E. The importance of weakly even periodic ring spectra is as follows.

Theorem: There is an equivalence between the category of Landweber exact formal groups (flat maps \mathrm{Spec}(R)\to\mathcal{M}_\mathrm{FG}) and the homotopy category of weakly even periodic spectra. The inverse functor sends E to the formal group \mathrm{Spec}(\pi_0 E)\to \mathcal{M}_\mathrm{FG}.

This is already a very interesting relationship between the stable homotopy category and the moduli stack of formal groups; but it goes deeper. At every prime p, we can stratify the moduli stack \mathcal{M}_\mathrm{FG}\times\mathrm{Spec}(\mathbf{Z}_{(p)}) by substacks \mathcal{M}_\mathrm{FG}^{\geq n} that parametrize formal groups of height \geq n, and the locally closed strata defined as \mathcal{M}_\mathrm{FG}^n = \mathcal{M}_\mathrm{FG}^{\geq n} - \mathcal{M}_\mathrm{FG}^{\geq n-1}. For example, a formal group law f has height 0 if and only if p is invertible in R, and height \geq 1 if and only if p = 0 in R. Hence, \mathcal{M}_\mathrm{FG}^0 = \mathcal{M}_\mathrm{FG}\times\mathrm{Spec}(\mathbf{Q}), which is isomorphic to the classifying stack B\mathbf{G}_m (where \mathbf{G}_m is the multiplicative formal group) over \mathrm{Spec}(\mathbf{Q}), and \mathcal{M}_\mathrm{FG}^{\geq 1} = \mathcal{M}_\mathrm{FG}\times\mathrm{Spec}(\mathbf{Z}/p\mathbf{Z}).

For example: denote by G^+ the algebraic group with G^+(R) = \{f\in R[[t]]|g(t) = b_0t+b_1t^2+\cdots,b+0\in R^\times\}. Then \mathbf{G}_m is the subgroup of G^+ with those b_0t+b_1t^2+\cdots such that b_i = 0 if i>0, and G is the subgroup of G^+ with b_0 = 1. We then see that \mathcal{M}_\mathrm{FG}^{\geq n} = \mathrm{Spec}(L_{(p)}/(v_0,\cdots,v_{n-1}))/G^+ where L_{(p)} \cong \mathbf{Z}_{(p)}\otimes L and \mathcal{M}_\mathrm{FG}^n = \mathrm{Spec}(L_{(p)}[v_n^{-1}]/(v_0,\cdots,v_{n-1}))/G^+ (exercise). Another example: if R is a ring of characteristic p, then a formal group law of height \infty over R is isomorphic to the additive formal group law. Hence \mathcal{M}_\mathrm{FG}^\infty = \mathcal{M}_\mathrm{FG}^{\geq \infty} = B\mathrm{Aut}(\mathbf{G}_a).

Consider, now, the cohomology theory MU; then MU_\ast(X) is an L-module, i.e., a quasicoherent sheaf on \mathcal{M}_\mathrm{FGL}. It turns out that the action of G can be lifted to an action on the sheaf MU_\ast(X), so MU_\ast(X) is really a quasicoherent sheaf on \mathcal{M}_\mathrm{FG}. In other words, MU takes the stable homotopy category to \mathbf{QCoh}(\mathcal{M}_\mathrm{FG}). What if we localize at a prime? To understand how we can “complete” the stable homotopy category with respect to the above-defined strata of \mathcal{M}_\mathrm{FG}, we need to define Morava K-theories and Morava E-theories. Let \mathbf{G}_0 be a formal group over a field k. A deformation of \mathbf{G}_0 is a local Artin ring R, with residue field k, with a formal group \mathbf{G} over R such that the restriction to k under the quotient map R\to k is \mathbf{G}_0. Lubin and Tate proved:

Proposition: Suppose \mathbf{G}_0 is a formal group law of height n over a perfect field k of characteristic p (for example, the finite field \mathbf{Z}/p\mathbf{Z}). There is a universal deformation of \mathbf{G}_0 over W(k)[[v_1,\cdots,v_{n-1}]] with residue field k, where W(k) is the Witt vectors of k, such that if R is any local Artin ring, with residue field k, then a deformation of \mathbf{G}_0 over R is equivalent to specifying a map:

deformation fgl

It can be shown that this universal formal group \mathbf{G} is Landweber exact, and hence gives a spectrum E(k,\mathbf{G}), called Morava E-theory. Goerss-Hopkins-Miller proves that this spectrum is an \mathbf{E}_\infty-ring spectrum. However, \mathbf{G}_0, the original formal group, need not be Landweber exact. Suppose v_0 = p\in W(k)[[v_1,\cdots,v_{n-1}]]\cong \pi_0E(k,\mathbf{G}); then let M(k) denote the cofiber of the map E(k,\mathbf{G})\to E(k,\mathbf{G}) given by multiplication by v_k\in \pi_0 E(k,\mathbf{G}). Consider the tensor product in \mathbf{Mod}_{E(k,\Gamma)}(\mathbf{Sp}): K(n) = \bigotimes^{n-1}_{k=0}M(k), which is independent of v_1,\cdots,v_n\in W(k)[[v_1,\cdots,v_{n-1}]]; thus \pi_\ast K(n) \cong k[\beta^{\pm 1}] where \beta\in \pi_2 K(n). This is called Morava K-theory. (Note that this is 2-periodic, and not 2(p^n-1)-periodic, Morava K-theories (the latter is what is more standard, I think).)

Another spectrum that can be formed is as follows. Consider the complement \mathcal{M}_\mathrm{FG}^{\leq n} of \mathcal{M}_\mathrm{FG}^{\geq n+1}. The inclusion of \mathcal{M}_\mathrm{FG}^{\leq n} into \mathcal{M}_\mathrm{FG} satisfies the conditions of the last theorem we stated, and hence gives a cohomology theory, E(n+1), called the Johnson-Wilson spectrum. This has coefficient ring \pi_\ast E(n+1)\cong\mathbf{Z}_{(p)}[v_1,\cdots,v_{n1},v_{n+1}^{\pm 1}] where |v_i| = 2(p^i-1). This is Bousfield equivalent to K(0)\vee K(1)\vee\cdots\vee K(n), i.e., E(n)_\ast(X)\cong 0 if and only if (K(0)\vee K(1)\vee\cdots\vee K(n))_\ast(X)\cong 0 where X is a spectrum.

Briefly, since we will need it for discussion below, this is what localization is. Let E be a spectrum. A spectrum X is said to be E-acyclic if \pi_\ast E\otimes X\cong 0, and a spectrum Y is E-local if a map X\to Y is nullhomotopic whenever X is E-acyclic. Define a localization functor L_E from spectra to spectra such that: L_EX is E-local for every spectrum X, and there is a natural localization map X\to L_EX that is an isomorphism on E-homology.

Geometrically, one can interpret localization at E(k,\mathbf{G}) as follows: the localization L_{E(k,\mathbf{G})} is the restriction to the complement \mathcal{M}_\mathrm{FG}^{\leq n} of \mathcal{M}_\mathrm{FG}^{\geq n+1}, and L_{K(n)} is completing along \mathcal{M}_\mathrm{FG}^n. When k=\overline{\mathbf{F}_p} and \mathbf{G} is of height n (this is unique (called the Honda formal group), since Lazard proved that formal groups over an algebraically closed field are classified completely by their height), we denote E(k,\mathbf{G}) as E_n.

Let’s now prove something interesting about localization.

Proposition: Let E, F, and X be spectra such that L_F X is E-acyclic. Then we claim that L_{E\vee F}X arises as the (homotopy) pullback of L_F X\leftarrow L_FL_E X\rightarrow L_E X.

To see this, suppose Y is the pullback of the diagram; then we have an obvious map L_{E\vee F} X\to Y. Now, the map Y\to L_F X is an equivalence in F-homology, and the map Y\to L_E X is an equivalence in E-homology. Consider, now, the map X\to L_{E\vee F} X; this is an equivalence in E and F-homology. Composing this with the map L_{E\vee F} X\to L_F X and L_{E\vee F} X\to L_E X gives an equivalence in F and E-homology, respectively. By the definition of localization, the map Y\to L_F X and Y\to L_E X splits through L_{E\vee F} X, so Y must be isomorphic to L_{E\vee F} X. As a special case, we get the incredibly important chromatic fracture square (important because it relates different heights in stable homotopy theory).

Corollary (chromatic fracture square): L_{E(n)}X is the homotopy pullback of L_{E(n-1)} X\leftarrow L_{E(n-1)}L_{K(n)} X\rightarrow L_{K(n)} X.

To see this, since E(n) is Bousfield equivalent to K(0)\vee K(1)\vee\cdots\vee K(n), it suffices to show that if K(n)\wedge X\simeq 0 then K(n)\wedge L_{E(n-1)}X\simeq 0. But it is known (not proved here) that L_{E(n)} is smashing, i.e., L_{E(n)} X\simeq X\wedge L_{E(n)}S where S is the sphere spectrum, so we’re done.

Let us conclude with two (really just one) computations, the first of which is used in Hopkins-Kuhn-Ravenel character theory (which I plan on learning sometime soon), namely E_n^\ast(B\mathbf{Z}/n\mathbf{Z}). Let x\in E_n^2(B\mathbf{Z}/n\mathbf{Z}) be the first Chern class of a generator of \mathrm{Hom}(\mathbf{Z}/n\mathbf{Z},U(1)), and consider the fibration S^1\to B\mathbf{Z}/n\mathbf{Z}\to BU(1). Then the Gysin sequence is a long exact sequence:

\cdots\to E_n^\ast[[t]] \xrightarrow{\cdot[n](t)} E_n^\ast[[t]]\to E_n^\ast(B\mathbf{Z}/n\mathbf{Z})\to\cdots

Since [n](t) is not a zero divisor, we see that E_n^\ast(B\mathbf{Z}/n\mathbf{Z}) is E_n^\ast[[t]]/([n](t)). In exactly the same way, we can compute K(n)^\ast(B\mathbf{Z}/p^k\mathbf{Z}) = K(n)^\ast(K(\mathbf{Z}/p^k\mathbf{Z},1)) to be K(n)^\ast[[t]]/([p^k](t))\cong K(n)^\ast[[t]]/(x^{p^{nk}}).


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