# 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 $\mathbf{CP}^\infty$:

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:

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 $K(n)^\ast[[t]]/([p^k](t))\cong K(n)^\ast[[t]]/(x^{p^{nk}})$.