Let’s pick up from where we were before. Just to recall, in the previous post, we showed that every complex-oriented cohomology theory gives rise to a formal group law over . In addition, formal group laws are classified by the Lazard ring , that is isomorphic as a graded ring to , so every complex orientation of gives rise to a formal group law over . The question we posed at the end of the previous post was the following. Given an -module we can form the functor , but this isn’t necessarily a cohomology theory. Our goal is to find the conditions on for which 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 gives a map of schemes ; the latter is infinite-dimensional affine space. Define to be . However, if we are considering formal groups, i.e., formal groups modulo the group of coordinate changes. This group is the group of power series . The composite map gives the formal group underlying the formal group law . Thus, define , where acts on by sending . Returning back to our question of whether is a cohomology theory, it turns out that it suffices to show that the composite is flat. This motivates:
Definition: A formal group over is Landweber exact if the map 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 be a commutative ring and and -module. A sequence of elements is said to be regular for if is not a zero divisor on , is not a zero divisor on , is not a zero divisor on , etc (so that is not a zero divisor for ). Then the Landweber exact functor theorem is:
Theorem: A -module is flat over if and only if for every prime the elements is a regular sequence for , where the elements are defined as the coefficient of in the -series of the universal formal group law over .
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 be a ring spectrum. Say that it is weakly even periodic if vanishes for odd and the map is an isomorphism. (Therefore, .) is even periodic if vanishes for odd and there is an invertible such that multiplication by gives isomorphisms .
If 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 is such that the map is surjective, then the Atiyah-Hirzebruch spectral sequence degenerates (at least for , so that we can compute ). Then consider the image of in . This is equivalent to a pointed map , and a complex orientation of is an extension to :
To solve this extension problem, we can use obstruction theory by filtering as . The obstruction to extending the map to lies in . Therefore, if for all , then is complex-orientable, and hence if is weakly even periodic, then it is complex-orientable. In addition, if is weakly even periodic, then the formal group law of reduces to a formal group law over , in the sense that splits as . This is because . 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 ) and the homotopy category of weakly even periodic spectra. The inverse functor sends to the formal group .
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 , we can stratify the moduli stack by substacks that parametrize formal groups of height , and the locally closed strata defined as . For example, a formal group law has height if and only if is invertible in , and height if and only if in . Hence, , which is isomorphic to the classifying stack (where is the multiplicative formal group) over , and .
For example: denote by the algebraic group with . Then is the subgroup of with those such that if , and is the subgroup of with . We then see that where and (exercise). Another example: if is a ring of characteristic , then a formal group law of height over is isomorphic to the additive formal group law. Hence .
Consider, now, the cohomology theory ; then is an -module, i.e., a quasicoherent sheaf on . It turns out that the action of can be lifted to an action on the sheaf , so is really a quasicoherent sheaf on . In other words, takes the stable homotopy category to . 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 , we need to define Morava K-theories and Morava E-theories. Let be a formal group over a field . A deformation of is a local Artin ring , with residue field , with a formal group over such that the restriction to under the quotient map is . Lubin and Tate proved:
Proposition: Suppose is a formal group law of height over a perfect field of characteristic (for example, the finite field ). There is a universal deformation of over with residue field , where is the Witt vectors of , such that if is any local Artin ring, with residue field , then a deformation of over is equivalent to specifying a map:
It can be shown that this universal formal group is Landweber exact, and hence gives a spectrum , called Morava E-theory. Goerss-Hopkins-Miller proves that this spectrum is an -ring spectrum. However, , the original formal group, need not be Landweber exact. Suppose ; then let denote the cofiber of the map given by multiplication by . Consider the tensor product in : , which is independent of ; thus where . This is called Morava K-theory. (Note that this is -periodic, and not -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 of . The inclusion of into satisfies the conditions of the last theorem we stated, and hence gives a cohomology theory, , called the Johnson-Wilson spectrum. This has coefficient ring where . This is Bousfield equivalent to , i.e., if and only if where is a spectrum.
Briefly, since we will need it for discussion below, this is what localization is. Let be a spectrum. A spectrum is said to be -acyclic if , and a spectrum is -local if a map is nullhomotopic whenever is -acyclic. Define a localization functor from spectra to spectra such that: is -local for every spectrum , and there is a natural localization map that is an isomorphism on -homology.
Geometrically, one can interpret localization at as follows: the localization is the restriction to the complement of , and is completing along . When and is of height (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 as .
Let’s now prove something interesting about localization.
Proposition: Let , , and be spectra such that is -acyclic. Then we claim that arises as the (homotopy) pullback of .
To see this, suppose is the pullback of the diagram; then we have an obvious map . Now, the map is an equivalence in -homology, and the map is an equivalence in -homology. Consider, now, the map ; this is an equivalence in and -homology. Composing this with the map and gives an equivalence in and -homology, respectively. By the definition of localization, the map and splits through , so must be isomorphic to . 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): is the homotopy pullback of .
To see this, since is Bousfield equivalent to , it suffices to show that if then . But it is known (not proved here) that is smashing, i.e., where 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 . Let be the first Chern class of a generator of , and consider the fibration . Then the Gysin sequence is a long exact sequence:
Since is not a zero divisor, we see that is . In exactly the same way, we can compute to be .