Morava K-Theory
Published:
I’m not a homotopy theorist but ever since the paper by Abouzaid-McLean-Smith was released which showed the power of homotopy theory towards answering questions in symplectic geometry, I’ve been interested in stable homotopy theory. In particular, what the role of the Morava K-theories play. Of course, the Abouzaid-Blumberg paper is an earlier application of Morava K-theory to the Arnold conjecture. In that paper, one would like to define Hamiltonian Floer theory over $\mathbb{F}_p$ but because our counts of curves ought to divide by the order of finite stabilizer groups, it is too naive to do mod $p$ counts since $p$ would not be invertible. Instead, one uses the hierarchy of generalized cohomology theories $K_p(n)$ which approximates $\mathbb{F}_p$ as $n$ gets larger. The notation here with the subscript $p$ is unconventional but I wanted to emphasize that we’ve fixed a prime $p$. I’ll continue to do this for a while but will eventually suppress that part of the notation.
On a first pass, Morava K-theories are associative ring spectra constructed algebraically; $K_p(n)$ where $p$ is prime, at height $n$ (if $p >2$, then its an homotopy commutative ring spectrum). If $n=1$, we have $KU$ modulo $p$ (localized at $p$, I think). If $n >1$, then its not usually something geometric (as far as we know). Moreover, the ring $K_p(n)(pt) = \pi_* K_p(n)=\mathbb{F}_p[v,v^{-1}]$, $|v| = 2(p^n -1)$. If $n$ is very large, as I wrote above, we have an approximation of $\mathbb{F}_p$ because there is an isomorphism $H_*(X,K_p (n))\cong H_*(X,\mathbb{F}_p) \otimes_{\mathbb{F}_p} \mathbb{F}_p[v,v^-1]$.
Morava K-theory is not really about vector bundles but maybe for the following reason, people named it as K-theory. Over complex K-theory $KU$, there is a multiplication by $p$ map and we can quotient by the image of the map to get mod $p$ complex K-theory $KU/p$. It splits into $p -1$ direct summands where any of these summands is equivalent to (a shift of) $K(1)$.
Abelian Groups and $\text{Spec}(\mathbb{Z})$
But this sort of description doesn’t yet tell us the role of Morava K-theories. To describe this, let’s first think about abelian groups which are $\mathbb{Z}$-modules. If we want to study finitely generated abelian groups, we know that they are always isomorphic to $\mathbb{Z}^r \oplus \bigoplus_i \mathbb{Z}/p_i$ for finitely many $i$. If we study $\text{Spec}(\mathbb{Z})$, the closed points are maximal ideals given by $(p)$ where $p$ is prime and it has a non-closed point (which we can think of as being in the neighborhood of all the other points) given by the non-maximal prime ideal $(0)$. Corresponding to the primes, we have fields $\mathbb{Z}/p$ but we can also localize at $(p)$ and complete to get the $p$-adic integers $\mathbb{Z}_p$. If we localize at the prime ideal $(0)$, this means we invert everything and get $\mathbb{Q}$. This describes the “geography” of $\text{Spec}(\mathbb{Z})$ and in a sense, describes abelian groups.
What about for the category of spectra? A spectrum always gives a generalized (co)homology theory; for example, $H\mathbb{Z}$ gives us singular cohomology with $\mathbb{Z}$ coefficients and $H\mathbb{F}_p$ with $\mathbb{F}_p$ coefficients. If we impose the dimension axiom onto a generalized cohomology theory, it can only be singular cohomology with coefficients in an abelian group. So in some sense, the category of spectra incorporates abelian groups. Now, if we have some spaces $X,Y$ and a field $\mathbb{K}$, then $H_*(X\times Y, \mathbb{K}) \cong H_*(X,\mathbb{K}) \otimes_\mathbb{K} H_*(Y,\mathbb{K})$ by the Künneth formula. But if we use a different abelian group as coefficients, there are complications. For example, if we use a PID $R$, then some tor groups come into play that fit into a short exact sequence with the two groups above. More generally, there is a spectral sequence.
So fields have special properties among abelian groups. What spectra give us cohomology theories that have Künneth formulas? Above, we had that $\mathbb{K} \cong H_*(pt,\mathbb{K})$ so the tensoring that appears in the formula should be modified slightly. If we’re talking about a cohomology theory $E$, we want to know when $E^*(X\times Y) \cong E^*(X)\otimes_{E^*(pt)} E^*(Y)$.
The ones that do might be righly called a “field” in the category of spectra. As it turns out, if we want the spectra that are irreducible in some sense (don’t decompose into wedge products, for example) and aren’t just constructed by tensoring such as how $H^*(-,\mathbb{Q}) \otimes_\mathbb{Q} \mathbb{C} \cong H^*(-,\mathbb{C})$, then there is a classification. They are exactly the Morava K-theories where we’ll say that $K_p(\infty)=H\mathbb{F}_p$ and also $K_p(0) = H\mathbb{Q}$. So the “geography” of spectra is similar to this drawing of $\text{Spec}(\mathbb{Z})$ but now, between $\mathbb{Q}$ and $\mathbb{Z}/p$, we have a whole family of $K_p(n)$ interpolating between the two. The larger $n$ is, the more mod $p$ information is captured. (I learned this from a talk by Jacob Lurie. There he says more about Morava K-theories).
So sometimes, people say that Morava K-theories $K(n)$ (with a prime $p$ chosen but notationally suppressed) are to spectra as $\mathbb{Z}/p$ is to $\mathbb{Q}$ in abelian groups. Another view is: the prime ideals of the sphere spectrum are classified by the Morava K-theories and you can understand a finite spectrum by picking out the pieces each $K(n)$ sees and gluing them back together. But maybe it’s better to call the Morava K-theories the “fields” because of the Künneth formula analogy. But there is actually more to this story about fields. Michael Mandell told me (I may very well have made lots of mistakes in trasncribing what he said): One can think of Morava K-theories as fields in the sense that if we have a graded module spectrum over a Morava K-theory, it must be a free graded module.
In order to relate this to the discussion about primes, note that if we localize at a prime ideal and then quotient out that prime ideal, we get a field. For example, if we localize at $p$ in $\mathbb{Z}$, we invert all primes other than $p$; then if we quotient out the prime ideal, we’re left with a ring in which everything has been inverted and hence, is a field. Also, if we’re looking at a classical ring $R$, then we can study the category of chain complexes and the subcategory of those chain complexes quasi-isomorphic to bounded chain complexes of finitely generated projective modules. A perfect chain complex with only one nonzero term is called a perfect module. I think one can think of perfect modules as being built from $R$ using direct sums/summands, (de)suspensions, retracts, and that sort of thing. Mandell says that the subcategory of perfect modules for a Noetherian ring $R$ is more or less equivalent to the prime ideals of the ring $R$. The reference for this is the paper but it originally appears in a paper by Mike Hopkins (but is hard to find a free version). I’ll write the actual statement of the theorem below. But one can say that Morava K-theories are like the primes of spectra (every spectrum is an $\mathbb{S}$-module), then they sort of correspond to the perfect modules of $\mathbb{S}$ somehow.
Theorem (Hopkins): Let $R$ be a Noetherian ring (take as part of the definition that it’s commutative). Let $D^b(R)$ be the derived category of bounded complexes of finitely generated projective $R$-modules. Define the following sets: $T={\text{Triangulated full subcategories of } D^b(R) \text{ closed under direct sums}}$ and $S={\text{Subsets of Spec}\,(R) \text{ closed under specialization}}$.
Then, there are maps of sets $f:T \to S$ and $g:S \to T$ where $f(L) = {p \in \text{Spec}(R): p \in \text{supp} \, X \text{ for some } X \in L}$ and $g(P) = {\text{the smallest triangulated category closed under specialization containing } R/p, \forall p \in P }$.
The maps are inverse isomorphisms.
Ambidexterity and $K(n)$-Local Spectra
Let’s now take a digression and give some definitions that aren’t exclusively about Morava K-theory but are useful to know. In this section, we’ll eventually get to ambidexterity which I want to mention because Abouzaid-McLean-Smith were interested in generalized cohomology theories that were ambidexterous for the classifying space $BG$ when $G$ is a finite group (we want a Poincaré duality when it comes to any sort of Gromov-Witten theory but we need to contend with group actions and orbifolds).
Definition: Let $\mathcal{C}$ be a category with limits and colimits and $X$ a topological space. Then a local system $\mathcal{L}$ on $X$ with values in $\mathcal{C}$ has the following data.
- For every point $x \in X$, we have an object $\mathcal{L}_x \in \mathcal{C}$.
- For every path $p:[0,1] \to X$, there is an isomorphism $\mathcal{L}_p:\mathcal{L}_{p(0)} \to \mathcal{L}_{p(1)}$.
- For every 2-simplex $\sigma$ that has paths $p,q,r$ between points $x,y,z$, we have that $\mathcal{L}_r = \mathcal{L}_q \circ \mathcal{L}_p$.
The last condition tells us that $\mathcal{L}_p$ only depends on the homotopy class of $p$ (with fixed endpoints). And more obviously, when we concatenate paths, the isomorphisms also compose. An example would be to let $\mathcal{C}$ be a category of vector spaces and the data of a flat connection would give us a local system. Note that an equivalent definition of a local system is to begin with the fundamental groupoid $\pi_{\leq 1}X$ whose objects are points of $X$ and morphisms are homotopy classes of paths. Then, a local system is a functor $\mathcal{L}: \pi_{\leq 1}X \to \mathcal{C}$. Moreover, if we fix a basepoint $x \in X$ in a connected space, then we really just need to know what a particular $\mathcal{L}$ does at $x$ and then know the action of $\pi_1(X,x)$. Put another way, a local system on $X$ with values in $\mathcal{C}$ is equivalent to an assignment of an object of $\mathcal{C}$ to a basepoint $x$ and also an action of $\pi_1(X,x)$. So if $\mathcal{C}$ is the vategory of complex vector spaces, then local systems are given by complex representations of $\pi_1(X,x)$.
Let $\mathcal{C}^X$ be the category of local systems on $X$ with values in $\mathcal{C}$ where the objects are local systems (viewed as functors) and the morphisms are natural transformations. Note that a map $f:X \to Y$ of spaces induces a functor $f^*:\mathcal{C}^Y \to \mathcal{C}^X$ by pulling back local systems on $Y$ to $X$. When $\mathcal{C}$ has limits and colimits, then $f^*$ has a left-adjoint, denoted $f_!$, and right-adjoint $f_*$. If it turns out that these two functors are equal, we call this phenomenon ambidexterity.
So let’s say we have a map $f:X \to Y$ and that it has a norm map $Nm_f:f_! \cong f_*$. We’ll call $f$ and ambidexterous map. For example, if $f$ is a homotopy equivalence, then $f^*:\mathcal{C}^Y \to \mathcal{C}^X$ is an equivalence of categories. Whenever we have an equivalence of categories, it has a left and right adjoint and both of those are just the inverse equivalence. Another example is when $f:X \to Y$ is a fibration (and all maps can be converted into fibrations by replacing a space with homotopy equivalence spaces) and suppose the diagonal map $\Delta:X \to X \times X$ is ambidexterous, with $\alpha:\Delta_! \cong \Delta_*$. We can use $\alpha$ to construct a norm map for $f$ and if it’s an isomorphism, then $f$ is ambidexterous. We’ll say that a space is ambidexterous if the map to a point $X \to *$ is ambidexterous. This means that for any local system $\mathcal{L}$ on $X$, we’ll get a norm map that gives an isomorphism $H_0(X,\mathcal{L}) \to H^0(X,\mathcal{L})$. One nice thing about fibrations is that if $f:X \to Y$ is an ambidexterous fibration, then every fiber is an ambidexterous space. The converse is almost true. If every fiber is an ambidexterous space and we have some upper bound on their homotopy groups, then the whole fibration is ambidexterous. Another fact is that ambidexterity gives us a way to “integrate” $X$-families of morphisms between pairs of objects in $\mathcal{C}$ in a canonical way.
We’ll now assume a prime $p$ has been fixed.
Definition: Let a spectrum $X$ be $K(n)$-acyclic if $X \wedge K(n)\simeq 0$. A $K(n)$-local spectrum $Y$ is such that for all $K(n)$-acyclic $X$, the mapping spectrum is $F(X,Y) \simeq 0$. Let $\textbf{Sp}^{K(n)}$ be the $\infty$-category of $K(n)$-local spectra.
Remark: This definition could work for other types of spectra as well; just pick a spectrum $Z$ and declare that $X$ is $Z$-acyclic if $X \wedge Z \simeq 0$, for example.
Also, compare the situation to that of abelian groups. Let $A$ be an abelian group such that $A \otimes_\mathbb{Z} \mathbb{F}_p =0$. For example, $A=\mathbb{Z}[1/p]$ has no information mod $p$ and is considered $p$-acyclic. If $q\neq p$ are two primes, then the localization $\mathbb{Z}_{(q)}$ contains $1/p$ and so is $p$-acylic. Note that a nontrivial ring map $f:\mathbb{Z}_{(p)} \to \mathbb{Z}_{(q)}$ would have to preserve the unit and hence, $f(1) = f(q)f(1/q) = qf(1/q)$ and $f(1/q)=q^{-1}$ which doesn’t exist. So all these ring maps are trivial. Thus, $\mathbb{Z}_{(p)}$ is a good candidate for being a $p$-local ring from this point of view (and in fact, is).
But if we want to think of what it means to be $p$-local and what the corresponding category is, we really should take chain complexes of abelian groups which have been $p$-adically completed. Well, with spectra, $\mathbb{F}_p = K_p(\infty)$ so $K_p(n)$ should be viewed as some kind of mix between localizing and completing. If $n$ is small, it’s more like localizing and less like completing. For example, $n=0$ gives $H\mathbb{Q}$ and remember, $\mathbb{Q}$ is the localization of $\mathbb{Z}$ at the prime ideal $(0)$. For $n$ large, it’s more like completing, less like localizing. This is, of course, a very loose and informal viewpoint.
A topological space is called $\pi$-finite if it has only finitely many nontrivial homotopy groups and each one of the homotopy groups is finite. For example, if $G$ is a discrete group, then $BG = K(G,1)$ is $\pi$-finite. Other Eilenberg-MacLane spaces are also $\pi$-finite.
Theorem (Hopkins-Lurie): Let $0 \leq n < \infty$ and $\mathcal{C}=\textbf{Sp}^{K(n)}$ be the $\infty$-category of $K(n)$-local spectra. Then every $\pi$-finite space is ambidexterous with respect to $\textbf{Sp}^{K(n)}$.
So if $X$ is a $\pi$-finite space, then we get some norm map that gives us an isomorphism $K(n)_*(X) \to K(n)^*(X)$. Ravenel-Wilson gave some computations of these things and $K(n)_*(X)$ is really just the dual of $K(n)^*(X)$. So ambidexterity gives us an isomorphism of the dual objects.
One manifestation of this theorem is to let $G$ be a finite $p$-group; i.e. all its elements are of order $p^k$ for some $k$. Let $X = BG = K(G,1)$ and recall that $K(1)$ is constructs complex K-theory mod $p$. If we take the $p$-adically completed complex K-theory $K^\wedge_p(BG)$, it is isomorphic to the representation ring of $G$, $p$-adically completed: $\text{Rep}(G)\otimes_\mathbb{Z} \mathbb{Z}_p$. Then, ambidexterity says that there is an isomorphism between this ring and its dual; that’s exactly the data of a Frobenius algebra. Thus, one can guess that this is related to 2D TQFTs because any 2D TQFT is really just the data of a commutative Frobenius algebra. In particular, this is related to a type of TQFT called Djikgraaf-Witten theory which is sort of about counting $G$-bundles over manifolds.
Chromatic Homotopy Theory
For some exposition on this topic, see this paper by Colin Ni titled “Lubin-Tate and Algebraic Geometry in Chromatic Homotopy Theory.” I wanted to include this section just because I find it fascinating (and there is much more to say)
Recall that a generalized cohomology theory is complex oriented if it admits a theory of Chern classes (so it should have some splitting principle, for example). A formal group law is a formal power series $F(X,Y)$ in two variables over a commutative ring $A$ such that $F(X, Y ) = F(Y, X) = X + Y + \text{higher order terms}$ and $F(X, F(Y, Z)) = F(F(X, Y ), Z)$. A formal group law is exactly the sort of object we can use to try and develop a theory of Chern classes for complex line bundles. For example, think of $X,Y$ as vector bundles and we want to generalize the relations we see from singular cohomology.
Now, complex cobordism $MU^*$ is a generalized cohomology theory that is universal in the sense that every complex oriented cohomology theory is classified by a map related to complex cobordism. By Quillen’s work, we know that $\pi_* MU \cong L$, where $L$ is the Lazard ring which is a universal ring that classifies formal group laws; i.e. given a formal group law over a ring $R$, it is determined by a morphism $L \to R$.
On the other hand, complex cobordism is very hard to understand. What one can do is perform localization of $MU$ at a prime $p$ (we get the $p$-localization $MU(p)$ ). This, in fact, decomposes as a direct sum and the direct summands are called the Brown-Peterson spectra. If I recall, correctly, Akira Tominaga (back in 2022 Vancouver), told me that we can use Morava K-theories to understand these Brown-Peterson spectra better and hence, understand complex cobordism better.
Here’s a fact: For each $n > 0$ and fixed prime $p$, there is a field spectrum $K(n)$ called Morava K-theory satisfying $\pi_* K(n) = \mathbb{F}_p[v_n, v^{-1}_n]$ where $v_n$ lives in degree $2(p^n-1)$, so that the formal group law of $K(n)$ has height exactly $n$. By convention, we’ve said earlier that $K(0) = H\mathbb{Q}$ which is the Eilenberg-Maclane spectra that gives rational cohomology. In odd degree, the rationalization $S^{2n+1}_\mathbb{Q} \simeq K(\mathbb{Q},2n+1)$ by Serre’s work. Serre also showed that for an even sphere $S^{2n}$, the only homotopy groups of $S^{2n}$ which are not torsion are $\pi_{2n}(S^{2n})$ and $\pi_{4n-1}(S^{2n})$. Another fact about $K(n)$ is that it is an associative ring spectrum for each $p$ and each $n$; so we call it an $\mathbb{E}_1 = A_\infty$-ring.
Anyways, back to complex oriented theories. Given a formal group law classified by a map $F:L \to R$ where $R$ is a commutative ring, we might ask if it arises from a cohomology theory $E$ where $\pi_*E = R$. For example, we could just try tensoring $MU_* \otimes_L R$ to form our theory; $R$ is an $L$-module via $F$. However, there’s the pesky problem that tensoring $-\otimes_L R$ isn’t an exact functor and so our candidate theory lacks the sorts of properties we may want. Luckily, there is a condition known as Landweber exactness which we can check on $R$ (a purely algebraic condition) and $MU_* \otimes_L R$ is a homology theory if and only if $R$ is Landweber exact.
There is more excitment. It turns out that one can actually form the moduli space of formal group laws over characteristic $p$ into a stack $\mathcal{M}_{fg}$ that is stratified by levels $\mathcal{M}^n_{fg}$. I haven’t really defined height but the stratification comes from that. The rough idea of height is that we iteratively take $F(F(F…),X)$ $p$ times; we’ll obtain a power series whose lowest term with nonvanishing coefficient is $X^{p^n}$; we take $n$ as the height.
Now, above, we talked about $K(n)$-local spectra. There is actually a procedure for taking a spectrum and turning it into a $K(n)$-local spectra; this is Bousfield localization. The procedure is even functorial so we’ll write the functor $L_{K(n)}$. We can ask how this functor acts on complex oriented cohomology theories. To address this, we can basically study the formal group laws. An amazing result is that (roughly speaking), one can understand $L_{K(n)}$ acting on the theories of height $n$ in terms of the geometry of the stratum $\mathcal{M}_{fg}^n$. More specifically, it acts like completion along this locally closed substack.
There is a lot more to say about this circle of ideas such as how this relates to Lubin-Tate deformation theory as a way to understand the geometry of the stack of formal group laws. I recommend taking a look at the survey paper linked above.