nLab Morava K-theory

Morava K-theory

Context

Higher algebra

higher algebra

universal algebra

Morava K-theory

Idea

For each prime $p$, the Morava K-theories are a tower $\{K(n)\}_{n \in \mathbb{N}}$ of complex oriented cohomology theories whose coefficient ring is

$\pi_\bullet\left(K\left(n\right)\right) \simeq \mathbb{F}_p [v_n, v_n^{-1}]$

where $v_n$ is in degree $2(p^n-1)$.

Hence with $p = 2$ for $n = 1$ $v_1$ is a Bott element of degree 2 and $K(1)$ is closely related to complex K-theory, while for $n= 2$ $v_2$ is then a Bott element of degree 6 and $K(2)$ is closely related to elliptic cohomology.

There is also integral Morava K-theory which instead has coefficient ring

$\pi_\bullet\left(K\left(n\right)\right) \simeq \mathbb{Z}_{(p)} [v_n, v_n^{-1}] \,,$

where $\mathbb{Z}_{(p)}$ is the p-adic integers.

Integral Morava K-theory can be obtained as a localization of a quotient $MU/I$ of complex cobordism cohomology theory $MU$ (Buhné 11).

Definition

We need the following standard notation throughout this entry.

Definition

For $p \in \mathbb{N}$ a prime number, we write

• $\mathbb{F}_p = \mathbb{Z}/(p)$ for the field with $p$ elements;

• $\mathbb{Z}_{(p)}$ for the localization ring of the integers at $p$;

• $\mathbb{Z}_p$ for the p-adic integers.

Axiomatic characterization

Proposition/Definition

For each prime integer $p$ there exists a sequence of multiplicative generalized cohomology/homology theories

$\{K(n)\}_{n \in \mathbb{N}}$

with the following properties:

1. $K(0)_\ast(X)=H_\ast(X;\mathbb{Q})$ and $\overline{K(0)}_\ast(X)=0$ when $\overline{H}_\ast(X)$ is all torsion.

2. $K(1)_\ast(X)$ is one of $p-1$ isomorphic summands of mod-$p$ complex topological K-theory.

3. $K(0)_\ast(pt.)=\mathbb{Q}$ and for $n\neq 0$, $K(n)_\ast(pt.)=\mathbb{F}_p[v_n,v_n^{-1}]$ where $\vert v_n\vert=2p^n-2$.

(This ring is a graded field in the sense that every graded module over it is free. $K(n)_\ast(X)$ is a module over $K(n)_\ast(pt.)$, see below)

4. There is a Künneth isomorphism: $K(n)_\ast(X\times Y)\cong K(n)_\ast(X)\otimes_{K(n)_\ast(pt.)}K(n)_\ast(Y).$

5. Let $X$ be a p-local finite CW-complex. If $\overline{K(n)}_\ast(X)$ vanishes then so does $\overline{K(n-1)}_\ast(X)$.

6. If $X$ as above is not contractible then $\overline{K(n)}_\ast(X)=K(n)_\ast(pt.)\otimes \overline{H}_\ast(X;\mathbb{Z}/(p))$.

These are called the Morava K-theories.

Due to the third point one may regard $K(n)$ as a ∞-field among the A-infinity rings. See below.

Properties

Universal characterization

Proposition

For each prime number $p$ and each $n \in \mathbb{N}$, the Morava K-theory $K(n)$ is, up to equivalence, the unique spectrum underlying an homotopy associative spectrum which is

1. whose formal group has height exactly $n$;

2. whose homotopy groups are $\pi_\bullet \simeq \mathbb{F}_p[v_n^\pm]$. (with $v_n$ defined as at height).

For instance (Lurie, lecture 24, prop. 11).

Ring structure

Proposition

$K(n)$ admits the structure of an A-∞ algebra, in fact of an $MU_{(p)}$-A-∞ algebra.

Due to Robinson (and Andrew Baker at $p = 2$). (See e.g. Lurie 10, lecture 22, lemma 2)

Remark

With the exception of the extreme case of $n=0$, the fields $K(n)$ do not admit E-∞-ring multiplicative structures. However, when $p\neq 2$, the multiplication is homotopy commutative. For $p = 2$ it is not even homotopy commutative. Nevertheless, for many spaces $X$, the $K(n)$-generalized cohomology at the prime $2$ of $X$ forms a commutative ring.

As $A_\infty$-fields

Proposition

If $E$ is an ∞-field then $E \otimes K(n) \neq 0$ and $E$ admits the structure of a $K(n)$-module.

This appears for instance as (Lurie, lecture 24, prop. 9, remark 13)

Remark

This means that the Morava $A_\infty$-rings $K(n)$ are essentially the only ∞-fields in the stable homotopy category.

As the primes in the $\infty$-category of spectra

The Morava K-theories label the prime spectrum of a symmetric monoidal stable (∞,1)-category of the (∞,1)-category of spectra for p-local and finite spectra . This is the content of the thick subcategory theorem.

Relation to chromatic homotopy theory

The layers in the chromatic tower capture periodic phenomena in stable homotopy theory, corresponding to the Morava K-theory $E_\infty$-fields.

Specifically the Bousfield localization of spectra $L_{K(n)}$ acts on complex oriented cohomology theories like completion along the locally closed substack

$\mathcal{M}^n_{FG} \hookrightarrow \mathcal{M}_{FG}$

of the moduli stack of formal groups at those of height $n$.

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$HZR-theory
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$KR-theory
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomologyelliptic spectrum $Ell_E$
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
3 …10K3 cohomologyK3 spectrum
$n$$n$th Morava K-theory$K(n)$
$n$th Morava E-theory$E(n)$BPR-theory
$n+1$algebraic K-theory applied to chrom. level $n$$K(E_n)$ (red-shift conjecture)
$\infty$complex cobordism cohomologyMUMR-theory

Relation to Bousfield lattice

It is known that in the Bousfield lattice of the stable homotopy category, the Bousfield classes of the Morava K-theories are minimal. It is conjectured by Mark Hovey and John Palmieri that the Boolean algebra contained in the Bousfield lattice is atomic and generated by the Morava K-theories and the spectra $A(n)$ which measure the failure of the telescope conjecture.

Orientation

The orientation of integral Morava K-theory is discussed in (Sati-Kriz 04, Buhné 11). It is essentially given by the vanishing of the seventh integral Stiefel-Whitney class $W_7$.

Notice that this is in higher analogy to how orientation in complex K-theory is given by the vanishing third integral Stiefel-Whitney class $W_3$ (spin^c-structure).

$\infty$-Group rings and twists

Write $gl_1(K(n))$ for the ∞-group of units of the (a) Morava K-theory spectrum

Proposition

For $p = 2$ and all $n \in \mathbb{N}$, there is an equivalence

$Maps(B^{n+1}U(1), B gl_1(K(n))) \simeq \mathbb{Z}_2$

between

and

Remark

By the discussion at (∞,1)-vector bundle this means that for each such map there is a type of twist of Morava K-theory (at $p = 2$).

References

Morava K-theory originates in unpublished preprints by Jack Morava in the early 1970s.

A first published account appears in (see at Johnson-Wilson spectrum):

• Doug Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Annals of Mathematics Studies 128, Princeton University Press (1992).

Textbook account:

A discussion with an eye towards category theoretic general abstract properties of localized stable homotopy theory is in

A survey of the theory is in

• Urs Würgler, Morava K-theories: a survey, Algebraic topology Poznan 1989, Lecture Notes in Math. 1474, Berlin: Springer, pp. 111 138 (1991) (doi:10.1007/BFb0084741)

In

• Jacob Lurie, Chromatic Homotopy Theory Lecture notes, (pdf)

Lecture 22 Morava E-theory and Morava K-theory (pdf)

Lecture 23 The Bousfield Classes of $E(n)$ and $K(n)$ (pdf)

Lecture 24 Uniqueness of Morava K-theory (pdf)

the explicit definition via formal group laws is in lecture 22 and the abstract characterization in lecture 24.

The $E_\infty$-algebra structure over $\widehat{E(n)}$ is comment on in

based on

• Neil Strickland, Products on $MU$-modules, Trans. Amer. Math. Soc. 351 (1999), 2569-2606.

Discussion in relation to the Arnold conjecture in symplectic topology:

On the Morava K-theory of iterated loop spaces of n-spheres:

The orientation of integral Morava K-theory is discussed in

Some twists of Morava K-theory/maps into its ∞-group of units as well as the Atiyah-Hirzebruch spectral sequence for Morava $K$ and Morava $E$ are discussed in

For a review in the context of M-theory see

Last revised on December 21, 2021 at 17:21:39. See the history of this page for a list of all contributions to it.