# nLab Morava K-theory

Morava K-theory

cohomology

### Theorems

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### 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 $p$ the Morava K-theories are the sequences

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

of multiplicative generalized cohomology/homology theories 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 $X$ is as above, then $\overline{K(n)}_\ast(X)=K(n)_\ast(pt.)\otimes \overline{H}_\ast(X;\mathbb{Z}/(p))$ for $n$ sufficiently large.

###### Remark

Due to the third point in , 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):

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 May 14, 2024 at 16:23:17. See the history of this page for a list of all contributions to it.