nLab 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.

Construction from complex cobordism

(e.g. Lurie 10, lecture 22, def. 5)

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.

(Ravenel 1992, Prop. 1.5.2)

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. complex oriented;

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

3. 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.

(e.g. Lurie 10, lecture 22, warning 6)

As $A_\infty$-fields

Proposition

If $E$ is an ∞-field and $E \otimes K(n) \neq 0$, then $E$ 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.

See (Lurie, lecture 24, remark 13)

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 $n$.

(Lurie 10, lecture 29)

chromatic homotopy theory

chromatic level	complex oriented cohomology theory	E-∞ ring/A-∞ ring	real oriented cohomology theory
0	ordinary cohomology	Eilenberg-MacLane spectrum $H \mathbb{Z}$	HZR-theory
	0th Morava K-theory	$K(0)$
1	complex K-theory	complex K-theory spectrum $KU$	KR-theory
	first Morava K-theory	$K(1)$
	first Morava E-theory	$E(1)$
2	elliptic cohomology	elliptic spectrum $Ell_E$
	second Morava K-theory	$K(2)$
	second Morava E-theory	$E(2)$
	algebraic K-theory of KU	$K(KU)$
3 …10	K3 cohomology	K3 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 cohomology	MU	MR-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

• the mapping space from the classifying space for circle (n+1)-bundles to the delooping of the ∞-group of units of $K(n)$

and

• the 2-adic integers.

(Sati-Westerland 11, theorem 1)

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 $p = 2$).

Related concepts

• equivariant Morava K-theory

• 2-periodic Morava K-theory

• K-theory, topological K-theory, algebraic K-theory

• chromatic homotopy theory

• K(n)-local stable homotopy theory

• integral Morava K-theory

• Morava E-theory

  • Lubin-Tate theory

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):

• David Copeland Johnson, W. Stephen Wilson, BP operations and Morava’s extraordinary K-theories, Math. Z. 144 (1): 55−75, (1975) (pdf)

see also

• Doug Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Annals of Mathematics Studies 128, Princeton University Press (1992) [ISBN:9780691025728, pdf, webpage]

Textbook account:

• Yuli Rudyak, Section IX.7 in: On Thom Spectra, Orientability, and Cobordism, Springer 1998 (doi:10.1007/978-3-540-77751-9)

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

• Mark Hovey, Neil Strickland, Morava K-theories and localization, American Mathematical Soc., Jan 1, 1999 (pdf)

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

• Andrew Baker, Brave new Hopf algebroids (pdf)

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:

• Mohammed Abouzaid, Andrew J. Blumberg, Arnold Conjecture and Morava K-theory, (arXiv:2103.01507)

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

• Douglas Ravenel, What we still don’t know about loop spaces of spheres, in: Mark Mahowald, Stewart Priddy (eds.), Homotopy Theory via Algebraic Geometry and Group Representations, Contemporary Mathematics 220, AMS 1998 (pdf, pdf, doi:10.1090/conm/220)

The orientation of integral Morava K-theory is discussed in

• Igor Kriz, Hisham Sati, M-theory, type IIA superstrings, and elliptic cohomology, Adv. Theor.

  Math. Phys. 8 (2004), no. 2, 345–394 (arXiv:hep-th/0404013)

• Lukas Buhné, Properties of Integral Morava K-Theory and the Asserted Application to the Diaconescu-Moore-Witten Anomaly, Diploma thesis Hamburg (2011)

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

• Hisham Sati, Craig Westerland, Twisted Morava K-theory and E-theory (arXiv:1109.3867)

For a review in the context of M-theory see

• Hisham Sati, Geometric and topological structures related to M-branes (2010)