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Morava K-theory

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Morava K-theory

Idea

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

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

where v nv_n is in degree 2(p n1)2(p^n-1).

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

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

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

where (p)\mathbb{Z}_{(p)} is the localization of the integers at the given prime.

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

Definition

We need the following standard notation throughout this entry.

Definition

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

Construction from complex cobordism

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

Axiomatic characterization

Proposition/Definition

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

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

with the following properties:

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

  2. K(1) *(X)K(1)_\ast(X) is one of p1p-1 isomorphic summands of mod-pp complex topological K-theory.

  3. K(0) *(pt.)=K(0)_\ast(pt.)=\mathbb{Q} and for n0n\neq 0, K(n) *(pt.)=𝔽 p[v n,v n 1]K(n)_\ast(pt.)=\mathbb{F}_p[v_n,v_n^{-1}] where |v n|=2p n2\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) *(X)K(n)_\ast(X) is a module over K(n) *(pt.)K(n)_\ast(pt.), see below)

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

  5. Let XX be a p-local finite CW-complex. If K(n)¯ *(X)\overline{K(n)}_\ast(X) vanishes then so does K(n1)¯ *(X)\overline{K(n-1)}_\ast(X).

  6. If XX as above is not contractible then K(n)¯ *(X)=K(n) *(pt.)H¯ *(X;/(p))\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)K(n) as a ∞-field among the A-infinity rings. See below.

Properties

Universal characterization

Proposition

For each prime number pp and each nn \in \mathbb{N}, the Morava K-theory K(n)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 nn;

  3. whose homotopy groups are π 𝔽 p[v n ±]\pi_\bullet \simeq \mathbb{F}_p[v_n^\pm]. (with v nv_n defined as at height).

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

Ring structure

Proposition

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

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

Remark

With the exception of the extreme case of n=0n=0, the fields K(n)K(n) do not admit E-∞-ring multiplicative structures. However, when p2p\neq 2, the multiplication is homotopy commutative. For p=2p = 2 it is not even homotopy commutative. Nevertheless, for many spaces XX, the K(n)K(n)-generalized cohomology at the prime 22 of XX forms a commutative ring.

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

As A A_\infty-fields

Proposition

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

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

Remark

This means that the Morava A A_\infty-rings K(n)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 E_\infty-fields.

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

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

of the moduli stack of formal groups at those of height nn.

(Lurie 10, lecture 29)

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyelliptic spectrum Ell EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
3 …10K3 cohomologyK3 spectrum
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex 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)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 7W_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 3W_3 (spin^c-structure).

\infty-Group rings and twists

Write gl 1(K(n))gl_1(K(n)) for the ∞-group of units of the (a) Morava K-theory spectrum

Proposition

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

Maps(B n+1U(1),Bgl 1(K(n))) 2 Maps(B^{n+1}U(1), B gl_1(K(n))) \simeq \mathbb{Z}_2

between

and

(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=2p = 2).

References

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

A first published account appears in

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

see also

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

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)

In

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

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

based on

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

The orientation of integral Morava K-theory is discussed in

Some twists of Morava K-theory/maps into its ∞-group of units are discussed in

For a review in the context of M-theory see

Revised on April 3, 2014 22:23:19 by Urs Schreiber (82.169.114.243)