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For each prime , the Morava K-theories are a tower of complex oriented cohomology theories whose coefficient ring is
where is in degree .
Hence with for is a Bott element of degree 2 and is closely related to complex K-theory, while for is then a Bott element of degree 6 and is closely related to elliptic cohomology.
There is also integral Morava K-theory which instead has coefficient ring
where is the localization of the integers at the given prime.
Integral Morva K-theory can be obtained as a localization of a quotient of complex cobordism cohomology theory (Buhné 11).
We need the following standard notation throughout this entry.
Construction from complex cobordism
(e.g. Lurie 10, lecture 22, def. 5)
For each prime integer there exists a sequence of multiplicative generalized cohomology/homology theories
with the following properties:
and when is all torsion.
is one of isomorphic summands of mod- complex topological K-theory.
and for , where .
(This ring is a graded field in the sense that every graded module over it is free. is a module over , see below)
There is a Künneth isomorphism:
Let be a p-local finite CW-complex. If vanishes then so does .
If as above is not contractible then .
These are called the Morava K-theories.
Due to the third point one may regard as a ∞-field among the A-infinity rings. See below.
For instance (Lurie, lecture 24, prop. 11).
Due to Robinson (and Andrew Baker at ). (See e.g. Lurie 10, lecture 22, lemma 2)
(e.g. Lurie 10, lecture 22, warning 6)
If is an ∞-field then and admits the structure of a -module.
This appears for instance as (Lurie, lecture 24, prop. 9, remark 13)
See (Lurie, lecture 24, remark 13)
As the primes in the -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 -fields.
Specifically the Bousfield localization of spectra acts on complex oriented cohomology theories like completion along the locally closed substack
of the moduli stack of formal groups at those of height .
(Lurie 10, lecture 29)
chromatic homotopy 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 which measure the failure of the telescope conjecture.
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 .
Notice that this is in higher analogy to how orientation in complex K-theory is given by the vanishing third integral Stiefel-Whitney class (spin^c-structure).
-Group rings and twists
Write for the ∞-group of units of the (a) Morava K-theory spectrum
For and all , there is an equivalence
(Sati-Westerland 11, theorem 1)
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)
- 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)
the explicit definition via formal group laws is in lecture 22 and the abstract characterization in lecture 24.
The -algebra structure over is comment on in
- Neil Strickland, Products on -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 as well as the Atiyah-Hirzebruch spectral sequence for Morava and Morava are discussed in
For a review in the context of M-theory see