group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
symmetric monoidal (∞,1)-category of spectra
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
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
where $\mathbb{Z}_{(p)}$ is the localization of the integers at the given prime.
Integral Morva K-theory can be obtained as a localization of a quotient $MU/I$ of complex cobordism cohomology theory $MU$ (Buhné 11).
We need the following standard notation throughout this entry.
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.
(e.g. Lurie 10, lecture 22, def. 5)
For each prime integer $p$ there exists a sequence of multiplicative generalized cohomology/homology theories
with the following properties:
$K(0)_\ast(X)=H_\ast(X;\mathbb{Q})$ and $\overline{K(0)}_\ast(X)=0$ when $\overline{H}_\ast(X)$ is all torsion.
$K(1)_\ast(X)$ is one of $p-1$ isomorphic summands of mod-$p$ complex topological K-theory.
$K(0)_\ast(pt.)=\mathbb{Q}$ 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)
There is a Künneth isomorphism: $K(n)_\ast(X\times Y)\cong K(n)_\ast(X)\otimes_{K(n)_\ast(pt.)}K(n)_\ast(Y).$
Let $X$ be a p-local finite CW-complex. If $\overline{K(n)}_\ast(X)$ vanishes then so does $\overline{K(n-1)}_\ast(X)$.
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.
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
whose formal group has height exactly $n$;
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).
$K(n)$ 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)
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)
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)
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)
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.
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
of the moduli stack of formal groups at those of height $n$.
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 | $Ell_E$ | |
second Morava K-theory | $K(2)$ | ||
second Morava E-theory | $E(2)$ | ||
algebraic K-theory of KU | $K(KU)$ | ||
$n$ | $n$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 |
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.
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).
Write $gl_1(K(n))$ for the ∞-group of units of the (a) Morava K-theory spectrum
For $p = 2$ and all $n \in \mathbb{N}$, there is an equivalence
between
and
(Sati-Westerland 11, theorem 1)
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 $p = 2$).
Morava K-theory originates in unpublished preprints by Jack Morava in the early 1970s.
A first published account appears in
see also
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
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
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