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\section*{Morava K-theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{morava_ktheory}{}\section*{{Morava K-theory}}\label{morava_ktheory}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{construction_from_complex_cobordism}{Construction from complex cobordism}\dotfill \pageref*{construction_from_complex_cobordism} \linebreak
\noindent\hyperlink{axiomatic_characterization}{Axiomatic characterization}\dotfill \pageref*{axiomatic_characterization} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{universal_characterization}{Universal characterization}\dotfill \pageref*{universal_characterization} \linebreak
\noindent\hyperlink{ring_structure}{Ring structure}\dotfill \pageref*{ring_structure} \linebreak
\noindent\hyperlink{AsInfinityFields}{As $A_\infty$-fields}\dotfill \pageref*{AsInfinityFields} \linebreak
\noindent\hyperlink{as_the_primes_in_the_category_of_spectra}{As the primes in the $\infty$-category of spectra}\dotfill \pageref*{as_the_primes_in_the_category_of_spectra} \linebreak
\noindent\hyperlink{relation_to_chromatic_homotopy_theory}{Relation to chromatic homotopy theory}\dotfill \pageref*{relation_to_chromatic_homotopy_theory} \linebreak
\noindent\hyperlink{relation_to_bousfield_lattice}{Relation to Bousfield lattice}\dotfill \pageref*{relation_to_bousfield_lattice} \linebreak
\noindent\hyperlink{Orientation}{Orientation}\dotfill \pageref*{Orientation} \linebreak
\noindent\hyperlink{group_rings_and_twists}{$\infty$-Group rings and twists}\dotfill \pageref*{group_rings_and_twists} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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

\begin{displaymath}
\pi_\bullet\left(K\left(n\right)\right)
  \simeq
  \mathbb{F}_p [v_n, v_n^{-1}]
\end{displaymath}
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

\begin{displaymath}
\pi_\bullet\left(K\left(n\right)\right)
  \simeq
  \mathbb{Z}_{(p)} [v_n, v_n^{-1}]
  \,,
\end{displaymath}
where $\mathbb{Z}_{(p)}$ is the [[localization]] of the integers at the given [[prime]].

Integral Morava K-theory can be obtained as a localization of a quotient $MU/I$ of [[complex cobordism cohomology theory]] $MU$ (\hyperlink{Buhne11}{Buhn\'e{} 11}).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

We need the following standard notation throughout this entry.

\begin{defn}
\label{}\hypertarget{}{}
For $p \in \mathbb{N}$ a [[prime number]], we write

\begin{itemize}%
\item $\mathbb{F}_p = \mathbb{Z}/(p)$ for the [[field]] with $p$ elements;


\item $\mathbb{Z}_{(p)}$ for the [[localization of a ring|localization ring]] of the [[integers]] at $p$;


\item $\mathbb{Z}_p$ for the [[p-adic integers]].



\end{itemize}
\end{defn}
\hypertarget{construction_from_complex_cobordism}{}\subsubsection*{{Construction from complex cobordism}}\label{construction_from_complex_cobordism}

(e.g. \hyperlink{LurieLecture}{Lurie 10, lecture 22, def. 5})

\hypertarget{axiomatic_characterization}{}\subsubsection*{{Axiomatic characterization}}\label{axiomatic_characterization}

\begin{prop}
\label{}\hypertarget{}{}
For each [[prime number|prime integer]] $p$ there exists a sequence of [[multiplicative cohomology theory|multiplicative]] [[generalized cohomology theories|generalized cohomology]]/[[generalized homology theories|homology theories]]

\begin{displaymath}
\{K(n)\}_{n \in \mathbb{N}}
\end{displaymath}
with the following properties:

\begin{enumerate}%
\item $K(0)_\ast(X)=H_\ast(X;\mathbb{Q})$ and $\overline{K(0)}_\ast(X)=0$ when $\overline{H}_\ast(X)$ is all [[torsion]].


\item $K(1)_\ast(X)$ is one of $p-1$ isomorphic summands of mod-$p$ complex [[topological K-theory]].


\item $K(0)_\ast(pt.)=\mathbb{Q}$ 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 module|free]]. $K(n)_\ast(X)$ is a module over $K(n)_\ast(pt.)$, see \hyperlink{AsInfinityFields}{below})


\item There is a [[Künneth isomorphism]]: $K(n)_\ast(X\times Y)\cong K(n)_\ast(X)\otimes_{K(n)_\ast(pt.)}K(n)_\ast(Y).$


\item Let $X$ be a p-local finite [[CW-complex]]. If $\overline{K(n)}_\ast(X)$ vanishes then so does $\overline{K(n-1)}_\ast(X)$.


\item If $X$ as above is not contractible then $\overline{K(n)}_\ast(X)=K(n)_\ast(pt.)\otimes \overline{H}_\ast(X;\mathbb{Z}/(p))$.



\end{enumerate}
These are called the \emph{Morava K-theories}.

\end{prop}
Due to the third point one may regard $K(n)$ as a [[∞-field]] among the [[A-infinity rings]]. See \hyperlink{AsInfinityFields}{below}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{universal_characterization}{}\subsubsection*{{Universal characterization}}\label{universal_characterization}

\begin{prop}
\label{}\hypertarget{}{}
For each [[prime number]] $p$ and each $n \in \mathbb{N}$, the Morava K-theory $K(n)$ is, up to [[equivalence in an (∞,1)-category|equivalence]], the unique [[spectrum]] underlying an [[H-space|homotopy associative]] spectrum which is

\begin{enumerate}%
\item [[complex oriented cohomology theory|complex oriented]];


\item whose [[formal group]] has [[height of a formal group|height]] exactly $n$;


\item whose [[homotopy groups]] are $\pi_\bullet \simeq \mathbb{F}_p[v_n^\pm]$. (with $v_n$ defined as at \emph{[[height of a formal group|height]]}).



\end{enumerate}
\end{prop}
For instance (\hyperlink{LurieLecture}{Lurie, lecture 24, prop. 11}).

\hypertarget{ring_structure}{}\subsubsection*{{Ring structure}}\label{ring_structure}

\begin{prop}
\label{}\hypertarget{}{}
$K(n)$ admits the structure of an [[A-∞ algebra]], in fact of an $MU_{(p)}$-[[A-∞ algebra]].

\end{prop}
Due to Robinson (and [[Andrew Baker]] at $p = 2$). (See e.g. \hyperlink{LurieLecture}{Lurie 10, lecture 22, lemma 2})

\begin{remark}
\label{}\hypertarget{}{}
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 \emph{not} even homotopy commutative. Nevertheless, for many spaces $X$, the $K(n)$-[[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology]] at the prime $2$ of $X$ forms a commutative ring.

\end{remark}
(e.g. \hyperlink{LurieLecture}{Lurie 10, lecture 22, warning 6})

\hypertarget{AsInfinityFields}{}\subsubsection*{{As $A_\infty$-fields}}\label{AsInfinityFields}

\begin{prop}
\label{}\hypertarget{}{}
If $E$ is an [[∞-field]] then $E \otimes K(n) \neq 0$ and $E$ admits the structure of a $K(n)$-[[module spectrum|module]].

\end{prop}
This appears for instance as (\hyperlink{LurieLecture}{Lurie, lecture 24, prop. 9, remark 13})

\begin{remark}
\label{}\hypertarget{}{}
This means that the Morava $A_\infty$-rings $K(n)$ are essentially the only [[∞-fields]] in the [[stable homotopy category]].

\end{remark}
See (\hyperlink{LurieLecture}{Lurie, lecture 24, remark 13})

\hypertarget{as_the_primes_in_the_category_of_spectra}{}\subsubsection*{{As the primes in the $\infty$-category of spectra}}\label{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-localization|p-local]] and [[finite spectra]] . This is the content of the \emph{[[thick subcategory theorem]]}.

\hypertarget{relation_to_chromatic_homotopy_theory}{}\subsubsection*{{Relation to chromatic homotopy theory}}\label{relation_to_chromatic_homotopy_theory}

The layers in the [[chromatic homotopy theory|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

\begin{displaymath}
\mathcal{M}^n_{FG}
  \hookrightarrow
  \mathcal{M}_{FG}
\end{displaymath}
of the [[moduli stack of formal groups]] at those of [[height of a formal group|height]] $n$.

(\hyperlink{LurieLecture}{Lurie 10, lecture 29})

[[!include chromatic tower examples - table]]

\hypertarget{relation_to_bousfield_lattice}{}\subsubsection*{{Relation to Bousfield lattice}}\label{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]].

\hypertarget{Orientation}{}\subsubsection*{{Orientation}}\label{Orientation}

The [[orientation in generalized cohomology|orientation]] of [[integral Morava K-theory]] is discussed in (\hyperlink{SatiKriz04}{Sati-Kriz 04}, \hyperlink{Buhne11}{Buhn\'e{} 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 generalized cohomology|orientation]] in [[complex K-theory]] is given by the vanishing third [[integral Stiefel-Whitney class]] $W_3$ ([[spin{\tt \symbol{94}}c-structure]]).

\hypertarget{group_rings_and_twists}{}\subsubsection*{{$\infty$-Group rings and twists}}\label{group_rings_and_twists}

Write $gl_1(K(n))$ for the [[∞-group of units]] of the (a) Morava K-theory spectrum

\begin{prop}
\label{}\hypertarget{}{}
For $p = 2$ and all $n \in \mathbb{N}$, there is an [[equivalence in an (infinity,1)-category|equivalence]]

\begin{displaymath}
Maps(B^{n+1}U(1), B gl_1(K(n)))
  \simeq
  \mathbb{Z}_2
\end{displaymath}
between

\begin{itemize}%
\item the [[mapping space]] from the [[classifying space]] for [[circle n-bundle|circle (n+1)-bundles]] to the [[delooping]] of the [[∞-group of units]] of $K(n)$

\end{itemize}
and

\begin{itemize}%
\item the [[p-adic integers|2-adic integers]].

\end{itemize}
\end{prop}
(\hyperlink{SatiWesterland11}{Sati-Westerland 11, theorem 1})

\begin{remark}
\label{}\hypertarget{}{}
By the discussion at [[(∞,1)-vector bundle]] this means that for each such map there is a type of [[twisted cohomology|twist]] of Morava K-theory (at $p = 2$).

\end{remark}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[equivariant Morava K-theory]]


\item [[2-periodic Morava K-theory]]


\item [[K-theory]], [[topological K-theory]], [[algebraic K-theory]]


\item [[chromatic homotopy theory]]


\item [[K(n)-local stable homotopy theory]]


\item [[integral Morava K-theory]]


\item [[Morava E-theory]]

\begin{itemize}%
\item [[Lubin-Tate theory]]

\end{itemize}


\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

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

A first published account appears in

\begin{itemize}%
\item David Copeland Johnson, ; W. Stephen Wilson, \emph{BP operations and Morava's extraordinary K-theories.}, Math. Z. 144 (1): 55$-$75, (1975)

\end{itemize}
see also

\begin{itemize}%
\item [[Doug Ravenel]], \emph{Nilpotence and Periodicity in Stable Homotopy Theory}, Annals of Mathematics Studies 128, Princeton University Press (1992).

\end{itemize}
A discussion with an eye towards [[category theory|category theoretic]] general abstract properties of localized stable homotopy theory is in

\begin{itemize}%
\item [[Mark Hovey]], [[Neil Strickland]], \emph{Morava K-theories and localization}, American Mathematical Soc., Jan 1, 1999 (\href{http://www.math.rochester.edu/people/faculty/doug/otherpapers/kn.pdf}{pdf})

\end{itemize}
A survey of the theory is in

\begin{itemize}%
\item Urs W\"u{}rgler, \emph{Morava K-theories: a survey}, Algebraic topology Poznan 1989, Lecture Notes in Math. 1474, Berlin: Springer, pp. 111 138 (1991)

\end{itemize}
In

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Chromatic Homotopy Theory]]} Lecture notes, (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture22.pdf}{pdf})

Lecture 22 \emph{Morava E-theory and Morava K-theory} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture22.pdf}{pdf})

Lecture 23 \emph{The Bousfield Classes of $E(n)$ and $K(n)$} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture23.pdf}{pdf})

Lecture 24 \emph{Uniqueness of Morava K-theory} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture24.pdf}{pdf})



\end{itemize}
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

\begin{itemize}%
\item [[Andrew Baker]], \emph{Brave new Hopf algebroids} (\href{http://www.maths.gla.ac.uk/~ajb/dvi-ps/brave-ha.pdf}{pdf})

\end{itemize}
based on

\begin{itemize}%
\item [[Neil Strickland]], \emph{Products on $MU$-modules}, Trans. Amer. Math. Soc. 351 (1999), 2569-2606.

\end{itemize}
The [[orientation in generalized cohomology|orientation]] of [[integral Morava K-theory]] is discussed in

\begin{itemize}%
\item [[Igor Kriz]], [[Hisham Sati]], \emph{M-theory, type IIA superstrings, and elliptic cohomology}, Adv. Theor. Math. Phys. 8 (2004), no. 2, 345--394 (\href{http://arxiv.org/abs/hep-th/0404013}{arXiv:hep-th/0404013})

\end{itemize}
\begin{itemize}%
\item [[Lukas Buhné]], \emph{[[Properties of Integral Morava K-Theory and the Asserted Application to the Diaconescu-Moore-Witten Anomaly]]}, Diploma thesis Hamburg (2011)

\end{itemize}
Some [[twisted cohomology|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

\begin{itemize}%
\item [[Hisham Sati]], [[Craig Westerland]], \emph{Twisted Morava K-theory and E-theory} (\href{http://arxiv.org/abs/1109.3867}{arXiv:1109.3867})

\end{itemize}
For a review in the context of [[M-theory]] see

\begin{itemize}%
\item [[Hisham Sati]], \emph{[[Geometric and topological structures related to M-branes]]} (2010)

\end{itemize}
[[!redirects Morava K-theories]]



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