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
There are several cohomology theories that are being called Morava E-theory at times:
$B P\langle n\rangle$, the truncated Brown-Peterson spectrum;
$E(n)$, the Johnson-Wilson spectrum, a localization of $B P \langle n\rangle$ at $v_n$;
$\widehat{E(n)}$ the complete Johnson-Wilson spectrum
$E(k,\Gamma)$ the Lubin-Tate spectrum associated to the universal deformation of a formal group law $\Gamma$ over $k$.
Choose
$k$ be a perfect field of characteristic $p$;
$f$ be a formal group of height $n$ over $k$.
Write
for the Lubin-Tate ring of $f$, classifying its universal deformation.
By the discussion there, this is Landweber exact, hence defines a cohomology theory. Therefore by the Landweber exact functor theorem there is an even periodic cohomology theory $E(n)^\bullet$ represented by a spectrum $E(n)$ with the property that its homotopy groups are
for $\beta$ of degree 2. This is called alternatively $n$th Morava E-theory, or Lubin-Tate theory or Johnson-Wilson theory.
(e.g. Lurie, lect 22)
There is a Snaith theorem for the homotopy fixed points of the Morava E-theory spectrum $E_n$ for the canonical action of a certain group, which identifies these with a localization of the ∞-group ∞-ring on the (n+1)-group $B^{n+1} \mathbb{Z}_p$. (Westerland 12, theorem 1.2)
See at Snaith-like theorem for Morava E-theory for more.
The Bousfield localization of spectra $L_{E(n)}$ at $n$th Morava E-theory is called chromatic localization. It behaves on complex oriented cohomology theories like the restriction to the closed substack
of the moduli stack of formal groups on those of height $\geq n+1$.
(e.g. Lurie, lect 22, above theorem 1)
In this way the localization tower at the Morava E-theories exhibits the chromatic filtration in 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 |
A version of the smash product theorem
For $X$ a homotopy type/spectrum and for all $n$, there is a homotopy pullback
where $L_{K(n)}$ denotes the Bousfield localization of spectra at $n$th Morava K-theory and similarly $L_{E(n)}$ denotes localization at Morava E-theory.
(Lurie 10, lect 23, theorem 4)
For all $n$, $E(n)$ is Bousfield equivalent to $E(n-1) \times K(n)$, where the last factor is $n$th Morava K-theory.
orthogonal form: EO(n)
Not to be confused with C*-algebra-E-theory.
Named after Jack Morava (see at Morava K-theory).
Relevant background lecture notes include
and more specifically see the lectures
also
Discussion of the $E_\infty$-algebra structure over $B P$ is in
based on
Discussion of twists of Morava E-theory is in
A Snaith theorem-like characterization of Morava E-theory is given in
Morava E-theory of configuration spaces of points:
Last revised on November 8, 2022 at 21:36:20. See the history of this page for a list of all contributions to it.