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:
, the truncated Brown-Peterson spectrum;
, the Johnson-Wilson spectrum, a localization of at ;
the complete Johnson-Wilson spectrum
the Lubin-Tate spectrum associated to the universal deformation of a formal group law over .
Choose
be a perfect field of characteristic ;
be a formal group of height over .
Write
for the Lubin-Tate ring of , 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 represented by a spectrum with the property that its homotopy groups are
for of degree 2. This is called alternatively 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 for the canonical action of a certain group, which identifies these with a localization of the ∞-group ∞-ring on the (n+1)-group . (Westerland 12, theorem 1.2)
See at Snaith-like theorem for Morava E-theory for more.
The Bousfield localization of spectra at 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 .
(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.
A version of the smash product theorem
For a homotopy type/spectrum and for all , there is a homotopy pullback
where denotes the Bousfield localization of spectra at th Morava K-theory and similarly denotes localization at Morava E-theory.
(Lurie 10, lect 23, theorem 4)
For all , is Bousfield equivalent to , where the last factor is 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 -algebra structure over 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.