Morava E-theory





Special and general types

Special notions


Extra structure



Stable Homotopy theory

Higher algebra



There are several cohomology theories that are being called Morava E-theory at times:




RW(k)[[v 1,,v n1]] R \coloneqq W(k)[ [ v_1, \cdots, v_{n-1} ] ]

for the Lubin-Tate ring of ff, 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) E(n)^\bullet represented by a spectrum E(n)E(n) with the property that its homotopy groups are

π (E(n))W(k)[[v 1,,v n1]][β ±1] \pi_\bullet(E(n)) \simeq W(k)[ [v_1, \cdots, v_{n-1} ] ] [ \beta^{\pm 1} ]

for β\beta of degree 2. This is called alternatively nnth Morava E-theory, or Lubin-Tate theory or Johnson-Wilson theory.

(e.g. Lurie, lect 22)


As a localization of the \infty-group \infty-ring on B n+1 pB^{n+1}\mathbb{Z}_p

There is a Snaith theorem for the homotopy fixed points of the Morava E-theory spectrum E nE_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 pB^{n+1} \mathbb{Z}_p. (Westerland 12, theorem 1.2)

See at Snaith-like theorem for Morava E-theory for more.

Bousfield localization and chromatic filtration

The Bousfield localization of spectra L E(n)L_{E(n)} at nnth Morava E-theory is called chromatic localization. It behaves on complex oriented cohomology theories like the restriction to the closed substack

FG n+1 FG×Spec (p) \mathcal{M}_{FG}^{\leq n+1} \hookrightarrow \mathcal{M}_{FG} \times Spec \mathbb{Z}_{(p)}

of the moduli stack of formal groups on those of height n+1\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 homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyelliptic spectrum Ell EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
3 …10K3 cohomologyK3 spectrum
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex cobordism cohomologyMUMR-theory

Smash product theorem

A version of the smash product theorem

For XX a homotopy type/spectrum and for all nn, there is a homotopy pullback

L E(n)X L K(n)X L E(n1)X L E(n1)L K(n)X, \array{ L_{E(n)}X &\longrightarrow& L_{K(n)}X \\ \downarrow && \downarrow \\ L_{E(n-1)}X &\longrightarrow& L_{E(n-1)}L_{K(n)}X } \,,

where L K(n)L_{K(n)} denotes the Bousfield localization of spectra at nnth Morava K-theory and similarly L E(n)L_{E(n)} denotes localization at Morava E-theory.

(Lurie 10, lect 23, theorem 4)

Bousfield equivalence class

For all nn, E(n)E(n) is Bousfield equivalent to E(n1)×K(n)E(n-1) \times K(n), where the last factor is nnth Morava K-theory.

(Lurie 10, lect. 23, prop. 1)

Not to be confused with C*-algebra-E-theory.


Relevant background lecture notes include

and more specifically see the lectures


  • Report of EE-theory conjectures seminar (2013) (pdf)

Discussion of the E E_\infty-algebra structure over BPB P is in

based on

  • Neil Strickland, Products on MUMU-modules, Trans. Amer. Math. Soc. 351 (1999), 2569-2606.

Discussion of twists of Morava E-theory is in

A Snaith theorem-like characterization of Morava E-theory is given in

Last revised on March 25, 2019 at 19:09:40. See the history of this page for a list of all contributions to it.