nLab
Morava E-theory

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Higher algebra

Contents

Idea

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

Definition

Choose

Write

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)

Properties

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.

References

Relevant background lecture notes include

and more specifically see the lectures

also

  • 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.