nLab Morava E-theory

Idea

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 $k$.

Definition

Choose

• $k$ be a perfect field of characteristic $p$;

• $f$ be a formal group of height $n$ over $k$.

Write

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

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

$$\pi_\bullet(E(n)) \simeq W(k)[ [v_1, \cdots, v_{n-1} ] ] [ \beta^{\pm 1} ]$$

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)

Properties

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

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.

Bousfield localization and chromatic filtration

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

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

of the moduli stack of formal groups on those of height $\geq n+1$.

(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 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

Smash product theorem

A version of the smash product theorem

For $X$ a homotopy type/spectrum and for all $n$, there is a homotopy pullback

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

where $L_{K(n)}$ 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)

Bousfield equivalence class

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.

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

Related concepts

• Morava K-theory

• orthogonal form: EO(n)

• real form ER-theory

• Morava E-theoretic Chern character

• chromatic homotopy theory

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

References

Named after Jack Morava (see at Morava K-theory).

Relevant background lecture notes include

• Mike Hopkins, Complex oriented cohomology theories and the language of stacks, course notes (pdf)

and more specifically see the lectures

• Jacob Lurie, Chromatic Homotopy Theory Lecture notes, (pdf), Lecture 22 Morava E-theory and Morava K-theory (pdf)

• Jacob Lurie, Chromatic Homotopy Theory Lecture notes, (pdf) Lecture 23 The Bousfield Classes of $E(n)$ and $K(n)$ (pdf)

• Jacob Lurie, Chromatic Homotopy Theory Lecture notes, ([pdf] Lecture 24 Uniqueness of Morava K-theory (pdf)

also

• Report of $E$-theory conjectures seminar (2013) (pdf)

Discussion of the $E_\infty$-algebra structure over $B P$ is in

• Andrew Baker, Brave new Hopf algebroids (pdf)

based on

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

Discussion of twists of Morava E-theory is in

• Hisham Sati, Craig Westerland, Twisted Morava K-theory and E-theory (arXiv:1109.3867)

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

• Craig Westerland, A higher chromatic analogue of the image of J (arXiv:1210.2472)

Morava E-theory of configuration spaces of points:

• Lukas Brantner, Jeremy Hahn, Ben Knudsen, The Lubin-Tate Theory of Configuration Spaces: I (arXiv:1908.11321)