# nLab Morava E-theory

Contents

cohomology

### Theorems

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## 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 $\geq n+1$.

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 $H \mathbb{Z}$HZR-theory
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$KR-theory
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomologyelliptic spectrum $Ell_E$
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
3 …10K3 cohomologyK3 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 cohomologyMUMR-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.

### 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.

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

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

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

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

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

Morava E-theory of configuration spaces of points: