Morava E-theory

In the theory of spectra in mathematics, a Morava E-theory (also called Lubin–Tate theory or Johnson–Wilson theory) is a particular spectrum or equivalently generalized (co)homology theory. For a prime number $p$ (obmitted in notation), these form a sequence $E(n)$ of spectra, hence one for each natural number $n\in \mathbb {N}$ . Morava E-theories form the foundation of chromatic homotopy theory, in which a spectrum is separated into so-called (mono)chromatic layers by using its various Bousfield localizations with respect to the different Morava E-theories, so-called chromatic localizations $L_{E(n)}$ . Morava E-theory is named after Jack Morava, who introduced it in unpublished preprints in the 1970s.

Definition

For a perfect field $\mathbb {K}$ , hence in which every irreducible polynomial is separable, of characteristic $p$ , let $W(\mathbb {K} )$ be its ring of Witt vectors.^[1] Let $R=W(\mathbb {K} )[v_{1},\ldots ,v_{n-1}]$ be the polynomial ring in $n-1$ variables, called Lubin–Tate ring. According to Landweber's exact functor theorem, a formal group law over the Lubin–Tate ring defines a generalized cohomology theory, which according to Brown's representability theorem is represented by a spectrum, which is Morava E-theory $E(n)$ . It fulfills:

\pi _{0}E(n)=R[\beta ^{\pm 1}]

for its ring of coefficients with $\beta \in R$ in second degree.

Properties

The chromatic localization $L_{E(n)}$ preserves smash products.^[2]
$E(n)$ is Bousfield equivalent to $E(n-1)\times K(n)$ .^[3] ( $K(n)$ is Morava K-theory. Two spectra are Bousfield equivalent, when their generalized homology theories vanish on the exact same topological spaces. Hence for every topological space $X$ , one has $E(n)_{*}(X)=0$ if and only if $E(n-1)_{*}(X)=0$ and $K(n)_{*}(X)=0$ .)
$E(n)$ is $K(n)$ -local.^[4] ( $K(n)$ is Morava K-theory. $K(n)$ -local means that every spectrum morphism $X\rightarrow E(n)$ from a $K(n)$ -acyclic spectrum $X$ is nullhomotopic. $K(n)$ -acyclic means that $K(n)\wedge X\simeq 0$ .)
A connected closed spin 10-manifold is generalized oriented in second Morava E-theory $E(2)$ (for $p=2$ ) if and only if its seventh integral Stiefel–Whitney class $W_{7}$ vanishes.

E(n)-local spheres

For the sphere spectrum $\mathbb {S}$ (with $\mathbb {S} _{n}=S^{n}$ ), its chromatic localizations $L_{E(n)}\mathbb {S}$ are also called $E(n)$ -local spheres.