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 $n\in \mathbb{\mathbb{N}}$. 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 Morava E-theory is named after Jack Morava, who introduced it in unpublished preprints in the 1970s.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. Let $R=W(\mathbb{𝕂})[v_1,\dots ,v_{n-1}]$ 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.
• $E(n)$ is Bousfield equivalent to $E(n-1)\times K(n)$. ($K(n)$ 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. ($K(n)$ is Morava K-theory. $K(n)$-local means that every spectrum morphism $X\to E(n)$ 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.

See also

• Morava K-theory

Literature

• Morava, Jack (1973). "Boarding School Lecture Notes". Princeton University. https://math.mit.edu/~hrm/manuscripts/bording-school.pdf. 
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External links

• Morava E-theory on the nLab