Morava E-theory

For a perfect field ${\displaystyle \mathbb {K} }$, hence in which every irreducible polynomial is separable, of characteristic ${\displaystyle p}$, let ${\displaystyle W(\mathbb {K} )}$ be its ring of Witt vectors.^[1] Let ${\displaystyle R=W(\mathbb {K} )[v_{1},\ldots ,v_{n-1}]}$ be the polynomial ring in ${\displaystyle n-1}$ 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 ${\displaystyle E(n)}$. It fulfills:

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

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

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

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

• Morava K-theory

• Morava, Jack (1973). "Boarding School Lecture Notes" (PDF). math.mit.edu. Princeton University. Retrieved 2026-08-21.
• Hopkins, Mike (1999). "Complex oriented cohomology theories and the language of stacks" (PDF). ncatlab.org. MIT. Retrieved 2026-08-21.
• Lurie, Jacob (2010-04-27). "Morava E-Theory and Morava K-Theory (Lecture 22)" (PDF). people.math.harvard.edu. Harvard University. Retrieved 2026-08-21.
• Lurie, Jacob (2010-04-27). "The Bousfield Classes of E(n) and K(n) (Lecture 23)" (PDF). people.math.harvard.edu. Harvard University. Retrieved 2026-08-21.
• Lurie, Jacob (2010-04-27). "The Image of J (Lecture 35)" (PDF). people.math.harvard.edu. Harvard University. Retrieved 2026-08-21.