There aren't that many Morava E-theories

TL;DR

This paper proves that the homotopy type of Morava E-theories, associated with formal group laws over perfect fields, does not depend on the specific formal group law chosen.

Contribution

It demonstrates that the underlying homotopy type of Morava E-theories is independent of the formal group law choice, clarifying their fundamental nature.

Findings

Homotopy type of E(n) is independent of formal group law.

Supports the robustness of Morava E-theories.

Provides insight into the structure of complex oriented cohomology theories.

Abstract

Let $k$ be a perfect field of characteristic $p$. Associated to any (1-dimensional, commutative) formal group law of finite height $n$ over $k$ there is a complex oriented cohomology theory represented by a spectrum denoted $E(n)$ and commonly referred to as Morava $E$-theory. These spectra are known to admit $E_\infty$-structures, and the dependence of the $E_\infty$-structure on the choice of formal group law has been well studied (cf.\ [GH], [R], [L], Section 5, [PV]). In this note we show that the underlying homotopy type of $E(n)$ is independent of the choice of formal group law.