Uniqueness of Morava K-theory

TL;DR

This paper proves that Morava K-theory spectrum K(n) has a unique S-algebra structure, but admits many MU or E_n-algebra structures, using spectral sequences and S-algebra invariants.

Contribution

It establishes the essential uniqueness of the S-algebra structure on K(n) and introduces a spectral sequence approach to analyze A-infinity structures.

Findings

Unique S-algebra structure on K(n)

Uncountably many MU or E_n-algebra structures on K(n)

Spectral sequence for homotopy groups of A-infinity structures

Abstract

We show that there is an essentially unique S-algebra structure on the Morava K-theory spectrum K(n), while K(n) has uncountably many MU or \hE{n}-algebra structures. Here \hE{n} is the K(n)-localized Johnson-Wilson spectrum. To prove this we set up a spectral sequence computing the homotopy groups of the moduli space of A-infinity structures on a spectrum, and use the theory of S-algebra k-invariants for connective S-algebras due to Dugger and Shipley to show that all the uniqueness obstructions are hit by differentials.