Galois extensions of structured ring spectra

TL;DR

This paper develops a Galois theory framework for structured ring spectra, introducing notions of Galois, separable, and etale extensions of commutative S-algebras with numerous examples and applications.

Contribution

It generalizes classical Galois theory to the setting of E_infty ring spectra, establishing main theorems and exploring separable closures and Hopf-Galois extensions.

Findings

The sphere spectrum S is separably closed.

Estimated the separable closure of localizations with respect to Morava K-theories.

Defined Hopf-Galois extensions and studied the spectrum MU as a model.

Abstract

We introduce the notion of a Galois extension of commutative S-algebras (E_infty ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological K-theory, Lubin-Tate spectra and cochain S-algebras. We establish the main theorem of Galois theory in this generality. Its proof involves the notions of separable and etale extensions of commutative S-algebras, and the Goerss-Hopkins-Miller theory for E_infty mapping spaces. We show that the global sphere spectrum S is separably closed, using Minkowski's discriminant theorem, and we estimate the separable closure of its localization with respect to each of the Morava K-theories. We also define Hopf-Galois extensions of commutative S-algebras, and study the complex cobordism spectrum MU as a common integral model for all of the local Lubin-Tate Galois extensions.