Galois extensions of structured ring spectra

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.

Final version, to appear in Memoirs of the A.M.S

References in corpus (4)

• Algebraic topology and modular forms
• Homology and Cohomology of E-infinity Ring Spectra
• Stably dualizable groups
• Toward a fundamental groupoid for the stable homotopy category

Cited by in corpus (31)

• Topological Hochschild homology and cohomology of A_\infty ring spectra
• Lifting homotopy T-algebra maps to strict maps
• Commutative ring objects in pro-categories and generalized Moore spectra
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• Logarithmic topological Hochschild homology of topological K-theory spectra
• Topological André-Quillen homology for cellular commutative $S$-algebras
• Brauer groups for commutative $S$-algebras
• Stably dualizable groups
• Towards an understanding of ramified extensions of structured ring spectra
• Toward a fundamental groupoid for the stable homotopy category
• Algebraic K-theory of the fraction field of topological K-theory
• Spectra for commutative algebraists
• Delta-discrete $G$-spectra and iterated homotopy fixed points
• Some Higher Coalgebra
• The homotopy limit problem and the cellular Picard group of Hermitian $K$-theory
• $L$-complete Hopf algebroids and their comodules
• Obtaining intermediate rings of a local profinite Galois extension without localization
• The Cosmic Galois group as Koszul dual to Waldhausen's A(pt)
• On the parametrized Tate construction
• Moduli stack of oriented formal groups and periodic complex bordism
• $MSp$ localized away from $2$ and odd formal group laws
• Algebraic K-theory of the first Morava K-theory
• Infinitesimal thickenings of Morava K-theories
• Invertible modules for commutative $\mathbb{S}$-algebras with residue fields
• Discrete G-Spectra and embeddings of module spectra
• Buildings, elliptic curves, and the K(2)-local sphere
• Homotopy theory of modules over a commutative $S$-algebra: some tools and examples
• An integral lift of the Gamma-genus
• Some properties of Lubin-Tate cohomology for classifying spaces of finite groups
• Homotopic Hopf-Galois extensions: foundations and examples
• Galois extensions of Lubin-Tate spectra