Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups

Introduces the notion of a Galois extension of commutative $S$-algebras ($E_/infty$ ring spectra), often localized with respect to a fixed homology theory. The author establishes the main theorem of Galois theory.

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The author introduces 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. He establishes 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. He shows that the global sphere spectrum $S$ is separably closed, using Minkowski's discriminant theorem, and he estimates the separable closure of its localization with respect to each of the Morava $K$-theories. He also defines Hopf-Galois extensions of commutative $S$-algebras and studies the complex cobordism spectrum $MU$ as a common integral model for all of the local Lubin-Tate Galois extensions.