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Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups
John Rognes, University of Oslo, Norway
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Memoirs of the American Mathematical Society
2008; 137 pp; softcover
Volume: 192
ISBN-10: 0-8218-4076-2
ISBN-13: 978-0-8218-4076-4
List Price: US$69 Individual Members: US$41.40
Institutional Members: US\$55.20
Order Code: MEMO/192/898

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 étale 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.

The author extends the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein and the $$p$$-complete study for $$p$$-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the $$E$$-local stable homotopy category, for any spectrum $$E$$.

Galois Extensions of Structured Ring Spectra
• Abstract
• Introduction
• Galois extensions in algebra
• Closed categories of structured module spectra
• Galois extensions in topology
• Examples of Galois extensions
• Dualizability and alternate characterizations
• Galois theory I
• Pro-Galois extensions and the Amitsur complex
• Separable and étale extensions
• Mapping spaces of commutative $$S$$-algebras
• Galois theory II
• Hopf-Galois extensions in topology
• References
Stably Dualizable Groups
• Abstract
• Introduction
• The dualizing spectrum
• Duality theory
• Computations
• Norm and transfer maps
• References
• Index