Memoirs of the American Mathematical Society 2008; 137 pp; softcover Volume: 192 ISBN10: 0821840762 ISBN13: 9780821840764 List Price: US$73 Individual Members: US$43.80 Institutional Members: US$58.40 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 EilenbergMac Lane spectra of commutative rings, real and complex topological \(K\)theory, LubinTate 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 GoerssHopkinsMiller 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 HopfGalois extensions of commutative \(S\)algebras and studies the complex cobordism spectrum \(MU\) as a common integral model for all of the local LubinTate 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 \(E\). Table of Contents 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
 ProGalois extensions and the Amitsur complex
 Separable and étale extensions
 Mapping spaces of commutative \(S\)algebras
 Galois theory II
 HopfGalois extensions in topology
 References
Stably Dualizable Groups  Abstract
 Introduction
 The dualizing spectrum
 Duality theory
 Computations
 Norm and transfer maps
 References
 Index
