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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Realizability of algebraic Galois extensions by strictly commutative ring spectra

Authors: Andrew Baker and Birgit Richter
Journal: Trans. Amer. Math. Soc. 359 (2007), 827-857
MSC (2000): Primary 55P42, 55P43, 55S35; Secondary 55P91, 55P92, 13B05
Published electronically: September 12, 2006
MathSciNet review: 2255198
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Abstract: We discuss some of the basic ideas of Galois theory for commutative $\mathbb {S}$-algebras originally formulated by John Rognes. We restrict our attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions to topological ones by applying obstruction theories of Robinson and Goerss-Hopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative $\mathbb {S}$-algebras. Examples such as the complex $K$-theory spectrum as a $KO$-algebra indicate that more exotic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones, and this leads to computable Harrison groups for such spectra. We end by proving an analogue of Hilbert’s theorem 90 for the units associated with a Galois extension.

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Additional Information

Andrew Baker
Affiliation: Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland
MR Author ID: 29540
ORCID: 0000-0002-9369-7702

Birgit Richter
Affiliation: Fachbereich Mathematik der Universität Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany

Keywords: Commutative $S$-algebra, Galois extension, $\Gamma$-homology, Kummer theory, Picard groups
Received by editor(s): December 23, 2004
Published electronically: September 12, 2006
Additional Notes: We would like to thank John Rognes, John Greenlees, Peter Kropholler, Stefan Schwede and the referee for helpful comments. We thank the Mathematics Departments of the Universities of Glasgow and Oslo for providing us with stimulating environments to pursue this work.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.