## Realizability of algebraic Galois extensions by strictly commutative ring spectra

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**359**(2007), 827-857 Request permission

## 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
- Email: a.baker@maths.gla.ac.uk
**Birgit Richter**- Affiliation: Fachbereich Mathematik der Universität Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany
- Email: richter@math.uni-hamburg.de
- 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.
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**359**(2007), 827-857 - MSC (2000): Primary 55P42, 55P43, 55S35; Secondary 55P91, 55P92, 13B05
- DOI: https://doi.org/10.1090/S0002-9947-06-04201-2
- MathSciNet review: 2255198