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The homotopy fixed point spectra of profinite Galois extensions


Authors: Mark Behrens and Daniel G. Davis
Journal: Trans. Amer. Math. Soc. 362 (2010), 4983-5042
MSC (2010): Primary 55P43; Secondary 55P91, 55Q51
DOI: https://doi.org/10.1090/S0002-9947-10-05154-8
Published electronically: April 14, 2010
MathSciNet review: 2645058
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Abstract: Let $ E$ be a $ k$-local profinite $ G$-Galois extension of an $ E_\infty$-ring spectrum $ A$ (in the sense of Rognes). We show that $ E$ may be regarded as producing a discrete $ G$-spectrum. Also, we prove that if $ E$ is a profaithful $ k$-local profinite extension which satisfies certain extra conditions, then the forward direction of Rognes's Galois correspondence extends to the profinite setting. We show that the function spectrum $ F_A((E^{hH})_k, (E^{hK})_k)$ is equivalent to the localized homotopy fixed point spectrum $ ((E[[G/H]])^{hK})_k$, where $ H$ and $ K$ are closed subgroups of $ G$. Applications to Morava $ E$-theory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action in terms of the derived functor of fixed points.


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Additional Information

Mark Behrens
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Daniel G. Davis
Affiliation: Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504

DOI: https://doi.org/10.1090/S0002-9947-10-05154-8
Received by editor(s): August 6, 2008
Received by editor(s) in revised form: July 3, 2009
Published electronically: April 14, 2010
Additional Notes: The first author was supported by NSF grant DMS-0605100, the Sloan Foundation, and DARPA
Part of the second author’s work on this paper was supported by an NSF VIGRE grant at Purdue University, a visit to the Mittag-Leffler Institute, and a grant from the Louisiana Board of Regents Support Fund.
Article copyright: © Copyright 2010 American Mathematical Society
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