The homotopy fixed point spectra of profinite Galois extensions
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- by Mark Behrens and Daniel G. Davis PDF
- Trans. Amer. Math. Soc. 362 (2010), 4983-5042 Request permission
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.References
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Additional Information
- Mark Behrens
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- MR Author ID: 690933
- Daniel G. Davis
- Affiliation: Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504
- 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. - © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2645058