The homotopy fixed point spectra of profinite Galois extensions
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Behrens_The Homotopy.pdf
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Author(s)
Behrens, Mark Joseph
Davis, Daniel G.
Date Issued
September 2009
Journal
Transactions of the American Mathematical Society
Publisher
American Mathematical Society
Citation
Behrens, Mark and Daniel G. Davis. "The homotopy fixed point spectra of profinite Galois extensions." Volume 362, Number 9, September 2010, p.4983–5042.
Version
Author's final manuscript
Abstract
Let E be a k-local profinite G-Galois extension of an E1-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 FA((E[superscript hH])k; (E[superscript hK])k) is equivalent
to the localized homotopy fixed point spectrum ((E[[G=H]])[superscript 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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