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Every $ K(n)$-local spectrum is the homotopy fixed points of its Morava module

Authors: Daniel G. Davis and Takeshi Torii
Journal: Proc. Amer. Math. Soc. 140 (2012), 1097-1103
MSC (2010): Primary 55P42, 55T15
Published electronically: July 18, 2011
MathSciNet review: 2869094
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Abstract: Let $ n \geq 1$ and let $ p$ be any prime. Also, let $ E_n$ be the Lubin-Tate spectrum, $ G_n$ the extended Morava stabilizer group, and $ K(n)$ the $ n$th Morava $ K$-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this paper show that if $ X$ is a finite spectrum, then the localization $ L_{K(n)}(X)$ is equivalent to the homotopy fixed point spectrum $ (L_{K(n)}(E_n \wedge X))^{hG_n}$, which is formed with respect to the continuous action of $ G_n$ on $ L_{K(n)}(E_n \wedge X)$. In this paper, we show that this equivalence holds for any ($ S$-cofibrant) spectrum $ X$. Also, we show that for all such $ X$, the strongly convergent Adams-type spectral sequence abutting to $ \pi_\ast(L_{K(n)}(X))$ is isomorphic to the descent spectral sequence that abuts to $ \pi_\ast((L_{K(n)}(E_n \wedge X))^{hG_n}).$

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Additional Information

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

Takeshi Torii
Affiliation: Department of Mathematics, Faculty of Science, Okayama University, Okayama 700-8530, Japan

Received by editor(s): December 17, 2010
Published electronically: July 18, 2011
Additional Notes: The first author was partially supported by a grant (LEQSF(2008-11)-RD-A-27) from the Louisiana Board of Regents.
Communicated by: Brooke Shipley
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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