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

Davis, Daniel; Torii, Takeshi

doi:10.1090/S0002-9939-2011-11189-4

Every $K(n)$–local spectrum is the homotopy fixed points of its Morava module
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by Daniel G. Davis and Takeshi Torii

Proc. Amer. Math. Soc. 140 (2012), 1097-1103

DOI: https://doi.org/10.1090/S0002-9939-2011-11189-4

Published electronically: July 18, 2011

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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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