Iterated homotopy fixed points for the Lubin-Tate spectrum

摘要

When G is a profinite group and H and K are closed subgroups, with H normal in K, it is not known, in general, how to form the iterated homotopy fixed point spectrum (Z~(hH))~(hK/H), where Z is a continuous G-spectrum and all group actions are to be continuous. However, we show that, if G = G_n, the extended Morava stabilizer group, and Z =L(E_n ∧ X), where L is Bousfield localization with respect to Morava K-theory, E_n is the Lubin-Tate spectrum, and X is any spectrum with trivial G_n-action, then the iterated homotopy fixed point spectrum can always be constructed. Also, we show that (E_n~(hH))~(hK/H) is just E_n~(hK), extending a result of Devinatz and Hopkins.