Iterated homotopy fixed points for the Lubin–Tate spectrum

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 h H ) h K / 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 ˆ ( E n ∧ X ) , where L ˆ 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 h H ) h K / H is just E n h K , extending a result of Devinatz and Hopkins.