Let $E_n$ be the Lubin-Tate spectrum and let $G_n$ be the $n$th extended Morava stabilizer group. Then there is a discrete $G_n$-spectrum $F_n$, with $L_{K(n)}(F_n) \simeq E_n$, that has the property that $(F_n)^{hU}\simeq E_n^{hU}$, for every open subgroup $U$ of $G_n$. In particular, $(F_n)^{hG_n} \simeq L_{K(n)}(S^0).$ More generally, for any closed subgroup $H$ of $G_n$, there is a discrete $H$-spectrum $Z_{n, H}$, such that $(Z_{n, H})^{hH} \simeq E_n^{hH}.$ These conclusions are obtained from results about consistent $k$-local profinite $G$-Galois extensions $E$ of finite vcd, where$L_k(-)$ is $L_M(L_T(-))$, with $M$ a finite spectrum and $T$ smashing. For example, we show that $L_k(E^{hH}) \simeq E^{hH}$, for every open subgroup $H$ of $G$.
Journal of Homotopy and Related Structures, Vol. 5(2010), No. 1, pp. 253-268