Homotopy fixed points for LK(n)(En∧X) using the continuous action

Abstract Let K ( n ) be the n th Morava K -theory spectrum. Let E n be the Lubin–Tate spectrum, which plays a central role in understanding L K ( n ) ( S 0 ) , the K ( n ) -local sphere. For any spectrum X , define E ∨ ( X ) to be L K ( n ) ( E n ∧ X ) . Let G be a closed subgroup of the profinite group G n , the group of ring spectrum automorphisms of E n in the stable homotopy category. We show that E ∨ ( X ) is a continuous G -spectrum, with homotopy fixed point spectrum ( E ∨ ( X ) ) hG . Also, we construct a descent spectral sequence with abutment π * ( ( E ∨ ( X ) ) hG ) .