Constructing the determinant sphere using a Tate twist

Following an idea of Hopkins, we construct a model of the determinant sphere S⟨ det ⟩ in the category of K(n)-local spectra. To do this, we build a spectrum which we call the Tate sphere S(1). This is a p-complete sphere with a natural continuous action of Zp×. The Tate sphere inherits an action of G_n via the determinant and smashing Morava E-theory with S(1) has the effect of twisting the action of G_n. A large part of this paper consists of analyzing continuous G_n-actions and their homotopy fixed points in the setup of Devinatz and Hopkins.