Constructing the determinant sphere using a Tate twist

Abstract

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 ${\mathbb{Z}}_p^{\times }$. The Tate sphere inherits an action of ${\mathbb{G}}_n$ via the determinant and smashing Morava $E$-theory with $S(1)$ has the effect of twisting the action of ${\mathbb{G}}_n$. A large part of this paper consists of analyzing continuous ${\mathbb{G}}_n$-actions and their homotopy fixed points in the setup of Devinatz and Hopkins.

Revised version, including a correction and a newly included example in the last section