Constructing the determinant sphere using a Tate twist

TL;DR

This paper constructs a model of the determinant sphere in the $K(n)$-local category by introducing the Tate sphere with a continuous group action, analyzing its properties and homotopy fixed points.

Contribution

It introduces the Tate sphere $S(1)$ with a $bZ_p^ imes$-action and constructs the determinant sphere $Sraket{det}$ using this new spectrum, extending Hopkins' ideas.

Findings

Constructed the Tate sphere $S(1)$ with a natural $bZ_p^ imes$-action.

Analyzed continuous $bG_n$-actions and their homotopy fixed points.

Established a model of the determinant sphere $Sraket{det}$ in the $K(n)$-local spectra category.

Abstract

Following an idea of Hopkins, we construct a model of the determinant sphere $S\langle det \rangle$ 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.