Constructing the determinant sphere using a Tate twist

2018/10/15 by Barthel, Tobias, Beaudry, Agnès, Goerss, Paul G. +1 · 1 citation
#55P42 #55P91 #55P92 #55Q51 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.06651

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 ℤ_p^×. The Tate sphere inherits an action of \mathbbG_n via the determinant and smashing Morava E-theory with S(1) has the effect of twisting the action of \mathbbG_n. A large part of this paper consists of analyzing continuous \mathbbG_n-actions and their homotopy fixed points in the setup of Devinatz and Hopkins.

Cited by

• Picard groups of quotient ring spectra

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• Lubin-Tate theory, character theory, and power operations