Abstract
Following an idea of Hopkins, we construct a model of the determinant sphere \(S\langle {\text {det}}\rangle \) 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.
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Notes
The quoted results only claims the vanishing for \(s > N\) where N depends only on n and p. To get \(N=n^2\) would require reworking the proof and using that \(\mathbb {G}\) has virtual Poincaré duality of dimension \(n^2\).
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Acknowledgements
We would like to thank Hans-Werner Henn, Mike Hopkins, Viet-Cuong Pham and Charles Rezk for helpful conversations and the referee for their comments
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This material is based upon work supported by the National Science Foundation under Grant No. DMS-1812122 and Grant No. DMS-1725563. Barthel was partially supported by the DNRF92 and the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 751794. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program Homotopy Harnessing Higher Structures when work on this paper was undertaken. This work was supported by EPSRC Grant Number EP/R014604/1.
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Barthel, T., Beaudry, A., Goerss, P.G. et al. Constructing the determinant sphere using a Tate twist. Math. Z. 301, 255–274 (2022). https://doi.org/10.1007/s00209-021-02864-x
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DOI: https://doi.org/10.1007/s00209-021-02864-x