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.
Similar content being viewed by others
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 \(n^2\).
References
Barthel, T., Beaudry, A.: Chromatic structures in stable homotopy theory. In: Miller, H. (ed.) Handbook of Homotopy Theory, Handbooks in Mathematics Series, pp. 163–220. CRC Press, Taylor & Francis Group, Boca Raton (2020)
Behrens, M., Davis, D.G.: The homotopy fixed point spectra of profinite Galois extensions. Trans. Am. Math. Soc. 362(9), 4983–5042 (2010)
Beaudry, A., Goerss, P.G., Henn, H.-W.: Chromatic splitting for the \(K(2)\)-local sphere at \(p=2\) (2017). arXiv:1712.08182
Devinatz, E.S., Hopkins, M.J.: Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups. Topology 43(1), 1–47 (2004)
Davis, D.G., Quick, G.: Profinite and discrete \(G\)-spectra and iterated homotopy fixed points. Algebraic Geom. Topol. 16(4), 2257–2303 (2016)
Davis, D.G., Torii, T.: Every \(K(n)\)-local spectrum is the homotopy fixed points of its Morava module. Proc. Am. Math. Soc. 140(3), 1097–1103 (2012)
Goerss, P., Henn, H.-W., Mahowald, M., Rezk, C.: A resolution of the \(K(2)\)-local sphere at the prime 3. Ann. Math. (2) 162(2), 777–822 (2005)
Goerss, P., Henn, H.-W., Mahowald, M., Rezk, C.: On Hopkins’ Picard groups for the prime 3 and chromatic level 2. J. Topol. 8(1), 267–294 (2015)
Hopkins, M.J., Gross, B.H.: The rigid analytic period mapping, Lubin–Tate space, and stable homotopy theory. Bull. Am. Math. Soc. (N.S.) 30(1), 76–86 (1994)
Hopkins, M.J., Mahowald, M., Sadofsky, H.: Constructions of elements in Picard groups. In: Topology and Representation Theory (Evanston, IL, 1992), Contemp. Math., vol. 158, pp. 89–126. Amer. Math. Soc., Providence (1994)
Hopkins, M.J., Smith, J.H.: Nilpotence and stable homotopy theory. II. Ann. Math. (2) 148(1), 1–49 (1998)
Hovey, M., Strickland, N.P.: Morava \(K\)-theories and localisation. Mem. Am. Math. Soc. 139(666), viii+100 (1999)
Heard, D., Stojanoska, V.: \(K\)-theory, reality, and duality. J. K-Theory 14(3), 526–555 (2014)
Peterson, E.: Coalgebraic formal curve spectra and spectral jet spaces. Geom. Topol. 24(1), 1–47 (2020)
Quick, G.: Continuous homotopy fixed points for Lubin–Tate spectra. Homol. Homotopy Appl. 15(1), 191–222 (2013)
Ravenel, D.C.: Nilpotence and periodicity in stable homotopy theory. In: Annals of Mathematics Studies, vol. 128, Appendix C by Jeff Smith. Princeton University Press, Princeton (1992)
Strickland, N.P.: Gross–Hopkins duality. Topology 39(5), 1021–1033 (2000)
Westerland, C.: A higher chromatic analogue of the image of \(J\). Geom. Topol. 21(2), 1033–1093 (2017)
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
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-021-02864-x