Advertisement

Computations of orbits for the Lubin–Tate ring

  • Agnès BeaudryEmail author
  • Naiche Downey
  • Connor McCranie
  • Luke Meszar
  • Andy Riddle
  • Peter Rock
Article

Abstract

We take a direct approach to computing the orbits for the action of the automorphism group \(\mathbb {G}_2\) of the Honda formal group law of height 2 on the associated Lubin–Tate rings \(R_2\). We prove that \((R_2/p)_{\mathbb {G}_2} \cong \mathbb {F}_p\). The result is new for \(p=2\) and \(p=3\). For primes \(p\ge 5\), the result is a consequence of computations of Shimomura and Yabe and has been reproduced by Kohlhaase using different methods.

Keywords

Honda formal group law Lubin-Tate ring Morava E-theory Morava stabilizer group Chromatic Vanishing Conjecture 

Notes

Acknowledgements

We thank some of the usual suspects for useful conversations: Tobias Barthel, Mark Behrens, Paul Goerss, Hans-Werner Henn, Mike Hopkins, Niko Naumann and Vesna Stojanoska. We also thank the referee and the editors their input.

References

  1. 1.
    Beaudry, A.: Towards the homotopy of the \(K(2)\)-local Moore spectrum at \(p=2\). Adv. Math. 306, 722–788 (2017).  https://doi.org/10.1016/j.aim.2016.10.020 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beaudry, A., Goerss, P.G., Henn, H.-W.: Chromatic splitting for the \(K(2)\)-local sphere at \(p=2\). arXiv e-prints (2017). arXiv:1712.08182
  3. 3.
    Bobkova, I., Goerss, P.G.: Topological resolutions in k(2)-local homotopy theory at the prime 2. Journal of Topology 11(4), 918–957 (2018).  https://doi.org/10.1112/topo.12076 CrossRefGoogle Scholar
  4. 4.
    Devinatz, E.S., Hopkins, M.J.: Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups. Topology 43(1), 1–47 (2004).  https://doi.org/10.1016/S0040-9383(03)00029-6 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Goerss, P.G., Hopkins, M.J.: Moduli spaces of commutative ring spectra. In: Baker, A., Richter, B. (eds.) Structured Ring Spectra. London Mathematical Society Lecture Note series, vol. 315, pp. 151–200. Cambridge University Press, Cambridge (2004).  https://doi.org/10.1017/CBO9780511529955.009
  6. 6.
    Goerss, P.G., Henn, H.-W., Mahowald, M.E.: The rational homotopy of the \(K(2)\)-local sphere and the chromatic splitting conjecture for the prime 3 and level 2. Doc. Math. 19, 1271–1290 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hazewinkel, M.: Formal Groups and Applications. Pure and Applied Mathematics, vol. 78. Academic Press, Inc. (Harcourt Brace Jovanovich Publishers), New York (1978)zbMATHGoogle Scholar
  8. 8.
    Henn, H.-W., Karamanov, N., Mahowald, M.E.: The homotopy of the \(K(2)\)-local Moore spectrum at the prime 3 revisited. Math. Z. 275(3–4), 953–1004 (2013).  https://doi.org/10.1007/s00209-013-1167-4 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hovey, M.: Bousfield localization functors and Hopkins’ chromatic splitting conjecture. In: Cenkl, M., Miller, H. (eds.) The Čech Centennial (Boston, MA, 1993). Contemporary Mathematics, vol. 181, pp. 225–250. American Mathematical Society, Providence, RI (1995).  https://doi.org/10.1090/conm/181/02036
  10. 10.
    Kohlhaase, J.: On the Iwasawa theory of the Lubin–Tate moduli space. Compos. Math. 149(5), 793–839 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lader, O.: Une résolution projective pour le second groupe de Morava pour \(p \ge 5\) et applications. Theses, Université de Strasbourg, October (2013). https://tel.archives-ouvertes.fr/tel-00875761
  12. 12.
    Lubin, J., Tate, J.: Formal moduli for one-parameter formal Lie groups. Bull. Soc. Math. France 94, 49–59 (1966). http://www.numdam.org/item?id=BSMF_1966__94__49_0
  13. 13.
    Morava, J.: Noetherian localisations of categories of cobordism comodules. Ann. Math. (2) 121(1), 1–39 (1985).  https://doi.org/10.2307/1971192 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Shimomura, K., Yabe, A.: The homotopy groups \(\pi _*(L_2S^0)\). Topology 34(2), 261–289 (1995).  https://doi.org/10.1016/0040-9383(94)00032-G MathSciNetCrossRefGoogle Scholar

Copyright information

© Tbilisi Centre for Mathematical Sciences 2018

Authors and Affiliations

  • Agnès Beaudry
    • 1
    Email author
  • Naiche Downey
    • 1
  • Connor McCranie
    • 1
  • Luke Meszar
    • 1
  • Andy Riddle
    • 1
  • Peter Rock
    • 1
  1. 1.Department of MathematicsUniversity of Colorado at BoulderBoulderUSA

Personalised recommendations