Roots of unity in 𝐾(𝑛)-local rings

Devalapurkar, Sanath

doi:10.1090/proc/14960

Roots of unity in $K(n)$-local rings
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by Sanath Devalapurkar

Proc. Amer. Math. Soc. 148 (2020), 3187-3194

DOI: https://doi.org/10.1090/proc/14960

Published electronically: February 26, 2020

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Abstract:

The goal of this paper is to address the following question: if $A$ is an $\mathbf {E}_{k}$-ring for some $k\geq 1$ and $f\colon \pi _0 A \to B$ is a map of commutative rings, when can we find an $\mathbf {E}_{k}$-ring $R$ with an $\mathbf {E}_{k}$-ring map $g\colon A \to R$ such that $\pi _0 g = f$? A classical result in the theory of realizing $\mathbf {E}_\infty$-rings, due to Goerss–Hopkins, gives an affirmative answer to this question if $f$ is étale. The goal of this paper is to provide answers to this question when $f$ is ramified. We prove a non-realizability result in the $K(n)$-local setting for every $n\geq 1$ for $H_\infty$-rings containing primitive $p$th roots of unity. As an application, we give a proof of the folk result that the Lubin–Tate tower from arithmetic geometry does not lift to a tower of $H_\infty$-rings over Morava $E$-theory.

References

R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger, $H_\infty$ ring spectra and their applications, Lecture Notes in Mathematics, vol. 1176, Springer-Verlag, Berlin, 1986. MR 836132, DOI 10.1007/BFb0075405
A. K. Bousfield, On $\lambda$-rings and the $K$-theory of infinite loop spaces, $K$-Theory 10 (1996), no. 1, 1–30. MR 1373816, DOI 10.1007/BF00534886
Andrew Baker and Birgit Richter, Realizability of algebraic Galois extensions by strictly commutative ring spectra, Trans. Amer. Math. Soc. 359 (2007), no. 2, 827–857. MR 2255198, DOI 10.1090/S0002-9947-06-04201-2
B. Dundas, A. Lindenstrauss, and B. Richter, Towards an understanding of ramified extensions of structured ring spectra, Mathematical Proceedings of the Cambridge Philosophical Society (2018), 1–20.
V. G. Drinfel′d, Elliptic modules, Mat. Sb. (N.S.) 94(136) (1974), 594–627, 656 (Russian). MR 0384707
P. G. Goerss and M. J. Hopkins, Moduli spaces of commutative ring spectra, Structured ring spectra, London Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, 2004, pp. 151–200. MR 2125040, DOI 10.1017/CBO9780511529955.009
Paul G. Goerss, Realizing families of Landweber exact homology theories, New topological contexts for Galois theory and algebraic geometry (BIRS 2008), Geom. Topol. Monogr., vol. 16, Geom. Topol. Publ., Coventry, 2009, pp. 49–78. MR 2544386, DOI 10.2140/gtm.2009.16.49
J. Hahn, On the Bousfield classes of $H_\infty$-ring spectra, https://arxiv.org/abs/1612.04386, December 2016.
Michael J. Hopkins, $K(1)$-local $E_\infty$-ring spectra, Topological modular forms, Math. Surveys Monogr., vol. 201, Amer. Math. Soc., Providence, RI, 2014, pp. 287–302. MR 3328537, DOI 10.1090/surv/201/16
Mark Hovey and Neil P. Strickland, Morava $K$-theories and localisation, Mem. Amer. Math. Soc. 139 (1999), no. 666, viii+100. MR 1601906, DOI 10.1090/memo/0666
Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich. MR 1876802
J. Lurie, Higher Algebra, 2016.
Jonathan D. Rogawski, Representations of $\textrm {GL}(n)$ and division algebras over a $p$-adic field, Duke Math. J. 50 (1983), no. 1, 161–196. MR 700135
M. Rapoport and Th. Zink, Period spaces for $p$-divisible groups, Annals of Mathematics Studies, vol. 141, Princeton University Press, Princeton, NJ, 1996. MR 1393439, DOI 10.1515/9781400882601
N. P. Strickland, Morava $E$-theory of symmetric groups, Topology 37 (1998), no. 4, 757–779. MR 1607736, DOI 10.1016/S0040-9383(97)00054-2
R. Schwänzl, R. M. Vogt, and F. Waldhausen, Adjoining roots of unity to $E_\infty$ ring spectra in good cases—a remark, Homotopy invariant algebraic structures (Baltimore, MD, 1998) Contemp. Math., vol. 239, Amer. Math. Soc., Providence, RI, 1999, pp. 245–249. MR 1718085, DOI 10.1090/conm/239/03606
Jared Weinstein, Semistable models for modular curves of arbitrary level, Invent. Math. 205 (2016), no. 2, 459–526. MR 3529120, DOI 10.1007/s00222-015-0641-5