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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Roots of unity in $K(n)$-local rings
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by Sanath Devalapurkar PDF
Proc. Amer. Math. Soc. 148 (2020), 3187-3194 Request permission

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.
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Additional Information
  • Sanath Devalapurkar
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • Email: sanathd@mit.edu
  • Received by editor(s): October 1, 2019
  • Received by editor(s) in revised form: December 1, 2019
  • Published electronically: February 26, 2020
  • Communicated by: Mark Behrens
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 3187-3194
  • MSC (2010): Primary 55P43, 55S12
  • DOI: https://doi.org/10.1090/proc/14960
  • MathSciNet review: 4099803