## 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

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## Bibliographic 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