Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

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


Author: Sanath Devalapurkar
Journal: Proc. Amer. Math. Soc. 148 (2020), 3187-3194
MSC (2010): Primary 55P43, 55S12
DOI: https://doi.org/10.1090/proc/14960
Published electronically: February 26, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 55P43, 55S12

Retrieve articles in all journals with MSC (2010): 55P43, 55S12


Additional Information

Sanath Devalapurkar
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: sanathd@mit.edu

DOI: https://doi.org/10.1090/proc/14960
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
Article copyright: © Copyright 2020 American Mathematical Society