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

Abstract

The goal of this paper is to address the following question: if $A$ is an ${\mathbf{E}}_k$-ring for some $k\ge 1$ and $f:{\pi }_{0}A\to B$ is a map of commutative rings, when can we find an ${\mathbf{E}}_k$-ring $R$ with an ${\mathbf{E}}_k$-ring map $g:A\to R$ 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 etale. 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 $n\ge 1$ 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.

7 pages; improved introduction; version accepted by Proc. Amer. Math. Soc

References in corpus (1)

• Towards an understanding of ramified extensions of structured ring spectra

Cited by in corpus (2)

• Chromatic Cyclotomic Extensions
• Adjoining roots in homotopy theory