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 $k\geq 1$ 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 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 $n\geq 1$ 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.