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

TL;DR

This paper investigates the conditions under which certain algebraic structures called $E_k$-rings can be realized with specific properties, especially when roots of unity are involved, revealing limitations in the $K(n)$-local setting.

Contribution

It provides new non-realizability results for $H_$-rings containing roots of unity in the $K(n)$-local context, extending classical realization theorems to ramified cases.

Findings

Non-realizability of $H_$-rings with roots of unity in $K(n)$-local setting

Proof that the Lubin--Tate tower cannot be lifted to $H_$-rings over Morava $E$-theory

Extension of classical realization results to ramified cases

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.