Ramification in $\mathbf{E}_{n}$-algebras

Abstract

We study $\mathbf{E}_n$-algebras with ramification in their $\pi_0$ and in particular show that the existence of ramification places strong limitations on their chromatic behavior. In particular, we show that $\pi_0$ of nonzero $K(1)$-local $\mathbf{E}_4$-$\mathrm{KU}$-algebras cannot have a map from the ring of integers in a ramified extension of $\mathbf{Q}_p$, and we show that a $p$-complete $\mathbf{E}_2$-ring with a map from a perfectoid ring into its $\pi_0$ is an associative $\mathbf{Z}_p$-algebra.