Localizations of Morava E-theory and deformations of formal groups

TL;DR

This paper explores the algebraic structures of Morava E-theory localizations, revealing their connections to formal group deformations and uncovering exotic $ ext{E}_ ext{infty}$ structures in certain localizations.

Contribution

It provides a modular interpretation of the coefficient rings of transchromatic localizations of Morava E-theory and identifies new exotic $ ext{E}_ ext{infty}$ structures.

Findings

Coefficient ring $ ext{pi}_0L_{K(n-1)}E_n$ represents deformations of formal groups.

Descriptions of cooperations algebra and $E_{n-1}$-homology for these spectra.

Existence of exotic $ ext{E}_ ext{infty}$ structures on $L_{K(1)}E_2$.

Abstract

We study the relationship between the transchromatic localizations of Morava $E$-theory, $L_{K(n-1)}E_n$, and formal groups. In particular, we show that the coefficient ring $\pi_0L_{K(n-1)}E_n$ has a modular interpretation, representing deformations of formal groups with certain extra structure, and derive similar descriptions of the cooperations algebra and $E_{n-1}$-homology of this spectrum. As an application, we show that $L_{K(1)}E_2$ has exotic $\mathcal{E}_\infty$ structures not obtained by $K(1)$-localizing the $\mathcal{E}_\infty$ ring $E_2$.