Localizations of Morava E-theory and deformations of formal groups
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 $E_2$.