Operations and co-operations in Morava $E$-theory

Abstract

Let $E=E_{n}$ denote the Morava $E$-theory spectrum, and let $\Gamma$ be the Morava stabilizer group of ring spectrum isomorphisms of $E$. We revisit the isomorphism $\pi_{*}L_{K(n)}(E\smash E)\cong C(\Gamma, E_{*})$ of graded formal Hopf algebroids, and its dual isomorphism $E^{*}E\cong E_{*}[[\Gamma]]$.