The duality resolution at $n=p=2$

Abstract

Working at the prime $2$ and chromatic height $2$, we construct a finite resolution of the homotopy fixed points of Morava $E$-theory with respect to the subgroup ${\mathbb{G}}_2^1$ of the Morava stabilizer group. This is an upgrade of the finite resolution of the homotopy fixed points of $E$-theory with respect to the subgroup ${\mathbb{S}}_2^1$ constructed in work of Goerss-Henn-Mahowald-Rezk, Beaudry and Bobkova-Goerss.