The duality resolution at $n=p=2$

TL;DR

This paper constructs a finite resolution of homotopy fixed points of Morava $E$-theory at prime 2 and height 2, refining previous resolutions by focusing on a specific subgroup of the Morava stabilizer group.

Contribution

It provides a new, more precise finite resolution of homotopy fixed points at prime 2 and height 2, enhancing the understanding of Morava $E$-theory's structure.

Findings

Constructed a finite resolution at prime 2, height 2.

Improved upon previous resolutions by focusing on subgroup $ extbf{G}_2^1$.

Enhances the understanding of homotopy fixed points in chromatic homotopy theory.

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.