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 $$\mathbb {G}_2^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> 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 $$\mathbb {S}_2^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> constructed in work of Goerss–Henn–Mahowald–Rezk, Beaudry and Bobkova–Goerss.