Cohomology of the Morava stabilizer group through the duality resolution at 𝑛=𝑝=2

Abstract

We compute the continuous cohomology of the Morava stabilizer group with coefficients in Morava E E -theory, H ∗ ( G 2 , E t ) H^*(\mathbb {G}_2, \mathbf {E}_t) , at p = 2 p=2 , for 0 ≤ t > 12 0\leq t > 12 , using the Algebraic Duality Spectral Sequence. Furthermore, in that same range, we compute the d 3 d_3 -differentials in the homotopy fixed point spectral sequence for the K ( 2 ) K(2) -local sphere spectrum. These cohomology groups and differentials play a central role in K ( 2 ) K(2) -local stable homotopy theory.