K-theory, reality, and duality

We present a new proof of Anderson's result that the real K -theory spectrum is Anderson self-dual up to a fourfold suspension shift; more strongly, we show that the Anderson dual of the complex K -theory spectrum KU is C 2 -equivariantly equivalent to Σ 4 KU , where C 2 acts by complex conjugation. We give an algebro-geometric interpretation of this result in spectrally derived algebraic geometry and apply the result to calculate 2-primary Gross-Hopkins duality at height 1. From the latter we obtain a new computation of the group of exotic elements of the K (1)-local Picard group.