The Stable Galois Correspondence for Real Closed Fields

In previous work [7], the authors constructed and studied a lift of the Galois correspondence to stable homotopy categories. In particular, if L/k is a nite Galois extension of elds with Galois group G, there is a functor c∗ L/k : SHG → SHk from the G-equivariant stable homotopy category to the stable motivic homotopy category over k such that c∗ L/k (G/H+) = Spec(L)+. The main theorem of [7] says that when k is a real closed eld and L = k[i], the restriction of c∗ L/k to the η-complete subcategory is full and faithful. Here we uncomplete this theorem so that it applies to c∗ L/k itself. Our main tools are Bachmann's theorem on the (2, η)-periodic stable motivic homotopy category and an isomorphism range for the map πR ?SR → π C2 ? SC2 induced by C2-equivariant Betti realization.