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 finite Galois extension of fields with Galois group G, there is a functor c#x002a;_L/k: SH^G → SH_k from the G-equivariant stable homotopy category to the stable motivic homotopy category over k such that c#x002a;_L/k (G/H₊) = Spec(L^H)+. The main theorem of [7] %citeHO:E2M says that when k is a real closed field and L = k[i], the restriction of c#x002a;_L/k to the η-complete subcategory is full and faithful. Here we “uncomplete” this theorem so that it applies to c#x002a;_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 (formula presented) induced by C₂-equivariant Betti realization.