The stable Galois correspondence for real closed fields

Abstract

In previous work [6], 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 * L/k : SH G → SH k from the G-equivariant stable homotopy category to the stable motivic homotopy category over k such that c * L/k (G/H + ) = Spec(L H ) + . The main theorem of [6] says that when k is a real closed field 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 π R ⋆ S R → π C 2 ⋆ S C 2 induced by C 2 -equivariant Betti realization.

Galois reconstruction of Artin-Tate $\mathbb{R}$-motivic spectra

“…It suffices to show that a : S 0,−1,0 p,η → S p,η is zero. This follows from the facts that c C/R is fully-faithful (see [HO18]) and that a σ = 0 in the C 2 -equivariant category after inverting 2 and η-completing.…”

“…Since they are symmetric monoidal left adjoints, it is easy to see that the composites Be•c : Sp → Sp and Be•c C/R : Sp C2 → Sp C2 are equivalent to the identity. In fact, upon restricting to the appropriate target category we uncover something more interesting: HO16,HO18]). The symmetric monoidal functors c and c C/R factor through the respective categories of Artin objects and provide equivalences,…”

Topological models for stable motivic invariants of regular number rings

“…The mathematical framework for motivic homotopy theory has been established over the last twentyfive years [Lev18]. An interesting aspect witnessed by the complex and real numbers, C, R, is that Betti realization functors provide mutual beneficial connections between the motivic theory and the corresponding classical and C 2 -equivariant stable homotopy theories [Lev14], [GWX18], [HO18], [BS19], [ES19], [Isa19], [IØ20]. We amplify this philosophy by extending it to deeper base schemes of arithmetic interest.…”

Real motivic and C2‐equivariant Mahowald invariants

“…The heart of our work is the deep connection between ‐motivic and ‐equivariant stable homotopy theory which has emerged over the past few decades [6, 9, 13, 18, 25, 26, 45, 55]. The key idea throughout our work on generalized Mahowald invariants is that all the relevant technology, such as stunted projective spectra, Lin's theorem, and Steenrod operations, is compatible under the comparison functors between the classical, motivic, and ‐equivariant stable homotopy categories.…”