Galois Equivariance and Stable Motivic Homotopy Theory

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] s...

The Homotopy Coniveau Filtration

We examine the “homotopy coniveau tower” for a general cohomology theory on smooth k-schemes, satisfying some natural axioms, and give a new proof that the layers of this tower for K-theory agree with motivic cohomology. We show how these constructions lead to a tower of functors on the Morel-Voevodsky stable homotopy category, and identify this stable homotopy coniveau tower with Voevodsky’s s...

The Homotopy Coniveau Tower

We examine the " homotopy coniveau tower " for a general cohomology theory on smooth k-schemes and give a new proof that the layers of this tower for K-theory agree with motivic cohomology. In addition, the homotopy coniveau tower agrees with Voevodsky's slice tower for S 1-spectra, giving a proof of a connect-edness conjecture of Voevodsky. The homotopy coniveau tower construction extends to a...

Open Problems in the Motivic Stable Homotopy Theory, I

Motivic Hopf elements and relations

We use Cayley–Dickson algebras to produce Hopf elements η, ν, and σ in the motivic stable homotopy groups of spheres, and we prove the relations ην = 0 and νσ = 0 by geometric arguments. Along the way we develop several basic facts about the motivic stable homotopy ring.