Galois equivariance and stable motivic homotopy theory

TL;DR

This paper investigates the relationship between Galois-equivariant stable homotopy theory and motivic homotopy theory, establishing conditions for full and faithful embeddings after certain completions, and exploring related realization and spectral sequence convergence results.

Contribution

It introduces conditions under which a Galois-equivariant functor becomes fully faithful in motivic homotopy theory, extending understanding of Galois actions in this context.

Findings

Full and faithful embedding after completion for real closed fields with L=k[i]

Necessary conditions on field extensions for functor faithfulness

Convergence results for the C_2-equivariant Adams spectral sequence

Abstract

For a finite Galois extension of fields L/k with Galois group G, we study a functor from the G-equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical Galois correspondence. We show that after completing at a prime and eta (the motivic Hopf map) this results in a full and faithful embedding whenever k is real closed and L = k[i]. It is a full and faithful embedding after eta-completion if a motivic version of Serre's finiteness theorem is valid. We produce strong necessary conditions on the field extension L/k for this functor to be full and faithful. Along the way, we produce several results on the stable C_2-equivariant Betti realization functor and prove convergence theorems for the p-primary C_2-equivariant Adams spectral sequence.