Primes and fields in stable motivic homotopy theory

TL;DR

This paper investigates the structure of the tensor triangular spectrum in stable motivic homotopy theory over fields of characteristic not 2, establishing key surjectivity results and exploring geometric properties.

Contribution

It proves the surjectivity of Balmer's comparison map for the homotopy category of compact motivic spectra and analyzes the tensor triangular geometry of cellular motivic spectra.

Findings

Surjectivity of Balmer's comparison map rho^* is established.

Identification of novel field spectra in the category of cellular motivic spectra.

Open questions posed about the structure of the tensor triangular spectrum.

Abstract

Let F be a field of characteristic different than 2. We establish surjectivity of Balmer's comparison map rho^* from the tensor triangular spectrum of the homotopy category of compact motivic spectra to the homogeneous Zariski spectrum of Milnor-Witt K-theory. We also comment on the tensor triangular geometry of compact cellular motivic spectra, producing in particular novel field spectra in this category. We conclude with a list of questions about the structure of the tensor triangular spectrum of the stable motivic homotopy category.