Primes and fields in stable motivic homotopy theory

Abstract

Let $F$ be a field of characteristic different from $2$. We establish surjectivity of Balmer’s comparison map

$${\rho }^{\bullet }:Spc\left({SH}^{{\mathbb{A}}^1}{\left(F\right)}^c\right)\to {Spec}^h\left(K_{\ast }^{MW}\left(F\right)\right)$$

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.