#### Volume 22, issue 4 (2018)

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Primes and fields in stable motivic homotopy theory

### Jeremiah Heller and Kyle M Ormsby

Geometry & Topology 22 (2018) 2187–2218
##### Abstract

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

${\rho }^{\bullet }:\phantom{\rule{0.3em}{0ex}}Spc\left({SH}^{{\mathbb{A}}^{1}\phantom{\rule{0.3em}{0ex}}}{\left(F\right)}^{c}\right)\to {Spec}^{h}\left({K}_{\ast }^{MW}\left(F\right)\right)$

from the tensor triangular spectrum of the homotopy category of compact motivic spectra to the homogeneous Zariski spectrum of Milnor–Witt $K\phantom{\rule{0.3em}{0ex}}$–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.

##### Keywords
tensor triangular geometry, stable motivic homotopy theory
##### Mathematical Subject Classification 2010
Primary: 14F42
Secondary: 19D45, 55P42, 18E30
##### Publication
Revised: 19 July 2017
Accepted: 29 August 2017
Published: 5 April 2018
Proposed: Stefan Schwede
Seconded: Mark Behrens, Haynes R Miller
##### Authors
 Jeremiah Heller Department of Mathematics University of Illinois at Urbana-Champaign Urbana IL United States Kyle M Ormsby Department of Mathematics Reed College Portland OR United States