If \(\mathcal{C}\) is a symmetric monoidal category and \(G \in \mathcal{C}\) is an invertible object, then one obtains a graded ring \(R(\mathcal{C}, G) := \bigoplus_{n \in \mathbb Z} \text{Hom}_\mathcal{C}(1, G^{\otimes n})\). Via Balmer’s tensor triangular geometry, if \(\mathcal{C}\) is a tensor triangulated category, then the homogeneous prime ideal spectrum \(\mathrm{Spec}^h(R(\mathcal{C}, G))\) is related to the set of thick subcategories in \(\mathcal{C}\) which are “prime ideals”.
This motivates the computation of \(\mathrm{Spec}^h(R(\mathcal{C}, G))\) for any \(\mathcal{C}\) of interest. The article under review takes up this computation for \(\mathcal{C} = SH(k)\), the motivic stable homotopy category over a field. In this case naturally \(G = \mathbb{G}_m\) and then by a theorem of morel \(R(SH(k), \mathbb{G}_m) = K_*^{MW}(k)\), where \(K_*^{MW}(k)\) is a graded ring called Milnor-Witt \(K\)-theory which can be presented explicitly. The authors describe the topological space \(\mathrm{Spec}^h(K_*^{MW}(k))\) completely in terms of orderings of \(k\). This is done without recourse to homotopy theory; instead they use algebraic arguments relying on the explicit presentation of Milnor-Witt \(K\)-theory.