On the Balmer spectrum of the Morel-Voevodsky category

We introduce the Morava-isotropic stable homotopy category and, more generally, the stable homotopy category of an extension $E\mathrm{/}k$. These “local” versions of the Morel-Voevodsky stable ${\mathbb{A}}^1$-homotopy category $SH(k)$ are analogues of local motivic categories introduced in [22], but with a substantially more general notion of “isotropy”. This permits to construct the, so-called, isotropic Morava points of the Balmer spectrum $\mathrm{Spc}(S{H}^c(k))$ of (the compact part of) the Morel-Voevodsky category. These analogues of topological Morava points are parametrized by the choice of Morava K-theory and a $K(p,m)$-equivalence class of extensions $E\mathrm{/}k$. This provides a large supply of new points, and substantially improves our understanding of the spectrum. An interesting new feature is that the specialization among isotropic points behaves differently than in topology.