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∕k. These “local” versions of the Morel–Voevodsky stable A1-homotopy category SH(k) are analogues of local motivic categories introduced in Vishik (2022) but with a substantially more general notion of “isotropy.” This permits to construct the so-called isotropic Morava points of the Balmer spectrum Spc(SHc(k)) of (the compact part of) the Morel–Voevodsky category. These analogues of topological Morava points are parameterized by the choice of Morava K-theory and a K(p,m)-equivalence class of extensions E∕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.