On the Balmer spectrum of the Morel-Voevodsky category

TL;DR

This paper constructs new isotropic Morava points in the Balmer spectrum of the Morel-Voevodsky category, enhancing understanding of its structure through generalized local motivic categories and Morava K-theory.

Contribution

It introduces the isotropic Morava points in the Balmer spectrum of the motivic stable homotopy category, generalizing local motivic categories with a broader notion of isotropy.

Findings

Construction of isotropic Morava points parametrized by Morava K-theory and field extensions

Provides a larger set of points in the spectrum, improving its understanding

Reveals different specialization behavior among isotropic points compared to topology

Abstract

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 ${\Bbb{A}}^1$-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 $\operatorname{Spc}(SH^c(k))$ 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 $E/k$. 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.