Isotropic and numerical equivalence for Chow groups and Morava K-theories

TL;DR

This paper proves that isotropic and numerical Chow groups coincide over flexible fields, and extends this equivalence to Morava K-theories, providing new insights into the structure of motivic and homotopic spectra.

Contribution

It establishes the equivalence of isotropic and numerical Chow groups and motives over flexible fields, and extends these results to Morava K-theories, enriching the understanding of motivic and homotopic spectra.

Findings

Isotropic Chow groups coincide with numerical Chow groups over flexible fields.

Hom groups between isotropic and numerical Chow motives are finite.

Provides new points for the Balmer spectrum of motivic and homotopic categories.

Abstract

In this paper we prove the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with ${\Bbb{F}}_p$-coefficients). This shows that Isotropic Chow motives coincide with Numerical Chow motives. In particular, homs between such objects are finite groups and $\otimes$ has no zero-divisors. It provides a large supply of new points for the Balmer spectrum of the Voevodsky motivic category. We also prove the Morava K-theory version of the above result, which permits to construct plenty of new points for the Balmer spectrum of the Morel-Voevodsky ${\Bbb{A}}^1$-stable homotopic category. This substantially improves our understanding of the mentioned spectra whose description is a major open problem.