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

Abstract

In this paper we prove the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with ${\mathbb{F}}_p$-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 ${\mathbb{A}}^1$-stable homotopic category. This substantially improves our understanding of the mentioned spectra whose description is a major open problem.

arXiv commentto appear in Inventiones Mathematicae

References in corpus (6)

• Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)
• Isotropic motives
• Three real Artin-Tate motives
• Primes and fields in stable motivic homotopy theory
• tt-geometry of Tate motives over algebraically closed fields
• Affine quadrics and the Picard group of the motivic category