On isotropic and numerical equivalence of cycles

TL;DR

This paper investigates the conjecture that isotropic and numerical Chow groups are equivalent over flexible fields, proving it for new cases and large primes, and exploring p-adic analogues to deepen understanding of motivic categories.

Contribution

It proves the isotropic and numerical Chow groups conjecture for new cases and large primes, and introduces p-adic analogues, advancing the understanding of motivic categories.

Findings

Conjecture holds for sufficiently large primes p.

Established p-adic analogue of the conjecture.

Interpreted integral numerically trivial classes as p-infinite anisotropic.

Abstract

We study the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with ${\Bbb{F}}_p$-coefficients). This conjecture is essential for understanding the structure of the isotropic motivic category and that of the tensor triangulated spectrum of Voevodsky category of motives. We prove the conjecture for the new range of cases. In particular, we show that, for a given variety $X$, it holds for sufficiently large primes $p$. We also prove the $p$-adic analogue. This permits to interpret integral numerically trivial classes in $CH(X)$ as $p^{\infty}$-anisotropic ones.