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 {F}₏ F p -coefficients). This shows that Isotropic Chow motives coincide with Numerical Chow motives. In particular, homs between such objects are finite groups and ⊗ 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 A^1 A 1 -stable homotopy category. This substantially improves our understanding of the mentioned spectra whose description is a major open problem.