On the $p$-primary and $p$-adic cases of the Isotropy Conjecture

Abstract

The purpose of this note is to show that, in contrast to the ${\mathbb{F}}_p$-case (proven in [7]), the $p$-primary and $p$-adic cases of the Isotropy Conjecture, claiming that the isotropic Chow groups with $\mathbb{Z}/p^r$, $r>1$, respectively, with ${\mathbb{Z}}_p$-coefficients over a flexible field coincide with the numerical ones, don't hold. We show that the $BP$-theory with $I(\infty )$-primary, respectively, $I(\infty )$-adic coefficients may serve as a regular substitute for $p$-primary, respectively, $p$-adic Chow groups, which permits to extend the results of [6] to arbitrary primes.

9 pages