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

TL;DR

This paper demonstrates that the $p$-primary and $p$-adic cases of the Isotropy Conjecture do not hold universally and proposes $BP$-theory as an alternative framework to extend previous results to all primes.

Contribution

It introduces $BP$-theory with $I( obreak ext{infty})$-primary and $I( obreak ext{infty})$-adic coefficients as substitutes for $p$-primary and $p$-adic Chow groups, extending prior results.

Findings

The $p$-primary and $p$-adic cases of the Isotropy Conjecture do not hold in general.

$BP$-theory with specific coefficients can replace Chow groups in this context.

The approach allows extension of results to arbitrary primes.

Abstract

The purpose of this note is to show that, in contrast to the ${\Bbb F}_p$-case (proven in [7]), the $p$-primary and $p$-adic cases of the Isotropy Conjecture, claiming that the isotropic Chow groups with ${\Bbb Z}/p^r$, $r>1$, respectively, with ${\Bbb Z}_p$-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.