On the $p$-primary and $p$-adic cases of the Isotropy Conjecture
Alexander Vishik
TL;DR
This paper disproves the extension of the Isotropy Conjecture to $p$-primary and $p$-adic Chow groups by exhibiting classes that are numerically trivial but not isotropic with ${\Bbb Z}/p^r$-coefficients, yet not liftable to numerically trivial $BP^*/I(\infty)^r$-classes; it then advocates $BP^*/I(\infty)^r$ (and its $I(\infty)$-adic completion $BP^*_{I(\infty)}$) as regular substitutes for $p$-primary and $p$-adic Chow theories, where the isotropy and numerical theories align. The authors demonstrate that the isotropy conjecture holds for these BP-based theories for arbitrary primes, extending results known for large primes, and provide explicit counterexamples to the CH$^*/p^r$-isotropy in the classical setting. They develop the $I(\infty)$-adic framework, establish isomorphisms between numerical and isotropic BP-theories, and prove structural properties such as finite generation and torsion behavior, thereby offering a robust, prime-agnostic alternative to $p$-adic Chow groups with potentially broad implications for oriented cohomology theories. The work clarifies the landscape of isotropy versus numerical equivalence in motivic contexts and provides a concrete toolset for extending isotropy-type results beyond prime cases.
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.