On the numerical triviality of $BP$-cycles

TL;DR

This paper investigates the conditions under which BP-cycles are numerically trivial modulo powers of an invariant ideal, linking this triviality to pure symbols in Milnor K-theory over certain field closures.

Contribution

It establishes a connection between the numerical triviality of BP-cycles and pure symbols in Milnor K-theory over the flexible closure of the base field for prime 2.

Findings

Numerical triviality of BP-cycles is governed by pure symbols in K^M_*/2.

The results apply specifically to the case of prime 2.

The work relates algebraic cycles to Milnor K-theory in a new way.

Abstract

We show that, in the case of a prime $2$, the numerical triviality of $BP$-cycles modulo various powers of the (augmentation) invariant ideal $I(\infty)$ is controlled by pure symbols in $K^M_*/2$ over the flexible closure of the base field.