On a base change conjecture for higher zero-cycles

TL;DR

This paper proves the surjectivity of a specialization map for higher zero-cycles on certain schemes, providing evidence for a broader conjecture about base change in motivic cohomology and its relation to torsion in Chow groups.

Contribution

It establishes the surjectivity of a specialization map for higher (0,1)-cycles on smooth projective schemes over henselian DVRs, supporting a conjecture on base change for motivic cohomology.

Findings

Surjectivity of the specialization map for higher (0,1)-cycles.

Evidence supporting the base change conjecture in motivic cohomology.

Connection between specialization maps and torsion in Chow groups.

Abstract

We show the surjectivity of a specialisation map on higher $(0,1)$-cycles for a smooth projective scheme over an excellent henselian discrete valuation ring. This gives evidence for a conjecture stated in an article of Kerz, Esnault and Wittenberg saying that base change holds for such schemes in general for motivic cohomology in degrees $(i,d)$ for fixed $d$ being the relative dimension over the base. Furthermore, the specialisation map we study is related to a finiteness conjecture for the $n$-torsion of $CH_0(X)$, where $X$ is a variety over a $p$-adic field.