On flat pullbacks for Chow groups

TL;DR

This paper provides a natural sheaf-theoretic proof that flat pullbacks preserve rational equivalence of cycles, ensuring the contra-functoriality of Chow groups in algebraic geometry.

Contribution

It introduces a sheaf-theoretic approach to prove the preservation of rational equivalence under flat pullback, clarifying a fundamental property of Chow groups.

Findings

Sheaf-theoretic proof of rational equivalence preservation

Enhanced understanding of flat pullback in Chow groups

Strengthened foundation for contra-functoriality of Chow groups

Abstract

It is a fundamental property of the Chow groups of algebraic schemes that they are contra-functorial with respect to flat morphisms between schemes. While the pullback homomorphism is easy to define at the level of algebraic cycles, the crucial step is to show that the pullback of cycles preserves rational equivalence, so that it descends to the Chow groups. The purpose of this note is to give a natural sheaf theoretic proof of the preservation of rational equivalence under flat pullback on cycles.