Chow Groups: A Structure Theorem, RIEMANN-ROCH without denominators and ARTIN approximation

TL;DR

This paper investigates the structure of Chow groups over ramified regular local rings, providing new characterizations, addressing factorial annihilation questions, and relating Chow groups of henselizations and completions.

Contribution

It introduces a new characterization of ramified regular local rings, explores factorial annihilation in Chow groups, and establishes relations between Chow groups of henselizations and completions.

Findings

Characterization of ramified regular local rings of finite type over a DVR.

Results on whether factorials annihilate certain Chow groups.

Constructive relations between Chow groups of henselizations and completions.

Abstract

The focus of this note is on the Chow group problem over ramified regular local rings $(R, m)$. Our goal is threefold: i) to introduce a characterization of a ramified regular local ring essentially of finite type over a dvr, ii) to address the question whether $(i-1)!$ $\mathbb{A}^i(U)=0$ for specific open subsets $U$ of Spec$R$ and iii) to establish a constructive relation between Chow groups of the henselization $(R^h, m^h)$ and Chow groups of the completion $(\hat{R}, \hat{m})$ of $(R, m)$.