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

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 $(R, m)$.