On Chow groups of complete regular local rings

Abstract

In this paper, we establish the validity of the Chow group problem for complete regular local rings $R$ of dimension up to 4. For dimension $n$ ($>4$) over ramified regular local ring $R$, we have two results: (1) When $I$ is an ideal of height 3 such that $R/I$ is a Gorenstein ring, then $[I]=0$ in $A_{n-3}(R)$. (2) We reduce any prime ideal of height $i$ to an almost complete intersection ideal of height $i$ and in some special cases of almost complete intersection ideal of height $i$, we show that all Chow groups except the top one vanish. A necessary and sufficient condition for the vanishing of Chow groups is also derived using Eisenstein extension.