Commun. Korean Math. Soc. 2002 Vol. 17, No. 2, 221-227 Printed June 1, 2002

Sichang Lee Korea Military Academy

Abstract : In this paper we will show that the followings ; (1) Let $R$ be a regular local ring of dimension $n$. Then $A_{n-2}(R)=0$. (2) Let $R$ be a regular local ring of dimension $n$ and $I$ be an ideal in $R$ of height 3 such that $R/I$ is a Gorenstein ring. Then $[I]=0$ in $A_{n-3}(R)$. (3) Let $R=V[[X_1$, $X_2$, $\cdots$, $X_{5}]]/(p+X_1^{t_1}$ $+ X_2^{t_2}$ $+ X_3^{t_3}$ $+ X_4^2$ $+ X_5^2)$, where $p \ne 2$, $t_1$, $t_2$, $t_3$ are arbitrary positive integers and $V$ is a complete discrete valuation ring with $(p)=m_V$. Assume that $R/m$ is algebraically closed. Then all the Chow group for $R$ is 0 except the last Chow group.

Keywords : Chow group, complete regular local ring, Gorenstein ideal of codimension 3, dimension 5, height 3 ideal