On Chow groups of complete regular local rings (Q2452312)

Let \(R\) be a complete regular local ring of dimension \(n > 0\). It is conjectured that the \(i\)th Chow group \(A_i(R)\) is zero for \(i < n\). This conjecture is true if \(R\) is equicharacteristic and unramified. In the present paper, the author studies this conjecture. First he shows that \(A_{n-2}(R) = 0\). Since it is known that \(A_i(R) = 0\) for \(i = 0\), \(1\), \(n-1\), the conjecture is true if \(n \leq 4\). Next he showsd that \([I] = 0\) in \(A_{n-3}(R)\) if \(I \subset R\) is a Gorenstein ideal of height \(3\). Finally he studies almost complete intersection ideals. If \(\mathfrak p \subset R\) is a prime ideal of height \(i\), then he shows that there is an almost complete intersection ideal \(I\) such that \([R/\mathfrak p] = [R/I]\) in \(A_{n-i}(R)\). Furthermore he gives a necessary and sufficient condition for \([R/I]\) to be \(0\) when \(R\) is ramified.