On the equations defining curves in a polynomial algebra

Abstract

Let $A$ be a commutative Noetherian ring of dimension $n$ ($n \ge 3$). Let $I$ be a local complete intersection ideal in $A[T]$ of height $n$. Suppose $I/{I^2}$ is free ${A[T]}/I$-module of rank $n$ and $({A[T]}/I)$ is torsion in $K_0(A[T])$. It is proved in this paper that $I$ is a set theoretic complete intersection ideal in $A[T]$ if one of the following conditions holds: (1) $n$ $\ge 5$, odd; (2) $n$ is even, and $A$ contains the field of rational numbers; (3) $n = 3$, and $A$ contains the field of rational numbers.