On the Equations De.ning Curves in a Polynomial Algebra

Abstract

Let A be a commutative Noetherian ring of dimension n (n ≥ 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 article that I is a set theoretic complete intersection ideal in A[T] if one of the following conditions holds: (1) n ≥ 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.

Key Words:

• Euler class group
• Set theoretic complete intersection

Mathematics Subject Classification:

• Primary 13C10
• Secondary 13C40

ACKNOWLEDGMENT

I sincerely thank my advisor, Professor N. Mohan Kumar, for his guidance and many valuable suggestions, that have made this article possible in its present form. I also wish to thank Dr. G.V. Ravindra for his encouragement.

Notes

Communicated by I. Swanson.