On the number of generators of ideals in polynomial rings
Résumé
For an ideal I in a noetherian ring R, let μ(I) be the minimal number of generators of I. It is well known that there is a sequence of inequalities μ(I/I2)≤μ(I)≤μ(I/I2)+1 that are strict in general. However, Murthy conjectured in 1975 that μ(I/I2)=μ(I) for ideals in polynomial rings whose height equals μ(I/I2). The purpose of this article is to prove a stronger form of the conjecture in case the base field is infinite of characteristic different from 2: Namely, the equality μ(I/I2)=μ(I) holds for any ideal I, irrespective of its height.