On the number of generators of ideals in polynomial rings

• 成果类型: Article
• 署名作者: Fasel, Jean
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2016.184.1.3
• 发表日期: 2016
• 页码: 315-331
• 关键词: projective-modules

摘要

For an ideal I in a noetherian ring R, let mu(I) be the minimal number of generators of I. It is well known that there is a sequence of inequalities mu(I/I-2)<=mu(I)<=mu(I/I-2)+1 that are strict in general. However, Murthy conjectured in 1975 that mu(I/I-2 = mu(I) for ideals in polynomial rings whose height equals mu(I/I-2) = mu(I) for ideals in polynomial rings whose height equals mu(I/I-2). 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 mu(I/I-2)=mu(I) holds for any ideal I, irrespective of its height.