Vasconcelos' conjecture on the conormal module

• 成果类型: Article
• 署名作者: Briggs, Benjamin
• 署名单位: Utah System of Higher Education; University of Utah
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01070-0
• 发表日期: 2022
• 页码: 415-428
• 关键词: homology; IDEALS; dependence; Algebra

摘要

For any ideal I of finite projective dimension in a commutative noetherian local ring R, we prove that if the conormal module I/I-2 has finite projective dimension over R/I, then I must be generated by a regular sequence. This resolves a conjecture of Vasconcelos. We prove a similar result for the first Koszul homology module of I. When R is a localisation of a polynomial ring over a field K of characteristic zero, Vasconcelos conjectured that R/I is a reduced complete intersection if the module Omega(R/I)/K of Kahler differentials has finite projective dimension; we prove this contingent on the Eisenbud-Mazur conjecture. The arguments exploit the structure of the homotopy Lie algebra associated to I in an essential way. By work of Avramov and Halperin, if every degree 2 element of the homotopy Lie algebra is radical, then I is generated by a regular sequence. Iyengar has shown that free summands of I/I-2 give rise to central elements of the homotopy Lie algebra, and we establish an analogous criterion for constructing radical elements, from which we deduce our main result.