论文标题
Vasconcelos在共素模块上的猜想
Vasconcelos' conjecture on the conormal module
论文作者
论文摘要
对于任何理想的$ i $ $ i $的有限投影尺寸，在通勤的noetherian本地环$ r $中，我们证明，如果Conormal模块$ I/I^2 $具有超过$ r/i $的有限投影尺寸，则$ i $必须按常规顺序产生。这解决了Vasconcelos的猜想。我们证明了第一个$ i $的第一个Koszul同源模块的结果。当$ r $是特征零的字段$ k $上的多项式戒指的本地化时，Vasconcelos认为，如果模块$ω_ {（R/I）/k} $差异的模块具有有限的投影尺寸，则$ r/i $是减少的完整交集；我们证明了对艾森布德 - 马祖尔的猜想的这一因素。 该论点利用同义的结构以$ i $相关的代数为基本的代数。通过Avramov和Halperin的工作，如果每个程度的$ 2 $ element lie代数是激进的，则$ i $由常规序列产生。 Iyengar表明，$ i/i^2 $的免费求和产生了同型lie代数的中心元素，我们建立了一个类似的标准来构建激进元素，从中我们推断出我们的主要结果。
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 $Ω_{(R/I)/K}$ of 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.
