Generalized stretched ideals and Sally Conjecture
Generalized stretched ideals and Sally Conjecture
复制标题
广义拉伸理想和莎莉猜想
DOI：
10.1016/j.jpaa.2015.08.013
复制
发表时间：
2011
期刊：
影响因子：
--
通讯作者：
Yupei Xie
中科院分区：
文献类型：
--
作者：
P. Mantero;Yupei Xie
在Noether局部环中引入了j-拉伸理想的概念。它推广了Sally和Rossi-Valla的拉伸m-准素理想的经典概念，以及Polini和Xie引入的极小和几乎极小j-重数理想的概念。我们的一个主要定理指出，对于一个j-拉伸理想，相关联的分次环是Cohen-Macaulay当且仅当理想的两个经典不变量，约化数和幂零指数相等。我们的第二个主要定理给出的数值条件，确保几乎Cohen-Macaulayness的相关分次环的j-拉伸理想，它提供了一个推广版本的萨利猜想。这一工作也适用于模，它统一了Rossi-Valla和Polini-Xie的方法，同时推广了Sally，Rossi-Valla，Wang，Elias，Corso-Polini-Vaz平托，Huckaba，Marley和Polini-Xie等人关于相关分次模的Cohen-Macaulay性或几乎Cohen-Macaulay性的结果.
We introduce the concept of j-stretched ideal in a Noetherian local ring. It generalizes to arbitrary ideals the classical notion of stretched m-primary ideal of Sally and Rossi–Valla, as well as the concept of ideal of minimal and almost minimal j-multiplicity introduced by Polini and Xie. One of our main theorems states that, for a j-stretched ideal, the associated graded ring is Cohen–Macaulay if and only if two classical invariants of the ideal, the reduction number and the index of nilpotency, are equal. Our second main theorem gives numerical conditions ensuring the almost Cohen–Macaulayness of the associated graded ring of a j-stretched ideal, and it provides a generalized version of Sally's conjecture. This work, which also holds for modules, unifies the approaches of Rossi–Valla and Polini–Xie and generalizes simultaneously results on the Cohen–Macaulayness or almost Cohen–Macaulayness of the associated graded module by several authors, including Sally, Rossi–Valla, Wang, Elias, Corso–Polini–Vaz Pinto, Huckaba, Marley and Polini–Xie.