Complexity and Tor on a Complete Intersection
Complexity and Tor on a Complete Intersection
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完整交叉点上的复杂性和 Tor
DOI：
10.1006/jabr.1998.7743
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发表时间：
1999
影响因子：
0.9
通讯作者：
David A. Jorgensen
中科院分区：
文献类型：
--
作者：
David A. Jorgensen
设(R, m)为完全交集，即局部环，其m进补全为正则局部环除以正则序列的商。假设是一个有限生成的 R 模块。已知Ma的偶数和奇数Betti序列最终由同次多项式给出； M 的复杂度是非负整数 n + 1。我们使用这个复杂度概念来研究有限生成模块 Man 和完整交集 R 上 TorRi(M, N) 的消失。我们证明了几个涉及 Tor 刚性的定理，这些定理是一般化的，并且在某些情况下是已知结果的改进。这些刚性定理的主要思想是，刚性定理假设中所需的连续消失 Tor 的数量更多地取决于 M 和 N 的复杂度的最小值，而不是 R 的余维数。我们举的例子表明这种依赖性是很强的。我们还表明，如果 M ⊗ RN 具有有限长度，那么，对于足够高的索引，两个连续消失的 Tors 会迫使所有更高的 Tors 消失。
Let (R, m) be a complete intersection, that is, a local ring whose m-adic completion is the quotient of a regular local ring by a regular sequence. SupposeMis a finitely generatedR-module. It is known that the even and odd Betti sequences ofMare eventually given by polynomials of the same degreen; the complexity ofMis the nonnegative integern + 1. We use this notion of complexity to study the vanishing of TorRi(M, N) for finitely generated modulesMandNover a complete intersectionR. We prove several theorems dealing with rigidity of Tor, which are generalizations and, in certain situations, improvements of known results. The main idea of these rigidity theorems is that the number of consecutive vanishing Tors required in the hypothesis of a rigidity theorem depends more on the minimum of the complexities ofMandNrather than on the codimension ofR. We give examples showing that this dependence is sharp. We also show that ifM ⊗ RNhas finite length, then, for sufficiently high indices, two consecutive vanishing Tors force the vanishing of all higher Tors.