Comparison of symbolic and ordinary powers of ideals
Comparison of symbolic and ordinary powers of ideals
复制标题
理想的象征力量与普通力量的比较
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
C. Huneke
中科院分区：
文献类型：
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作者：
M. Hochster;C. Huneke
本文给出的所有环都是交换环、结合恒等式和Notherian环。最近，L.Ein，R.Lazarsfeld和K.Smith[Els]发现了一个关于理想的符号幂在等特征0的仿射正则环中的行为的显著而令人惊讶的事实：如果h是i的一个伴随素数的最大高度，则i(Hn)⊆In对所有n≥0.这里，如果W是I的相关素数的并的并的补集，I(T)表示ItRW到R的收缩，其中RW是R在乘法系统W上的局部化。它们的证明依赖于乘子理想理论，包括渐近版本，尤其需要解决奇点和消失定理。我们想承认，如果没有他们的慷慨和迅速地分享他们的研究，这份手稿就不会存在。我们在这里的目标是给出更强的结果，这些结果可以用在某些方面更初级的方法来证明。我们的结果在等价特征0和正素数特征p中都是有效的，但依赖于对特征p的约化.我们使用紧闭包方法，因此，我们既不需要解奇点，也不需要解决可能在正特征中失败的消失定理.对于结果的最基本形式，紧闭包理论所需要的就是紧闭包的定义，以及在正则环中，每个理想都是紧闭的事实。我们注意到，这里的主要论点与[Hu，5.14-16，p.45]中的一个证明密切相关，即特征p中的正则局部环是UFD，
All given rings in this paper are commutative, associative with identity, and Noetherian. Recently, L. Ein, R. Lazarsfeld, and K. Smith [ELS] discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0: if h is the largest height of an associated prime of I , then I (hn) ⊆ I n for all n ≥ 0. Here, if W is the complement of the union of the associated primes of I , I (t) denotes the contraction of I t RW to R, where RW is the localization of R at the multiplicative system W . Their proof depends on the theory of multiplier ideals, including an asymptotic version, and, in particular, requires resolution of singularities as well as vanishing theorems. We want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist. Our objective here is to give stronger results that can be proved by methods that are, in some ways, more elementary. Our results are valid in both equal characteristic 0 and in positive prime characteristic p, but depend on reduction to characteristic p. We use tight closure methods and, in consequence, we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic. For the most basic form of the result, all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring, every ideal is tightly closed. We note that the main argument here is closely related to a proof given in [Hu, 5.14–16, p. 45] that regular local rings in characteristic p are UFDs,