摘要翻译：
我们研究了局部环$(a，\m)的第一Hilbert系数（重数后）$E_1$。$在各种情况下，从D.G.的工作开始，也称为局部环$A$的{\bf Chern数}。Northcott在60年代证明了几个结果，这些结果给出了Hilbert系数之间的一些关系，但总是假定基本环的Cohen-Macaulayness。S.Goto、K.Nishida、a.Corso和W.Vasconcelos最近的论文将这种兴趣推向了一个更普遍的背景。本文推广了S.Huckaba和T.Marley证明的E_1$的一个上界。因此，我们得到了环$a$的Cohen-Macauly，作为整数$E_1极值行为的结果。这个结果可以被认为是W.Vasconcelos的论文的一般哲学的证实，其中Chern数被推测为与$a的Cohen-Macaulyness距离的度量。这篇论文的主要结果是表面元素的一个很好的，也许是意想不到的性质的结果。它本质上是一种局部环的“萨利机”。在最后一节中，我们描述了这些结果的一个应用，关于一个不一定是Cohen-Macaulay的模的好过滤的Sally模的重数的一个上界。它是Vaz Pinto的一个结果的非Cohen-Macaulay情形的推广。
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英文标题：
《On the Chern number of a filtration》
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作者：
M.E. Rossi and G. Valla
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最新提交年份：
2008
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分类信息：

一级分类：Mathematics        数学
二级分类：Commutative Algebra        交换代数
分类描述：Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
交换环，模，理想，同调代数，计算方面，不变理论，与代数几何和组合学的联系
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  We study the first Hilbert coefficient (after the multiplicity) $e_1$ of a local ring $(A,\m). $ Under various circumstances, it is also called the {\bf Chern number} of the local ring $A.$ Starting from the work of D.G. Northcott in the 60's, several results have been proved which give some relationships between the Hilbert coefficients, but always assuming the Cohen-Macaulayness of the basic ring. Recent papers of S. Goto, K. Nishida, A. Corso and W. Vasconcelos pushed the interest toward a more general setting. In this paper we extend an upper bound on $e_1$ proved by S. Huckaba and T. Marley. Thus we get the Cohen-Macaulayness of the ring $A$ as a consequence of the extremal behavior of the integer $e_1.$ The result can be considered a confirm of the general philosophy of the paper of W. Vasconcelos where the Chern number is conjectured to be a measure of the distance from the Cohen-Macaulyness of $A.$ This main result of the paper is a consequence of a nice and perhaps unexpected property of superficial elements. It is essentially a kind of "Sally machine" for local rings. In the last section we describe an application of these results, concerning an upper bound on the multiplicity of the Sally module of a good filtration of a module which is not necessarily Cohen-Macaulay. It is an extension to the non Cohen-Macaulay case of a result of Vaz Pinto.
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PDF链接：
https://arxiv.org/pdf/0804.4438