滤波模的Hilbert函数

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摘要翻译：
在这篇文章中，我们将讨论局部环上模的Hilbert函数理论的一些方面，并且我们打算引导读者沿着过去三十年在这个动态数学活动领域的进展中的一条可能的路线前进。在这一领域日益增长的兴趣的推动下，我们的目标是将这一理论的许多新发展汇集到一个地方，并用一种统一的方法来展示它们，这种方法给出了自足的和更容易的证明。在本文中，我们将讨论不同作者的许多结果，基本上遵循J.Sally的开创性工作所代表的方向。我们对这个主题的个人观点最明显地通过第1章和第2章的介绍来表达，在这两章中我们讨论了表面元素和相关装置的使用。将强调基本技术，目的是通过使用更基本的方法来反驳最近的结果。在过去的几年里，出现了几篇论文，将Hilbert函数理论的经典结果推广到滤波模的情况。将该理论推广到模上的一般过滤情形还有一个更重要的动因。也就是说，我们在研究与过滤无关的分次代数上有有趣的应用，特别是纤维锥和Sally-模。我们在这里证明了这些代数中的每一个都适合于某些短的精确序列，以及与过滤相关的代数。因此，我们可以借助我们在过滤的情况下得到的知识来研究Hilbert函数和这些代数的深度。
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英文标题：
《Hilbert Functions of Filtered Modules》
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作者：
M.E. Rossi, G. Valla
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最新提交年份：
2009
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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：
  In this presentation we shall deal with some aspects of the theory of Hilbert functions of modules over local rings, and we intend to guide the reader along one of the possible routes through the last three decades of progress in this area of dynamic mathematical activity. Motivated by the ever increasing interest in this field, our goal is to gather together many new developments of this theory into one place, and to present them using a unifying approach which gives self-contained and easier proofs. In this text we shall discuss many results by different authors, following essentially the direction typified by the pioneering work of J. Sally. Our personal view of the subject is most visibly expressed by the presentation of Chapters 1 and 2 in which we discuss the use of the superficial elements and related devices. Basic techniques will be stressed with the aim of reproving recent results by using a more elementary approach. Over the past few years several papers have appeared which extend classical results on the theory of Hilbert functions to the case of filtered modules. The extension of the theory to the case of general filtrations on a module has one more important motivation. Namely, we have interesting applications to the study of graded algebras which are not associated to a filtration, in particular the Fiber cone and the Sally-module. We show here that each of these algebras fits into certain short exact sequences, together with algebras associated to filtrations. Hence one can study the Hilbert function and the depth of these algebras with the aid of the know-how we got in the case of a filtration.
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PDF链接：
https://arxiv.org/pdf/0710.2346

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关键词：hilbert Ber ert Hil Presentation into 数学 代数 many algebras