滤子模的Hilbert函数
Hilbert Functions of Filtered Modules
摘要：
在本介绍中，我们将处理的某些方面的理论希尔伯特功能的模块在局部环，我们打算引导读者沿着一条可能的路线，通过过去三十年的进展，在这一领域的动态数学活动。在这个领域的兴趣不断增加的动机，我们的目标是收集到一个地方这个理论的许多新的发展，并提出他们使用一个统一的方法，提供自包含和更容易的证明。在本文中，我们将讨论许多结果，由不同的作者，基本上以下的方向典型的开创性工作的J。我们个人对这一主题的看法在第1章和第2章的介绍中得到了最明显的表达，在这两章中，我们讨论了表面要素和相关手段的使用。基本的技术将强调与reprofiting最近的结果，通过使用一个更基本的方法的目的。在过去的几年中，出现了几篇论文，其中扩展经典的结果理论的希尔伯特函数的情况下，过滤模块。将该理论推广到模上的一般滤子的情形还有一个更重要的动机。也就是说，我们有有趣的应用程序的研究分次代数，这是不相关的过滤，特别是纤维锥和萨利模块。我们在这里表明，这些代数适合某些短的确切序列，连同代数相关的过滤。因此，人们可以研究希尔伯特函数和这些代数的深度与援助的诀窍，我们得到的情况下，过滤。
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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