Studies in Commutative Algebra and Algebraic Geometry

交换代数和代数几何研究

• 批准号: 0901145
• 负责人: Melvin Hochster
• 金额: $ 75.07万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901145&HistoricalAwards=false
• 关键词: Studies; Commutative; Algebra; Algebraic; Geometry

Hochster proposes to continue investigating several long standing questions in the theory of (primarily local) Noetherian rings and to explore the relationship of these questions with the explosively developing theory of tight closure, which was introduced by Hochster and Huneke. One main thrust is to explore several notions of closure in rings that do not contain a field, such as finitely generated algebras over the integers, with the hope of extending tight closure theory to such rings, thereby solving many open questions. Approaches to the problem of proving existence of big Cohen-Macaulay modules in mixed characteristic are also given. A new theory of quasi-length and content related to certain local cohomology modules will be studied. This theory has raised some difficult, central problems, and is one of several methods proposed to attack the long standing and important question of whether regular rings are direct summands of their module-finite extensions. The proposed research deals with commutative rings, which are abstract systems in which one has what might be thought of as artificial numbers. One can add, subtract, and multiply these artificial numbers: call them ring elements. The integers and real numbers are examples, but there are many other examples, including rings that contain only finitely many elements. In one example, one has only 0 and 1, and 1+1 = 0 (like the properties of even and odd integers: odd + odd =even). In studying systems of many equations in many unknowns, one can force the equations to hold in an abstract ring. The study of the properties of this ring gives information about the solutions. One can also study instead a sort of graph of the solutions, that exists in a high dimensional space. It is of great benefit to go back and forth between these algebraic and geometric points of view. A third method, which will be used frequently in the proposed research, is to study solutions that are somewhat like integers (but more general), but to do so while ignoring multiples of a prime number: elements are considered equivalent if they differ by a multiple of this prime. Doing this for many different prime numbers, possibly all prime numbers, can provide a huge amount of information about the solutions of the equations. The proposed research deals with a systematic method, called tight closure, for using this idea, as well as its extensions into new contexts.

Hochster建议继续研究(主要是局部的)Notherian环理论中的几个长期存在的问题，并探索这些问题与由Hochster和Huneke提出的爆炸性发展的紧闭包理论的关系。一个主要的目的是探索不含域的环中闭包的几个概念，例如整数上的有限生成代数，希望将紧闭包理论推广到这类环，从而解决许多公开的问题。文中还给出了混合特征大Cohen-Macaulay模存在性的证明方法。研究了与某些局部上同调模有关的拟长度和内容的新理论。这一理论提出了一些困难的核心问题，并且是为解决长期存在的重要问题而提出的几种方法之一，即正则环是否为其模-有限扩张的直和。这项拟议的研究涉及交换环，这是一种抽象系统，其中一个人拥有可能被认为是人造数字的东西。人们可以对这些人造数字进行加、减、乘：我们称它们为环元素。整数和实数都是例子，但还有很多其他例子，包括只包含有限多个元素的环。在一个例子中，只有0和1，并且1+1=0(类似于偶数和奇数整数的属性：奇数+奇数=偶数)。在研究由许多未知数的许多方程组成的系统时，人们可以强迫这些方程保持在一个抽象的环中。对这个环的性质的研究给出了关于解的信息。相反，人们也可以研究一种存在于高维空间中的解图。在这些代数和几何观点之间来回转换是很有好处的。第三种方法将在拟议的研究中经常使用，即研究有点像整数(但更一般)的解，但这样做时忽略素数的倍数：如果元素相差这个素数的倍数，则被认为是等价的。对于许多不同的素数，可能都是素数，这样做可以提供关于方程解的大量信息。这项拟议的研究涉及一种系统的方法，称为紧密闭合，用于使用这一想法，以及它在新背景下的扩展。