Studies in Commutative Algebra and Algebraic Geometry

交换代数和代数几何研究

• 批准号: 9970702
• 负责人: Melvin Hochster
• 金额: $ 50.11万
• 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: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970702&HistoricalAwards=false
• 关键词: Studies; Commutative; Algebra; Algebraic; Geometry

HOCHSTER, 9970702It is proposed to study several problems related to tight closure, including the long open question as to whether it commutes with localization, the relationship of tight closure with Hilbert-Kunz functions, the problem of characterizing tight closure in equal characteristic zero in new ways that will produce better insight into the phenomenon, and the central problem of extending the underlying idea to rings that do not necessarily contain a field, which would solve many open questions, including the long standing conjecture that regular rings are direct summands of their module-finite extension rings. Commutative algebra and algebraic geometry may be thought of as studying solutions of many equations in many unknowns when, typically, the solution is not unique. The set of solutions can then be viewed geometrically, but one can use instead encode all the pertinent information about the equations in abstract algebraic objects called commutative rings. The two points of view interact in beautiful and productive ways. A breakthrough method in commutative algebra that has had explosive development in this decade is the theory of tight closure. Part of the underlying idea is to consider the system of equations, after suitable modification, or the ring, modulo many different prime integers. This is a powerful technique that has solved many problems, while opening up vast new areas for research.

HOCHSTER，9970702提出研究与紧闭包有关的几个问题，包括它是否与局部化交换的长期开放问题，紧闭包与Hilbert-Kunz函数的关系，以新的方式以相等的特征零表征紧闭包的问题，这将产生对现象的更好的洞察，以及将基本思想扩展到不一定包含域的环的中心问题，这将解决许多悬而未决的问题，包括长期存在的猜想，即正则环是其模有限扩张环的直和项。交换代数和代数几何可以被认为是研究许多未知数中的许多方程的解，通常，解不是唯一的。解的集合可以被几何地观察，但是我们可以使用将所有关于方程的相关信息编码在抽象的代数对象中，称为交换环。这两种观点以美丽而富有成效的方式相互作用。交换代数的一个突破性方法是紧闭包理论，它在这十年中得到了爆炸性的发展。部分基本思想是考虑方程组，经过适当的修改，或环，模许多不同的素数。这是一种强大的技术，解决了许多问题，同时开辟了广阔的新研究领域。