Studies in Commutative Algebra and Algebraic Geometry

建议研究与紧闭相关的几个问题，包括它是否与局部化交换的长期开放问题，紧闭与Hilbert-Kunz函数的关系，以新的方式在相等特征零中表征紧闭的问题，这将更好地理解这一现象，以及将基本思想扩展到不一定包含场的环的中心问题。这将解决许多悬而未决的问题，包括长期存在的猜想，即规则环是它们的模有限扩展环的直接和。交换代数和代数几何可以被认为是研究许多方程在许多未知数中的解，通常情况下，解不是唯一的。然后可以从几何角度来看待解的集合，但是我们可以将所有关于方程的相关信息编码为抽象的代数对象，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。紧闭理论是交换代数中近十年来取得爆炸性发展的一种突破性方法。部分基本思想是考虑方程系统，经过适当的修改，或环，模许多不同的素数整数。这是一项强大的技术，解决了许多问题，同时开辟了广阔的新研究领域。

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.