Studies in Commutative Algebra and Algebraic Geometry

Hochster，9970702建议研究与紧闭包有关的几个问题，包括关于它是否与局部化交换的长时间公开问题，紧闭包与Hilbert-kunz函数的关系，以新的方式刻画紧闭包为等特征零的问题，以更好地理解这一现象，以及将基本思想推广到不一定包含域的环的中心问题，这将解决许多公开的问题，包括长期存在的猜想，正则环是它们的模有限扩张环的直和。交换代数和代数几何可以被认为是研究在许多未知数中的许多方程的解，当解通常不是唯一的时候。然后可以从几何角度查看这组解，但可以使用编码在称为交换环的抽象代数对象中编码有关方程的所有相关信息。这两种观点以美丽而富有成效的方式相互作用。紧闭包理论是交换代数的一种突破性方法，在这十年中得到了爆炸性的发展。基本思想的一部分是考虑方程组，经过适当的修改，或环，模许多不同的素数。这是一项强大的技术，已经解决了许多问题，同时为研究开辟了广阔的新领域。

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.