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Common threads in the theories of Local Cohomology, D-modules and Tight Closure and their interactions

局部上同调、D 模和紧闭理论的共同点及其相互作用

基本信息

批准号：
EP/J005436/1
负责人：
Mordechai Katzman
金额：
$ 28.86万
依托单位：
University of Sheffield
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ005436%2F1
关键词：
Common threads theories Local Cohomology

项目摘要

Many theorems in Commutative Algebra can be proved by showing that:(1) if the theorem fails, one can find a counter-example in a ring of prime characteristic p (i.e., a ring which contains the ring of integers modulo a prime number p), and(2) no such counter-example exists in characteristic p.Step (2) above is often much easier to prove than in characteristic zero because of the existence of the Frobenius function f(r) which raises r to the pth power. This functon is an endomorphism of the rings, i.e., it has the property that f(r+s)=f(r)+f(s), and surprisingly, gives a good handle on many problems in characteristic p.During the course of development of the study of commutative rings of prime characteristic, various notions and techniques were introduced, e.g., a certain tight-closure operation of ideals, certain structures on ``large'' objects called local cohomology modules, and differential operators acting on these rings. The objects and their associated techniques have proved to be very successful in tackling algebraic and geometric problems, and the interactions between these concepts turned out to be especially fertile.I propose to study these interactions further with the aid of a research assistant, and to apply the resulting techniques to the solution of several outstanding problems in my field.

交换代数中的许多定理可以通过证明：（1）如果定理失败，可以在素特征p的环中找到反例（即，一个环，它包含整数模一个素数p的环），以及（2）在特征p中不存在这样的反例。上面的步骤（2）通常比在特征零中更容易证明，因为存在将r提升到p次幂的Frobenius函数f（r）。这个函数是环的自同态，即，它具有f（r+s）=f（r）+f（s）的性质，令人惊讶的是，它很好地处理了特征p中的许多问题。在素特征交换环研究的发展过程中，引入了各种概念和技术，例如，理想的某种紧闭运算，称为局部上同调模的“大”对象上的某些结构，以及作用于这些环的微分算子。对象和他们的相关技术已被证明是非常成功的解决代数和几何问题，这些概念之间的相互作用原来是特别fertilized.I建议进一步研究这些相互作用的援助下的研究助理，并应用所产生的技术来解决几个突出的问题在我的领域。