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

• 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

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.

可以证明：（1）如果理论失败，可以证明许多定理的定理，可以在主要特征p的环中找到反例（即，包含整数ring ring on of the Integers a Modulo a prime d p），并且（2）在特征范围内（2）的特征（2）在特征上（2）的特征（2）在特征上（2）的特征（2） Frobenius函数F（R）将R提高到PTH功率。该功能子是环的内态性，即，它具有f（r+s）= f（r+s）= f（r）+f（s），令人惊讶的是，它可以很好地解决许多特征性的问题。共同体模块和差异操作员作用于这些环。事实证明，这些物体及其相关技术在解决代数和几何问题方面非常成功，这些概念之间的相互作用尤其肥沃。我建议在研究助理的帮助下进一步研究这些相互作用，并将所得技术应用于我领域的几个出色问题的解决方案。

项目成果

Annihilators of Artinian modules compatible with a Frobenius map

与 Frobenius 地图兼容的 Artinian 模块的歼灭者

• DOI: 10.48550/arxiv.1301.1468
• 发表时间: 2013
• 作者: Katzman M

The Support of Local Cohomology Modules

局部上同调模的支持

• DOI: 10.1093/imrn/rnx094
• 发表时间: 2017
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Katzman M

Global parameter test ideals

全局参数测试理想

• DOI: 10.1016/j.jalgebra.2017.05.010
• 发表时间: 2017
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Katzman M

Test, multiplier and invariant ideals

检验、乘数和不变理想

• DOI: 10.48550/arxiv.1407.4324
• 发表时间: 2014
• 期刊: arXiv e-prints
• 作者: Henriques

Rings of Frobenius operators

弗罗贝尼乌斯算子环

• DOI: 10.1017/s0305004114000176
• 发表时间: 2014
• 期刊: Mathematical Proceedings of the Cambridge Philosophical Society
• 影响因子: 0.8
• 作者: KATZMAN M