CAREER: Local Cohomology, de Rham Cohomology and D-modules

职业：局部上同调、de Rham 上同调和 D 模

• 批准号: 1752081
• 负责人: Wenliang Zhang
• 金额: $ 40万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1752081&HistoricalAwards=false
• 关键词: CAREER; Local; Cohomology; de; Rham

The proposed project is in the area of commutative algebra, which is a study of systems of algebraic (polynomial) equations with solutions in commutative rings. A commutative ring may be viewed as an abstract analogue of the integers, in which one can add, subtract and multiply elements. In the past few decades, commutative algebra has not only found remarkable interactions with other research areas in mathematics, but proven a valuable tool in other disciplines, such as engineering and computer science. Advising graduate students, mentoring postdocs, and organizing workshops and seminars are also part of this project.More specifically, the PI will study local cohomology, de Rham cohomology and D-modules. Projects include developing a theory of de Rham cohomology of graded D-modules in characteristic p, generalizing classic connectedness theorems due to Faltings and Hartshorne by studying homology of a simplicial complex associated a local ring, extending work on Lyubeznik numbers of projective schemes in characteristic p 0 to characteristic zero, extending his previous work on finiteness properties of local cohomology modules to ramified regular rings and his previous work on Zariski-closedness of support of local cohomology modules, and investigating singularities in characteristic p. Solutions to the problems addressed in this project will provide a theory of de Rham cohomology of graded D-modules in characteristic p, settle some long-standing open questions regarding finiteness properties of local cohomology modules, provide new insight into the structure of local cohomology modules, and lead to new connections between singularities in characteristic p and those in characteristic zero.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

拟议的项目是在交换代数领域，这是一个研究系统的代数（多项式）方程的解决方案，在交换环。一个交换环可以被看作是一个抽象的模拟的整数，其中可以加，减，乘元素。在过去的几十年里，交换代数不仅与数学中的其他研究领域有着显著的相互作用，而且在其他学科（如工程和计算机科学）中也被证明是一个有价值的工具。为研究生提供建议，指导博士后，并组织研讨会和研讨会也是该项目的一部分。更具体地说，PI将研究局部上同调，de Rham上同调和D-模。项目包括发展特征p中分次D-模的de Rham上同调理论，通过研究与局部环相关的单纯复形的同调来推广Faltings和哈茨霍恩的经典连通性定理，将特征p 0中投射方案的Lyubeznik数的工作扩展到特征零，将他以前关于局部上同调模的有限性性质的工作扩展到分歧正则环和他以前关于局部上同调模的支持的Zurkiki闭性的工作，本项目所解决的问题将提供特征p上分次D-模的de Rham上同调理论，解决局部上同调模有限性的一些长期悬而未决的问题，为局部上同调模的结构提供新的见解，该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

On completion of graded $D$-modules

完成分级的 $D$ 模块后

• 发表时间: 2018
• 期刊: Mathematical Research Letters
• 影响因子: 1
• 作者: N. Switala;Wenliang Zhang
• 通讯作者: Wenliang Zhang

On asymptotic socle degrees of local cohomology modules

局部上同调模的渐近底度

• 发表时间: 2021
• 期刊: Journal of pure and applied algebra
• 影响因子: 0.8
• 作者: Zhang, Wenliang

On Lyubeznik type invariants

关于 Lyubeznik 类型不变量

• 发表时间: 2022
• 期刊: Topology and its applications
• 影响因子: 0.6
• 作者: Reichelt, Thomas;Walther, Uli;Zhang, Wenliang
• 通讯作者: Zhang, Wenliang

Duality and de Rham cohomology for graded D-modules

• DOI: 10.1016/j.aim.2018.07.005
• 发表时间: 2017-05
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: N. Switala;Wenliang Zhang

F^e-modules with applications to D-modules

F^e 模块以及 D 模块的应用

• 发表时间: 2023
• 期刊: Journal of algebra
• 影响因子: 0.9
• 作者: Zhang, Wenliang