Symbolic Powers and p-Derivations

符号幂和 p 导数

基本信息

批准号：
2140355
负责人：
Eloísa Grifo
金额：
$ 16.29万
依托单位：
University of Nebraska-Lincoln
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2140355&HistoricalAwards=false
关键词：
Symbolic Powers Derivations

项目摘要

This is a project in commutative algebra, with connections to algebraic geometry, combinatorics, and arithmetic geometry. The research centers on the theme of symbolic powers, an active topic of research with connections to virtually all aspects of commutative algebra. This project aims to resolve two questions, on comparing the geometric notion of symbolic powers with the algebraic notion of natural powers, and on the application of the differential algebraic notion of p-derivations to commutative algebra. The investigator will direct undergraduate research experiences and organize a graduate workshop in commutative algebra.The unifying theme of this project is the study of symbolic powers of ideals. The research focuses primarily on two questions: the containment problem and applications of p-derivations to commutative algebra. While ideals in a polynomial ring correspond to the polynomials that vanish on a certain variety in affine space, their symbolic powers consist of the polynomials that vanish to a certain order on the given variety. This natural geometric notion has an algebraic description coming from the theory of primary decomposition, an ideal-theoretic version of the fundamental theorem of arithmetic. This is an old, rich, and ubiquitous topic, yet there is an abundance of questions about symbolic powers that are both easy to phrase and very difficult to solve. The containment problem attempts to compare symbolic powers, a natural geometric notion, with ordinary powers, a natural algebraic notion. This relates to other interesting questions, such as determining the minimal degrees of polynomials vanishing on a given variety. Another part of the project is to discover new connections between p-derivations and commutative algebra, with an eye towards a new singularity theory in mixed characteristic that combines p-derivations with differential operators in the spirit of the theory of F-singularities and its connections to D-modules.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.

这是一个交换代数项目，与代数几何、组合学和算术几何有关。该研究以符号幂为主题，这是一个活跃的研究主题，几乎与交换代数的所有方面都有联系。该项目旨在解决两个问题，即比较符号幂的几何概念与自然幂的代数概念，以及p-导数的微分代数概念在交换代数中的应用。研究者将指导本科生的研究经验，并组织交换代数的研究生研讨会。该项目的统一主题是理想的象征力量的研究。该研究主要集中在两个问题：包含问题和 p 导数在交换代数中的应用。虽然多项式环中的理想对应于在仿射空间中的某个簇上消失的多项式，但它们的符号幂由在给定簇上按一定顺序消失的多项式组成。这种自然的几何概念具有来自初级分解理论的代数描述，初级分解理论是算术基本定理的理想理论版本。这是一个古老、丰富且普遍存在的话题，但关于象征力量却存在大量易于表述却很难解决的问题。遏制问题试图将符号幂（一种自然的几何概念）与普通幂（一种自然的代数概念）进行比较。这涉及其他有趣的问题，例如确定给定品种上多项式消失的最小次数。该项目的另一部分是发现 p 导数和交换代数之间的新联系，着眼于混合特征中的新奇点理论，该理论本着 F 奇点理论及其联系的精神将 p 导数与微分算子结合起来该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。