The Non-Commutative Hodge Conjecture and Multiplicities of Modules and Complexes

非交换霍奇猜想以及模和复形的重数

• 批准号: 2200732
• 负责人: Mark Walker
• 金额: $ 28.26万
• 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: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200732&HistoricalAwards=false
• 关键词: Non; Commutative; Hodge; Conjecture; Multiplicities

This research project concerns topics in commutative and homological algebra and related fields. In commutative algebra, one studies formal systems in which the rules for manipulating equations are the same as in high school algebra but done in more general settings. The field is related to many other areas of pure mathematics, such as number theory, the study of properties of the integers, and algebraic geometry, the study of geometric properties of solutions to systems of polynomial equations. Homological algebra is a branch of algebra related to the field of algebraic topology and is the study of topological spaces, that is, "shapes." A central object of study in homological algebra is that of a complex of modules, which can be thought of an abstraction of the notion of a topological space. This project aims to settle various open conjectures, including one on the possible values of Euler characteristics of certain types of complexes. Here, the Euler characteristic of a complex is a generalization of the integer invariant for polyhedra. The grant will also support graduate students working on affiliated topics.The project involves four main topics: (1) lengths of modules of finite projective dimension and Dutta multiplicities of "tiny complexes," (2) Ulrich modules and lim Ulrich sequences of modules, (3) cones of Betti tables and cohomology tables, (4) the nc-Hodge-conjecture for matrix factorizations. The central goal of (1) is to prove the conjecture that a module of finite projective dimension over a local ring must have length at least as large as the multiplicity of the module. This conjecture admits a generalization involving Euler and Dutta multiplicities of "tiny complexes". Part (2) concerns a primary tool used in tackling these conjectures: Ulrich modules, which are maximal Cohen-Macaulay modules whose multiplicities equal their minimal numbers of generates, and lim Ulrich sequences of modules—sequences of modules that asymptotically approximate the former. The central goal is to construct such things for a larger class of rings than previously known. Part (3) concerns the cones of Betti tables of modules over local rings and cones of cohomology tables of coherent sheaves on projective varieties. Ulrich sheaves and lim Ulrich sequences of sheaves—sheaf theoretic analogues of the module versions of these notions—play an essential role here. The central aim of Part (4) is to prove the non-commutative analogue of the classical Hodge conjecture for the category of matrix factorizations of a hypersurface with an isolated singularity. Each part will be pursued in collaboration with other researchers.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.

该研究项目涉及交换和同源代数及相关领域的主题。在交换代数中，一种研究正式系统，其中操纵方程的规则与高中代数相同，但在更一般的环境中进行。该领域与纯数学的许多其他领域有关，例如数字理论，整数的性质研究以及代数几何，对多项式方程系统解决方案的几何特性的研究。同源代数是与代数拓扑领域相关的代数分支，并且是对拓扑空间的研究，即“形状”。同源代数研究的一个核心对象是一个模块的复合物，可以认为拓扑空间的概念的抽象。该项目旨在解决各种开放式猜想，其中包括某些类型的复合物的Euler特征的可能值。在这里，复合物的Euler特征是多面体整数不变的概括。 The grant will also support graduate students working on affiliate topics.The project involves four main topics: (1) lengths of modules of finite projective dimension and Dutta multiples of "tiny complexes," (2) Ulrich modules and lim Ulrich sequences of modules, (3) cones of Betti tables and cohomology tables, (4) the nc-Hodge-conjecture for matrix factors. （1）的中心目标是证明一个概念，即在本地环上的有限射击尺寸模块必须具有至少与模块的多重性一样大的概念。这种猜想承认了涉及“小复合物”的欧拉和杜塔倍数的概括。第（2）部分涉及用于解决这些概念的主要工具：Ulrich模块，这些模块是最大的Cohen-Macaulay模块，其倍数等于其最小生成数量和Lim Ulrich模块序列 - 模块的序列，这些模块的序列是不对称的。中心目标是为比以前已知的更大的戒指构建这些东西。第（3）部分涉及局部戒指上的模块表的锥锥，并在投影品种上相干滑轮的共同体表的锥。乌尔里希滑轮和林乌尔里希序列是这些音符的模块版本的sheaf理论类似物 - 在这里扮演着重要的作用。第（4）部分的核心目的是证明经典霍奇概念的非共同类似物对于具有孤立的奇异性的高度表面的基质因子类别。该奖项将与其他研究人员合作进行。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准来评估，被认为是珍贵的支持。

项目成果

Idempotent completions of equivariant matrix factorization categories

等变矩阵分解类别的幂等完成

• DOI: 10.1016/j.jalgebra.2023.07.023
• 发表时间: 2023
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Brown, Michael K.;Walker, Mark E.
• 通讯作者: Walker, Mark E.