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

本研究课题涉及交换代数、同调代数及相关领域。在交换代数中，人们研究形式系统，其中处理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，例如数论，整数性质的研究，代数几何，多项式方程系统解的几何性质的研究。同调代数是代数的一个分支，与代数拓扑领域有关，是对拓扑空间，即“形状”的研究。同调代数研究的一个中心对象是模的复合体，它可以被认为是拓扑空间概念的抽象。这个项目旨在解决各种开放的猜想，包括一个关于某些类型的配合物的欧拉特性的可能值。这里，复数的欧拉特征是多面体整数不变量的推广。该基金还将支持从事相关课题研究的研究生。该项目涉及四个主要主题：(1)有限射影维的模的长度和“微小复合体”的Dutta多重性，(2)模的Ulrich模和lim Ulrich模序列，(3)Betti表和上同表的锥，(4)矩阵分解的nc- hodge猜想。(1)的中心目标是证明局部环上有限射影维模的长度至少与模的多重数相等的猜想。这个猜想承认一个涉及“微小复合体”的欧拉和杜塔多重性的推广。第(2)部分涉及用于解决这些猜想的主要工具：Ulrich模块，它是最大Cohen-Macaulay模块，其多重性等于其最小生成数，以及lim Ulrich模块序列-渐近逼近前者的模块序列。中心目标是为比以前已知的更大的环类构建这样的东西。第(3)部分讨论了局部环上模的Betti表的锥和投影簇上相干束的上同表的锥。乌尔里希轴和利姆乌尔里希轴序列——这些概念的模块版本的轴理论类似物——在这里起着至关重要的作用。第(4)部分的中心目的是证明具有孤立奇点的超曲面的矩阵分解范畴的经典Hodge猜想的非交换类似。每个部分都将与其他研究人员合作进行。该奖项反映了美国国家科学基金会的法定使命，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。

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.