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

本研究计画主要探讨交换代数、同调代数及相关领域。 在交换代数中，人们研究的是形式系统，其中操作方程的规则与高中代数相同，但在更一般的设置中完成。该领域与纯数学的许多其他领域有关，如数论，研究整数的性质，代数几何，研究多项式方程组解的几何性质。同调代数是代数学的一个分支，与代数拓扑学有关，是研究拓扑空间即“形”的学科。同调代数的一个中心研究对象是模的复形，它可以被认为是拓扑空间概念的抽象。该项目旨在解决各种开放的问题，包括某些类型复形的欧拉特征的可能值。这里，复形的欧拉特征是多面体的整数不变量的推广。该项目包括四个主要课题：（1）有限投射维数模的长度和“微小复形”的Dutta重数，（2）模的Ulrich模和lim Ulrich序列，（3）Betti表的锥和上同调表，（4）矩阵分解的nc-Hodge猜想。（1）的中心目标是证明一个猜想，即局部环上的有限投射维数模的长度必须至少与模的重数一样大。这个猜想承认一个推广涉及欧拉和杜塔多重性的“微小的复合物”。第（2）部分涉及处理这些问题的主要工具：Ulrich模，它是极大的Cohen-Macaulay模，其重数等于它们的最小生成数，以及lim Ulrich模序列-渐近逼近前者的模序列。中心目标是为比以前已知的更大的一类环构造这样的东西。（3）局部环上模的Betti表的锥和投射簇上凝聚层的上同调表的锥。乌尔里希层和林乌尔里希序列的层理论类似的模块版本的这些概念在这里发挥了重要作用。第（4）部分的主要目的是证明经典Hodge猜想在具有孤立奇点的超曲面的矩阵分解范畴中的非交换类似。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。

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.