Some Questions in Commutative Algebra

这个方案中的问题集中在交换代数中四个不同但紧密相连的研究领域：局部上同调、代数D-模、F-奇点和同调猜想。局部上同调是共同的主线。例如，关于局部上同调模上的Frobenius作用的研究在F-奇异性的研究中已经取得了丰硕的成果，而由Hochster和Huneke引入的局部上同调的内容为同调猜想提供了许多新的方法，而在局部上同调的研究中，代数D-模理论被证明是不可或缺的。PI想要研究局部上同调的各个方面及其在F-奇点和同调猜想中的应用，并研究一个没有特征的完整概念。本课题主要研究交换代数中的问题及其在代数几何中的应用。代数集是多项式方程组的解的集合。交换代数研究代数集合上的函数(在交换代数中，这些代数集合可以在任何交换环上)；而代数几何研究代数集合的几何或拓扑方面(在代数几何中，这些代数集合经常在域上)。这两个领域密切相关，互惠互利。这个建议中的问题可以被认为是关于多项式方程组解的存在性以及解集的拓扑或几何性质的问题。

The questions in this proposal are centered around 4 different but deeply connected research areas in commutative algebra: local cohomology, algebraic D-modules, F-singularities, and homological conjectures. Local cohomology is the common thread. For example, the study of Frobenius actions on local cohomology modules has been fruitful in the study of F-singularities, and the content of local cohomology, a notion introduced by Hochster and Huneke, has provided a number of new approaches to homological conjectures, and in the study of local cohomology, algebraic D-module theory has proven to be indispensable. The PI wants to study various aspects of local cohomology and their applications to both F-singularities and homological conjectures and to investigate a characteristic-free notion of holonomicity. This project is mainly concerned with questions in commutative algebra and their applications to algebraic geometry. Algebraic sets are sets of solutions of systems of polynomial equations. Commutative algebra studies functions over algebraic sets (in commutative algebra, these algebraic sets can be over any commutative rings); while algebraic geometry studies the geometric or topological aspects of algebraic sets (in algebraic geometry, these algebraic sets are very often over a field). These two areas are closely related and are mutually beneficial. Questions in this proposal can be considered as questions about the existence of solutions to systems of polynomial equations and the topological or geometric nature of the solution sets.