Studies on Integrality of Ideals

这个项目研究交换代数中的问题。虽然这些问题涉及不同的主题，但有一条潜在的共同线索将它们统一起来：积分闭包理论。PI和她的合作者首先计划开发产生理想的积分闭包的方法。在与伯恩德·乌尔里希的联合工作中，提出者确立了链接是在一个完整的交叉口上捕获积分元素的一种方法。现在，研究人员希望将同样的程序扩展到Gorenstein连接和剩余交叉口。其次，PI旨在阐明理想的核心与Lipman的伴随(或Ein和Lazarsfeld的乘子理想)之间的关系，找到核心的显式公式，并计算单项理想的核心(这是Eisenbud和Sturmfels提出的一个问题)。研究人员打算使用链接论、剩余交理论和Groebner基理论。第三，研究特殊纤维环。从几何的角度来看，这个对象非常重要，因为它编码了关于吹气的特殊纤维的代数信息。这里的重点是找到迫使纤维变得不混合的条件，甚至是科恩-麦考利。最后，PI计划研究零维正规理想的希尔伯特函数(如果一个理想的所有幂都是整数闭的，那么它就是正规的)。作者的目的之一是证明这类理想的所有Hilbert系数都是非负的，而另一个目的是用理想本身或其幂的相关分次环的Cohen-Macaulay性来刻画Hilbert系数的消失。在基本层面上，交换代数是关于求解多项式方程组的技术。交换代数在过去的五十年里有了革命性的发展，因为它为理解纯数学和应用数学中的许多问题提供了工具。在许多应用问题中，多项式方程和交换代数起着至关重要的作用。过去应用交换代数结果的领域包括运筹学、计算机科学、机器人学、控制论、编码论和密码学。

This project deals with questions in commutative algebra. While thequestions address a variety of topics, there is an underlyingcommon thread that unifies them all: the theory of integral closure.The PI, together with her collaborators, first plans to developmethods that produce the integral closure of an ideal. In joint workwith Bernd Ulrich the proposer has established that linkage is amethod for capturing integral elements over a complete intersection. Nowthe investigator wants to extend this same procedure to Gorensteinlinkage and residual intersections. Second, the PI intends to clarify theconnection between the core of an ideal and the adjoint of Lipman (or themultiplier ideal of Ein and Lazarsfeld), find an explicit formula forthe core, and compute the core of monomial ideals (a question posedby Eisenbud and Sturmfels). The investigator intends to use linkagetheory, residual intersection theory and Groebner basis theory.Third, the investigator would like to study the special fiber ring. Thisobject is very important from a geometric point of view because itencodes algebraic information on the special fiber of the blowup. The focus here is on finding conditions that force the fiber to be unmixed oreven Cohen-Macaulay. Last, the PI plans to study the Hilbert function of zero-dimensional normal ideals (an ideal is normal if all its powers are integrally closed). One of the proposer's goals is to show that all the Hilbert coefficients of such ideals are non-negative, while another goal is to characterize the vanishing of the Hilbert coefficients in terms ofthe Cohen-Macaulayness of the associated graded ring of the idealitself or of one of its powers.The proposed research is concerned with problems in commutativealgebra. On a basic level commutative algebra is about techniquesfor solving systems of polynomial equations. Commutative algebrahad a revolutionary growth in the past fifty years as it providedthe tools for understanding many problems in pure and appliedmathematics. In many applied problems, polynomial equations andhence commutative algebra play a crucial role. Applied areas wherecommutative algebra results have been used in the past includeoperations research, computer science, robotics, control theory, codingtheory and cryptography to mention a few.