Studies on Integrality of Ideals

这个项目处理交换代数中的问题。虽然这些问题涉及各种各样的主题，但有一个潜在的共同线索将它们统一起来：整体闭包理论。PI和她的合作者们首先计划开发一种方法，使理想的整体闭合。在与Bernd Ulrich的联合工作中，提出了连杆是一种捕获完整交集上的积分元素的方法。现在研究者想把同样的程序扩展到Gorensteinlinkage和剩余交集。其次，PI旨在澄清理想的核心与Lipman的伴随（或Ein和Lazarsfeld的乘数理想）之间的联系，为核心找到一个明确的公式，并计算单项式理想的核心（这是Eisenbud和Sturmfels提出的问题）。本研究拟采用联系理论、残差交集理论和格罗布纳基础理论。第三，研究者想研究特殊的纤维环。从几何角度来看，这个物体非常重要，因为它在放大的特殊纤维上编码了代数信息。这里的重点是找到迫使纤维未混合的条件，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（如果理想的所有幂都是整闭的，那么理想就是正规的）。提议者的目标之一是证明这些理想的所有希尔伯特系数都是非负的，而另一个目标是描述希尔伯特系数在理想本身或它的一个幂的相关梯度环的科恩-麦考利性方面的消失。本文研究的是交换代数中的一些问题。交换代数在基本层面上是关于多项式方程组的求解技术。交换代数在过去的50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。交换代数结果在过去的应用领域包括运筹学、计算机科学、机器人、控制理论、编码理论和密码学等。

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.