Studies in Commutative Algebra

本研究项目是关于交换代数理论中问题的研究。这是一个研究代数变体的领域：几何对象是作为多项式方程系统的解给出的。例如，两个变量的单个多项式方程的解集可以几何地实现为平面上的一条曲线(例如，抛物线y=x^2)。一个重要的方面是了解这些解集的局部情况，这在许多科学和工程中都有应用。例如，抛物线是光滑的(这意味着它在局部看起来像一条直线)，而由y^2=x^3定义的曲线是平滑的，除了在原点处它是一个尖点。代数簇的奇异点或非光滑点具有丰富而微妙的局部结构，详细描述它们的性质是许多研究的关键部分。将要探索的项目集中在代数簇的局部性质上，例如它们的局部交集数和重数(这是衡量奇点有多糟糕的一个指标)。正在研究的几个问题由来已久，具有根本性的重要性。这个项目从新的角度研究了交换代数中的几个长期悬而未决的问题。一种是用Lim Cohen-Macaulay模列的新概念来反驳Serre关于交重数正性的猜想和Hochster的直和猜想，它的存在将建立这两个猜想。另一种是利用正特征方法的进展来反驳Lech关于Hilbert-Samuel多重性的猜想。研究人员最近用等特征证明了这个猜想在维度3上的正确性(该猜想以前只在维度小于或等于2的维度上已知)。这里的目标是改进方法，寻求更高维度的解决方案。其他研究项目包括研究正特征奇点，特别是研究它们在变形下的行为，在传递到一般连杆下的行为，以及它们与F-模和D-模理论的联系。一个关键的技巧是局部上同调模上的Frobenius结构。

This research project concerns the study of questions in the theory of commutative algebra. This is a field that deals with the study of algebraic varieties: geometric objects given as the solutions of a system of polynomial equations. For example, the solution set of a single polynomial equation in two variables can be geometrically realized as a curve in the plane (e.g., a parabola y=x^2). One important aspect is to understand the local picture of these solution sets, which has applications in many sciences and engineering. For example, the parabola is smooth (meaning that locally it looks like a line) while the curve defined by y^2=x^3 is smooth except at the origin, where it is a cusp. The singular or non-smooth points of an algebraic variety have rich and subtle local structure, and detailing their properties is a crucial part of many investigations. The projects that will be explored are focused on the local properties of algebraic varieties, such as their local intersection numbers and the multiplicities (which is a measure of how bad the singular points are). Several of the questions under study are longstanding and of fundamental importance. This project investigates several longstanding open questions in commutative algebra from new perspectives. One is to attack Serre's conjecture on positivity of intersection multiplicities and Hochster's direct summand conjecture using a new notion called lim Cohen-Macaulay sequences of modules, whose existence will establish both conjectures. Another is to attack Lech's conjecture on Hilbert-Samuel multiplicities using advances in positive characteristic methods. The investigator recently settled this conjecture in dimension three in equal characteristic (the conjecture was previously known only in dimension less than or equal to two). The goal here is to improve the methods and to seek solutions in higher dimension. Other research projects include studying singularities in positive characteristic, especially investigating their behavior under deformation, under passing to a generic linkage, and their connections with F-module and D-module theory. A crucial technique to be employed is the Frobenius structures on local cohomology modules.