Studies in Commutative Algebra

本研究计划涉及交换代数理论中的问题研究。这是一个涉及代数变量研究的领域：几何对象作为多项式方程组的解给出。例如，两个变量的单多项式方程的解集可以在几何上实现为平面上的曲线（例如抛物线y=x^2）。一个重要的方面是理解这些解决方案集的局部图像，这在许多科学和工程中都有应用。例如，抛物线是光滑的（意思是局部看起来像一条线），而由y^2=x^3定义的曲线是光滑的，除了在原点，它是一个尖点。代数类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变量的局部性质上，比如它们的局部交点数和多重性（这是衡量奇点有多糟糕的一种方法）。正在研究的几个问题是长期存在的，具有根本重要性。这个项目从新的角度研究了交换代数中几个长期存在的开放问题。一种是用一个新的概念——模的lim Cohen-Macaulay序列来攻击Serre关于交多重性的正性猜想和Hochster的直接和猜想，它的存在将建立两个猜想。另一种是利用正特征方法的进步来攻击莱赫关于希尔伯特-塞缪尔多重性的猜想。研究者最近解决了这个猜想在三维等特征（猜想以前只知道在小于或等于2维）。这里的目标是改进方法并寻求更高维度的解决方案。其他研究项目包括研究正特征中的奇异性，特别是研究它们在变形、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。其中一个关键的技术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.