Homological Questions in Commutative Algebra

这个项目的目的是研究Serre的交叉多重性的正性猜想。这个猜想涉及正则局部环上两个有限生成模的交多重性，它们的支点仅在环的最大理想处相交。它指出，当且仅当支撑的维数之和等于环的维数时，多重性是正的。近年来，混合特征奇异性分解理论的新发展为解决这一问题提供了新的思路，这些思路被用来证明交集多重性总是非负的。本建议概述了使用这些方法的正性猜想的方法。多年来，交点多重性的定义一直是交换代数中的一个重要问题。其基本思想是测量一个空间的两个子空间彼此相切的程度，其定义应满足各种条件，以确保它在简单情况下与通常的定义一致，并可用于计算。三十多年前，Serre提出了一个代数定义，它满足了许多必需的性质，但涉及到负项。当交叉多重度为零（消失），它们是否总是大于或等于零（非负性），以及它们严格大于零（正性）的精确条件时，还有几个问题没有解决。从那时起，消失猜想和最近的非负性猜想都得到了证明。这个提议涉及到用代数几何的新方法研究正性猜想。

ROBERTSDMS 98-02235The aim of this project is to study Serre's Positivity Conjecture for intersection multiplicities. This conjectureconcerns the intersection multiplicity of two finitely generated modules over a regular local ring which have the property that their supports intersect only at the maximal ideal of the ring. It states that the multiplicity is positive if and only if the sum of the dimensions of the supports is equal to the dimension of the ring. Recently new developments inthe theory of resolution of singularities in mixed characteristic have led to a new approach to this problem, and these ideas have been used to show that intersection multiplicities are always nonnegative. This proposal outlines an approach to the positivity conjecture using these methods.The question of how to define intersection multiplicities in a satisfactory way has been an important topic in Commutative Algebra for many years. The basic idea is to measure of the extenttwo which two subspaces of a space are tangent to each other, and thedefinition should satisfy various conditions to ensure that itagrees with usual definitions in simple cases and that it canbe used in computations. A little over thirty years ago Serre proposed an algebraic definition which satisfied many of the required properties but which involved negative terms. Therewere several questions left open concerning when the intersectionmultiplicities are zero (vanishing), whether they are always greater than or equal to zero (nonnegativity), and the precise conditions for them to be strictly greater than zero (positivity). Since then the vanishing and, more recently, the nonnegativity conjectures have been proven. This proposalconcerns a study of the positivity conjecture using new methodsfrom Algebraic Geometry.