Homological Questions in Commutative Algebra

ROBERTSDMS 98- 02235本项目的目的是研究交叉多重性的Serre正性猜想。 本文讨论正则局部环上两个支撑只在极大理想处相交的双生成模的交重数。它指出重数是正的当且仅当支撑的维数之和等于环的维数。 最近在混合特征线奇异性分解理论中的新发展为这个问题带来了新的解决途径，这些思想已被用来证明相交多重性总是非负的。 如何定义交重数是交换代数中的一个重要课题，而如何定义交重数是交换代数中的一个重要课题。其基本思想是测量一个空间的两个子空间相切的程度，定义应满足各种条件，以确保它与简单情况下的通常定义一致，并可用于计算。 一点超过30年前塞尔提出了一个代数定义，满足了许多所需的性质，但其中涉及负面条款。 有几个悬而未决的问题，当交叉多重性为零（消失），他们是否总是大于或等于零（非负性），以及他们严格大于零（正性）的精确条件。 从那时起，消失和最近的非负性假设已经被证明。本文利用代数几何中的新方法研究了正性猜想。

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.