Intersection Theory and Commutative Algebra

大约四十年前塞尔提出了一个代数定义的交叉多重性，满足了许多性质所需的这样一个定义，但留下了几个悬而未决的问题。 这些问题涉及，除其他事项外，是否相交的多重性总是大于或等于零（非负性）和精确的条件，他们消失（消失）或大于零（积极）。 这些问题已成为众所周知的塞尔的十字路口的多重性。 这项建议有两个部分。 第一部分是继续研究的主要调查员的积极性猜想，使用最近的进展，决议的奇异性，导致了解决方案的非负性猜想。 本文还将研究这些思想与交换代数中重数问题的关系。 第二部分研究非正则环上的有限投射维数模。 本文将利用主要研究者和V. Srinivas最近的一个结果来研究有限长和有限投射维数模的交性质，并将这些结果推广到研究两个有限投射维数模的消失猜想，主要研究代数与几何关系中的一些基本问题。 几何集合通常被定义为多项式方程的解的集合。在研究这些解的行为时，必须定义多重性，它给出了一个解应该被计数的次数。 这个概念推广了多项式根的重数，这对多项式的大多数应用至关重要。 对这些思想的研究导致了代数中的几个基本问题。 这些问题之一是确定这些多重性何时为正的问题。另一组问题涉及有限投射维数的模，这些模可以用有限分解来描述;这种描述的有限性用于计算不变量的计算机代数。 主要研究者将研究这些模块的属性及其与多重性问题以及代数其他分支的关系。

Around forty years ago Serre proposed an algebraic definition of intersection multiplicities which satisfied many of the properties required of such a definition but left several unanswered questions. These problems concern, among other things, whether the intersection multiplicities are always greater than or equal to zero (nonnegativity) and precise conditions for them to vanish (vanishing) or to be greater than zero (positivity). These problems have become known as Serre's conjectures for intersection multiplicities. This proposal has two parts. The first part is a continuation of research of the principal investigator on the positivity conjecture that uses recent advances on the resolution of singularities which have led to a solution of the nonnegativity conjecture. This investigation will also study relations between these ideas and other questions on multiplicities in commutative algebra. The second part concerns modules of finite projective dimension over nonregular rings. It will use a recent result of the principal investigator and V. Srinivas to study intersection properties of modules of finite length and finite projective dimension and extend these results to study the vanishing conjecture for two modules of finite projective dimension.In this proposal the principal investigator studies some fundamental questions in the relations between Algebra and Geometry. Geometric sets are often defined as sets of solutions to polynomial equations. In studying the behaviour of these solutions, one has to define multiplicities, which give the number of times a solution should be counted. This concept generalizes the multiplicity of a root of a polynomial, which is crucial to most applications of polynomials. The investigation of these ideas has led to several fundamental questions in Algebra. One of these questions is the problem of determining when these multiplicities are positive. Another group of questions concerns modules of finite projective dimension, which are modules which can be described by a finite resolution; the finiteness of this description is used, for example, in computer algebra for computing invariants. The principal investigator will study properties of these modules and their relation to questions on multiplicities as well as to other branches of Algebra.