Intersection Theory and Commutative Algebra

大约40年前，Serre提出了交重数的一个代数定义，它满足了这个定义所需的许多性质，但留下了几个悬而未决的问题。这些问题尤其涉及交集的重数是否总是大于或等于零(非负性)，以及它们消失(消失)或大于零(正性)的精确条件。这些问题被称为关于交集重数的Serre猜想。这项提议包括两个部分。第一部分是主要研究者对正性猜想的研究的继续，它利用了奇点分解的最新进展，这些奇点导致了非负性猜想的解。这项研究还将研究这些思想与交换代数中有关重数的其他问题之间的关系。第二部分是关于非正则环上的有限投射维模。它将利用主要研究者和V.Srinivas的最新结果来研究有限长度模和有限射影维模的交性质，并将这些结果推广到研究有限射影维模的零猜想。在这个建议中，主要研究者研究了代数和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，人们必须定义重数，它给出了一个解应该被计算的次数。这个概念推广了多项式的根的重数，这对多项式的大多数应用是至关重要的。对这些思想的研究导致了代数中的几个基本问题。其中一个问题是确定这些多重性何时为正的问题。另一组问题涉及有限射影维度的模，这是可以通过有限分解来描述的模；例如，在计算机代数中使用这种描述的有限性来计算不变量。主要研究人员将研究这些模的性质以及它们与重数问题以及与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.