Multiplicities and Local Cohomology in Commutative Algebra

本文研究交换代数中同调猜想中的两个问题。第一个是关于有限投射维模定义的交重数。这部分研究了这种模定义的抽象Chow群上的作用，主要研究者和V.Srinivas的一个定理在零维的情况下回答了这个问题，本研究将在更高维的情况下研究更复杂的情况。第二部分是关于局部上同调群的元素在有限扩张中是否存在小零化子的问题。这个问题是由Ray Heitmann最近的一个结果启发的，他证明了素数p的小幂在三维混合特征环中达到了这一点，并利用这一事实证明了在那种情况下的直和猜想。在这个方案中，我们考虑了混合特征的替代方法以及相似结果是否具有特征零。如何度量空间的子空间的相切或交的倍数是数学中经常出现的问题之一。例如，这些数字被用来计算给定方程有多少解或有多少特定类型的几何对象。虽然对于平面相交的曲线，计算相切顺序似乎很简单，但在更高的维度上，它可能会相当复杂。这项提议解决了试图用代数来衡量这些问题时出现的问题。更具体地说，这些方法使用了同调代数，并且它们已经引起了一系列被称为“同调猜想”的问题。这项建议中的特定问题涉及确定这些不变量是否为零，并研究它们对环和模的结构的影响。

This proposal studies two questions in the HomologicalConjectures in Commutative Algebra. The first is on intersection multiplicities defined by modules of finiteprojective dimension. This part of the proposal studiesthe action on thAbstracte Chow group defined by such a module.A theorem of the Principal Investigator and V. Srinivas answers this question in the case of dimension zero,and this research will investigate the more complicatedsituation in higher dimension. The second part concernsthe question of whether there are small annihilators of elementsof local cohomology groups in finite extensions. Thisquestion was inspired by a recent result of Ray Heitmann,who proved that small powers of the prime p achieved thisin rings of mixed characteristic in dimension three andused this fact to prove the Direct Summand Conjecture in thatcase. In this proposal we consider alternative approachesin mixed characteristic and whether similar results holdin characteristic zero.One of the problems that often arises in mathematicsis how to measure the order of tangency, or multiplicityof intersection, of subspaces of a space. These numbers areused, for example, to count how many solutions a given equation has or how many geometric objects of a certain type there are. While computing the order of tangency may seem simple for curves intersecting in the plane, in higher dimension it canbe quite complicated. This proposal deals with questions that arisefrom attempting to measure these algebraically. More specifically,the methods use Homological Algebra, and they havegiven rise to a set of questions known as the "Homological Conjectures."The particular questions in this proposal concern determiningwhether these invariants are zero and studying their consequences forthe structure of rings and modules.