Multiplicities and Local Cohomology in Commutative Algebra

本文研究了交换代数中的同调猜想中的两个问题。 第一个是关于有限投射维数模的交重数。 这部分研究了由这样一个模定义的抽象Chow群上的作用，主要研究者和V.Srinivas的一个定理在零维情形下回答了这个问题，本研究将研究更高维情形下的更复杂的情形. 第二部分讨论局部上同调群的有限扩张中是否存在元素的小零化子。 这个问题的灵感来自于Ray Heitmann最近的一个结果，他证明了素数p的小幂在三维混合特征的环中实现了这一点，并利用这一事实证明了这种情况下的直接和数猜想。在这一建议中，我们考虑了混合特征中的替代方法以及类似结果是否保持特征零。在代数学中经常出现的问题之一是如何度量空间的子空间的相切阶或交的重数。 例如，这些数字被用来计算一个给定的方程有多少个解，或者有多少个特定类型的几何对象。对于平面相交的曲线，计算切线的阶数似乎很简单，但在高维空间中，它可能相当复杂。 这个建议涉及的问题，从试图衡量这些代数。 更具体地说，这些方法使用了同调代数，它们产生了一系列被称为“同调猜想”的问题。“这个建议中的特殊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.