Homological Questions in Commutative Algebra

在这个建议中，研究的主要部分是关于交换代数中的一组猜想，通常被称为“同调猜想”。 首席研究员在前几年深入研究了这些猜想，最近雷·海特曼在三维空间中证明了直接和数猜想，重新引起了人们的兴趣。 这部分的建议将调查性质的局部上同调模的诺特环，特别是是否有可能找到元素的小价值，零化元素的局部上同调模。 该提案主要涉及发展的新方法的环的混合特性使用的想法算术几何。 此外，还将对与相交多重性相关的几个问题进行研究。代数在几何中的应用可以追溯到几个世纪前坐标系的引入。 近年来，交换代数领域中的许多基本问题都来源于由代数方程定义的几何空间的切序问题。 该提案调查了这一领域的几个问题，主要集中在所谓的算术情况下，这是类似的情况下，方程定义的整数，而不是一个领域，如复数。该提案的主要重点是研究开发新的工具，适用于这种情况。

The main part of the research in this proposal is concerned with a set of conjectures in Commutative Algebra, often referred to as the"`Homological Conjectures". The Principal Investigator has studied these conjectures intensively in previous years, and interest in them has recently been revived by the proof of the Direct Summand Conjecture in dimension three by Ray Heitmann. This part of the proposal will investigate properties of local cohomology modules of Noetherian rings, and in particular whether it is possible to find elements of small valuation that annihilate elements of the local cohomology modules. The proposal deals primarily with the development of new methods for rings of mixed characteristic using ideas from Arithmetic Geometry. In addition, research will be carried out work on several related questions on intersection multiplicities.Applications of Algebra to Geometry go back to the introduction of coordinate systems several centuries ago. In recent years many fundamental problems in the field of Commutative Algebra have come from the question of defining the order of tangency for geometricspaces defined by algebraic equations. This proposal investigatesseveral questions in this area, concentrating primarily on the so-called arithmetic case, which is analogous to the situation where equations are defined over the integers instead of a field such as the complex numbers.The main focus of the research in this proposal is to develop new tools to apply in this situation.