Homological Questions in Commutative Algebra

本文研究的主要部分是交换代数中的一组猜想，通常被称为“同调猜想”。前几年，首席研究员对这些猜想进行了深入的研究，最近，由于Ray Heitmann在三维空间中证明了直接求和猜想，对这些猜想的兴趣又恢复了。这一部分将研究Noetherian环的局部上同调模的性质，特别是是否有可能找到小值的元素湮灭局部上同调模的元素。本文主要讨论了利用算术几何思想发展混合特征环的新方法。此外，还将对交叉多重性的几个相关问题进行研究。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，交换代数领域的许多基本问题都来自于代数方程所定义的几何空间的切线阶的定义问题。本提案研究了这一领域的几个问题，主要集中在所谓的算术情况，这类似于在整数上定义方程而不是在复数等字段上定义方程的情况。本提案的主要研究重点是开发适用于这种情况的新工具。

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.