Mathematical Sciences: Number of Defining Equations, Local Cohomology

本项目涉及一些相互关联的研究 局部上同调、Etale上同调和 定义代数簇的方程。 了一些进展 最近，我们对这些问题进行了研究，但仍有许多工作要做。 它 仍然不知道仿射的每个非奇异子簇是否 空间是一个集合论上的完全交。 的结构 局部上同调模仍然是一个谜。 的关系 上同调维数和余上同调维数之间的关系 规模仍不清楚。 关于本地的一些技术问题 Etale上同调仍然是开放的。 主要研究者将 研究这些问题使用的方法已经成功， 同时也在发展一些新的方法。 代数几何是研究几何对象的科学 由多项式方程组的零点集合产生。 这是一个最古老的，目前最活跃的 数学的分支。 它具有广泛的应用， 数学、计算机科学和物理学。

This project is concerned with the study of some interrelated questions of local cohomology, etale cohomology and the number of defining equations of algebraic varieties. Some progress has been made on these problems recently, but much remains to be done. It is still unknown whether every non-singular subvariety of affine space is a set-theoretic complete intersection. The structure of local cohomology modules remains a mystery. The relationship between the cohomological dimension and the etale cohomological dimension is still unclear. Some technical questions about local etale cohomology remain open. The principal investigator will study these questions using methods that have been successful in the past as well as developing some new methods. Algebraic geometry is the study of the geometric objects arising from the sets of zeros of systems of polynomial equations. This is one of the oldest and currently one of the most active branches of mathematics. It has widespread applications in mathematics, computer science and physics.