Local Cohomology and Related Questions

9970277在过去的五年里，首席研究员在局部上同调方面的大部分研究都致力于与几个相当不同的数学领域的一些惊人的联系的研究，如表上同调，代数变的拓扑，微分算子和其他包括紧闭理论和（最近的）群上同调。虽然在这一思想循环方面已经取得了相当大的进展，但仍有许多工作要做。建议继续使用过去成功的方法以及开发一些新方法来研究这些（和其他一些）问题。给定一个多项式方程组，在任何具有相同解集的方程组中多项式方程的最小可能个数是多少？直到今天，这个问题还没有令人满意的答案。局部上同调模与这个问题有很大关系：如果多项式方程组的第n个局部上同调模不消失，则具有相同解集的每个多项式方程组由n个或更多方程组成。这只是局部上同调在一些非常自然的问题上的许多应用之一。本提案致力于研究局部上同，并特别强调与其他数学分支的一些迷人的相互关系。

9970277A large part of the Principal Investigator's research on local cohomology over the last five years has been devoted to the study of a number of striking connections with several quite diverse areas of mathematics, such as etale cohomology, topology of algebraic varieties, differential operators and others including the theory of tight closure and (very recently) cohomology of groups. While considerable progress on this circle of ideas has been made, a lot remains to be done. It is proposed to continue to study these (and some other) questions by using methods that have been successful in the past as well as developing some new methods.Given a system of polynomial equations, what is the minimum possible number of polynomial equations in any system with the same solution set? To this day there is no satisfactory answer to this question. Local cohomology modules have a lot to do with this problem: if the n-th local cohomology module of a system of polynomial equations does not vanish, then every system of polynomial equations with the same solution set consists of n or more equations. This is just one out of many applications of local cohomology to some very natural questions. The proposal is devoted to studying local cohomology with a special emphasis on some fascinating interrelations with other branches of mathematics.