Local Cohomology and Related Questions

局部上同调及相关问题

• 批准号: 9970277
• 负责人: Gennady Lyubeznik
• 金额: $ 13.5万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970277&HistoricalAwards=false
• 关键词: Local; Cohomology; Related; Questions

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.

9970277 过去五年来，首席研究员对局部上同调的研究很大一部分致力于研究与几个相当不同的数学领域的许多显着联系，例如 etale 上同调、代数簇拓扑、微分算子以及其他包括紧闭闭理论和（最近）群上同调的理论。 尽管这一思想圈已经取得了相当大的进展，但仍有许多工作要做。 建议通过使用过去成功的方法以及开发一些新方法来继续研究这些（以及其他一些）问题。给定一个多项式方程组，在具有相同解集的任何系统中，多项式方程的最小可能数量是多少？ 时至今日，这个问题还没有一个令人满意的答案。 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都由 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众多应用之一。 该提案致力于研究局部上同调，特别强调与其他数学分支的一些有趣的相互关系。