Arithmetic Cohomology

算术上同调

• 批准号: 0556263
• 负责人: Thomas Geisser
• 金额: $ 15.33万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0556263&HistoricalAwards=false
• 关键词: Arithmetic; Cohomology

Geisser continues the study of arithmetic cohomology for varieties of finite type over the integers, a cohomology theory which should be finitely generated and an integral model of l-adic cohomology, .For varieties over a finite field, Geisser previously constructed a goodcandidate, and propose to continue proving basic properties,such as Poincare-duality, and integral versions of various theorem and conjectures, such as Kato's conjecture.For varieties flat over the integers, Geisser proposes to examine ifLichtenbaum's working definition has good properties.Given a system of polynomial equations with integer coefficients,it is an important question to determine if it has solutions, andto count them if they exists. The number of solutions can be encoded ina function called "zeta-function". Relating the zeta-funtion to invariantsof the sytem of equations called "cohomology groups" helps to gain information on the number of solutions. Geisser proposes to continue to study a new type of cohomology groups, which has better properties than the cohomology groups used so far.

Geisser继续研究整数上的有限型簇的算术上同调，一个应该被生成的上同调理论和一个l-adic上同调的积分模型，对于有限域上的簇，Geisser以前构造了一个很好的候选者，并提出继续证明基本性质，如Poincare对偶，以及各种定理和定理的积分版本，对于整数上平坦的簇，Geisser提出检验Lichtenbaum的工作定义是否具有好的性质。给定一个整数系数的多项式方程组，确定它是否有解，如果有解，就计数是一个重要的问题。解决方案的数量可以编码在一个函数称为“zeta函数”。 将zeta函数与称为“上同调群”的方程组的不变量联系起来，有助于获得关于解的数量的信息。Geisser建议继续研究一种新型的上同调群，它比迄今为止使用的上同调群具有更好的性质。