Arithmetic Cohomology

Geisser继续研究整数上有限型的算术上同调，有限生成的上同调理论和l进上同调的积分模型。对于有限域上的变量，Geisser先前构造了一个很好的候选，并建议继续证明基本性质，如庞加莱对偶性，以及各种定理和猜想的积分版本，如加藤猜想。对于平放于整数之上的变量，Geisser建议检验flichtenbaum的工作定义是否具有良好的性质。给定一个系数为整数的多项式方程组，确定它是否有解，以及是否有解，是一个重要的问题。解的数量可以编码成一个叫做“ζ函数”的函数。将ζ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的上同群，它比目前使用的上同群具有更好的性质。

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.