Arithmetic Cohomology

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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.