Weil-etale topology over local rings

Abstract

In algebraic number theory, the study of Galois groups is one of the subjects mathematicians are most interested in. A fruitful set of results have been obtained in this field. Class field theory and some of its consequent theorems are certainly an important part of these results. In [1], E.Artin and Tate used the notion of class formations to reinterpret class field theory and proposed another interesting object, which is called the Weil group and has a strong relation with class formations, for us to study. Recently, research shows that Weil groups deserve more investigation and they behave better than the classical Galois groups in some cases. Lichtenbaum's paper [11] showed us evidence for the above mentioned opinion. In his paper, Lichtenbaum gave the definition of Weil-kale topology on schemes over finite fields and then the associated cohomology groups. He mainly showed that the arithmetic information from these cohomology groups can decipher the zeta functions' values at special points. Motivated by Lichtenbaum's paper, in this dissertation. we will try to use the similar idea to define_the Weil-etale topology and then prove an analogous duality theorem for sheaves on local rings. The main theorem is stated using derived category language which seems to be the natural language we should employ in this situation.