Hasse Principle and Cohomology of Groups

Abstract

In a recent article, Colliot−Thelene, Gille and Parimala have considered fieldsK of cohomological dimension 2, of geometric type, analogous to totally imaginary numbers fields. One standard example is the field C((x,y)). Using previous results of Borovoi and the author, they compute the cohomology of K in degree one and two with coefficients in a semi−simpleK−group. The aim of our paper is to extend their results to fields K of cohomological dimension 2 that are not of geometric type but satisfy the Hasse principle; by Efrat, extensions of PAC fields of relative transcendence degree 1 are examples of such fields. For such fields K, we show that it is possible to calculate the non abelian cohomology in degree two with coefficients in a semi−simple K−group (the cohomology in degree one is calculated by Serre′s conjecture about the fields of cohomological dimension 2). We also show, in the case that K is of transcendence degree 1 over a PAC field, that if the group is semi−simple and a direct factor of a K−rational variety, then its Shafarevitch group is trivial, thus getting an analog of a result of Sansuc for number fields. For the field C((x,y)), the analogous result was established by Borovoi−Kunyavskii.