Galois module structure of Étale cohomology groups

This thesis is concerned with proving a refined function field analogue of the Coates-Sinnott conjecture. The theorem we prove identifies precisely the Fitting ideal of a certain etale cohomology group. The techniques employed are directly inspired by recent work of Greither and Popescu in equivariant Iwasawa theory, both for number fields and function fields. They rest on an in-depth study of the Galois module structure of certain naturally defined 1-motives associated to a function field