Abstract: In these lectures, I will consider several aspects of the following question: to what extent is the cohomology of an l-adic sheaf on a variety over a field of chacteristic prime to l controlled by the local behaviour of that sheaf? It turns out that certain global invariants of an l-adic sheaf, such as the alternated rank of its cohomology groups, namely its Euler characteristic, or such as the alternated determinant of its cohomology groups, can be computed through local invariants of the sheaf. Incarnations of this principle include the Grothendieck-Ogg-Shafarevich formula, or Saito's generalization thereof, and Laumon's product formula. I will describe a general procedure giving such localization results, and apply it in order to factor the determinant of the cohomology of an l-adic sheaf as a finite product of local contributions.
