Stanford Number Theory Learning Seminar

In the course of Deligne’s proof of the purity theorem, he makes certain monodromy constructions which do not a priori yield legitimate `-adic sheaves. In order to make the proof work, it therefore becomes necessary to slightly enlarge the category of sheaves considered from `-adic (constructible) sheaves to Weil sheaves. These satisfy very similar formal properties to ordinary `-adic sheaves, and the Grothendieck-Lefschetz trace formula remains valid in this context. Then, using this formalism, it becomes possible to analyze the sheaves that are actually relevant to the proof, and to demonstrate that they are in fact `-adic sheaves in the ordinary sense. Now, fix a scheme X0 of finite type1 over a finite field k = Fq and an algebraic closure k2, and let X = X0 ×k k with π : X → X0 the projection morphism. Let F : α 7→ α be the geometric Frobenius automorphism of k, which topologically generates Gal(k/k).