Applications of de Jong’s Theorem on Alterations

Abstract

The purpose of this article is to give a survey of some of the applications of de Jong’s theorem on alterations [dJ]. Most of the applications fall into one of the following two categories:

The first type of application deals with contravariant functors F from some subcategory of the category of schemes to the category of ℚ-vector spaces with extra structure (e.g. Galois action), which are equipped with a trace map for finite étale morphisms. In a situation like this, for a given scheme X, F(X) will be a direct summand of F(X’) for an alteration X’ of X. This allows to deduce properties of F(X) for general X if one only knows the same property for smooth schemes. The following are examples of this kind of application: The independence of l in Grothendieck’s monodromy theorem, the p-adic monodromy theorem, finiteness of rigid cohomology, and a (conditional) vanishing theorem for motivic cohomology. All but the last application were already discussed by Berthelot in his Bourbaki talk, and we follow his exposition.