Mixed $\ell$-adic complexes for schemes over number fields

Abstract

If $X$ is a variety over a number field, Annette Huber has defined a category of “horizontal” (or “almost everywhere unramified”) $\ell$-adic complexes and $\ell$-adic perverse sheaves on $X$. For such objects, the notion of weights makes sense (in the sense of Deligne), just as in the case of varieties over finite fields. However, contrary to what happens in that last case, mixed perverse sheaves (or mixed locally constant sheaves) on $X$ do not have a weight filtration in general, even when $X$ is a point. The goal of this paper is to show how to avoid this problem by working directly in the derived category of the abelian category of perverse sheaves that do admit a weight filtration. As an application, the method of the author to calculate the intermediate extension of a pure perverse sheaf using weight truncations apply over any finitely generated field, and not just over a finite field.