  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 methods of a previous paper of the author to
calculate the intermediate extension of a pure perverse sheaf apply over any
finitely generated field, and not just over a finite field.
