Sur l'ind\'ependance de l en cohomologie l-adique sur les corps locaux

TL;DR

This paper extends Gabber's independence of l theorem from finite fields to local fields, introducing a new notion of independence for l-adic sheaves and proving its stability under key operations, with applications to algebraic stacks.

Contribution

It introduces a new notion of l-independence for complexes of l-adic sheaves over local fields and proves its stability under Grothendieck's six operations and nearby cycles.

Findings

Established stability of l-independence under six operations.

Provided a new proof of Gabber's theorem.

Generalized results to algebraic stacks.

Abstract

Gabber deduced his theorem of independence of $l$ of intersection cohomology from a general stability result over finite fields. In this article, we prove an analogue of this general result over local fields. More precisely, we introduce a notion of independence of $l$ for systems of complexes of $l$-adic sheaves on schemes of finite type over a local field, equivariant under finite groups. We establish its stability by Grothendieck's six operations and the nearby cycle functor. Our method leads to a new proof of Gabber's theorem. We also give a generalization to algebraic stacks. ----- Gabber a d\'eduit son th\'eor\`eme d'ind\'ependance de $l$ de la cohomologie l'intersection d'un r\'esultat g\'en\'eral de stabilit\'e sur les corps finis. Dans cet article, nous d\'emontrons un analogue sur les corps locaux de ce r\'esultat g\'en\'eral. Plus pr\'ecis\'ement, nous introduisons une notion d'ind\'ependance de $l$ pour les syst\`emes de complexes de faisceaux $l$-adiques sur les sch\'emas de type fini sur un corps local \'equivariants sous des groupes finis et nous \'etablissons sa stabilit\'e par les six op\'erations de Grothendieck et le foncteur des cycles proches. Notre m\'ethode permet d'obtenir une nouvelle d\'emonstration du th\'eor\`eme de Gabber. Nous donnons aussi une g\'en\'eralisation aux champs alg\'ebriques.