Constructibilit\'e et mod\'eration uniformes en cohomologie \'etale

TL;DR

This paper investigates the stability of constructibility and tameness conditions in étale cohomology under direct image functors, establishing uniform bounds on Betti numbers and extending previous results to more general settings.

Contribution

It demonstrates that a natural uniform tameness condition remains stable under direct images, even when the morphism is not proper, and establishes uniform bounds on Betti numbers for a broad class of sheaves.

Findings

Stability of Gabber's tameness condition under direct images

Existence of uniform bounds on Betti numbers for sheaves

Extension of constructibility results to non-proper morphisms

Abstract

Let S be a Noetherian scheme and f:X -> S a proper morphism. By SGA 4 XIV, for any constructible sheaf F of Z/nZ-modules on X, the sheaves of Z/nZ-modules R^if_*F obtained by direct image (for the etale topology) are also constructible: there is a stratification of S on whose strata these sheaves are locally constant constructible. After previous work of N. Katz and G. Laumon, or L. Illusie, on the special case in which S is generically of characteristic zero or the sheaves F are constant (with invertible torsion on S), here we study the dependency of the stratification on F. We show that a natural "uniform" tameness and constructibility condition satisfied by constant sheaves, which was introduced by O. Gabber, is stable under the functors R^if_*. If f is not proper, this result still holds assuming tameness at infinity, relatively to S. We also prove the existence of uniform bounds on Betti numbers, in particular for the stalks of the sheaves R^if_*Z/lZ, where l ranges through all prime numbers invertible on S.