$\ell$-adic realization of geometric triangular motives. I

Abstract

In this work, we provide an integral $\ell$-adic realization functor for Voevodsky's triangulated category of geometrical motives over a noetherian separated scheme. Our approach to the realization problem is to study finite correspondences from the Nisnevich and étale local point of view. We set the existence of a local decomposition for finite correspondences which implies the existence of local transfers. This result allows us to provide canonical transfers on the Godement resolution of a Nisnevich sheaf with transfers and then to carry out the construction of the $\ell$-adic realization functor. We also give a moderate $\ell$-adic realization functor in some geometrical situations.