#### DOCUMENTA MATHEMATICA, Vol. 12 (2007), 607-671

Florian Ivorra

Réalisation ell-Adique des Motifs Triangulés Géométriques I

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.

2000 Mathematics Subject Classification: Primary 19E15; Secondary 19F27, 14F42.

Keywords and Phrases: Mixed motives, $\ell$-adic realizations, algebraic cycles.

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