| Résume | The Yoga of motives is born with étale cohomology, as a conceptual way to explain elusive integral aspects of l-adic cohomologies. This has lead to deep independence of l problems that have found positive answers - from Deligne's proof of the Weil conjectures to the proof of Deligne's conjecture on companions over smooth algebraic varieties (Lafforgue, Drinfeld, Esnault and Kerz). The Tate conjecture, together with its variations due to Beilinson and Lichtenbaum remains open though. On the other hand, motives have become a full fledged theory that lead to the proof of the Bloch-Kato conjecture by Rost and Voevodsky. We will formulate categorified version of independence of l conjectures, in the language of motivic sheaves. To our knowledge, they are not equivalent to any of the classical conjectures, but interesting relations can be established. For instance, they follow from the Tate-Beilinson conjecture and imply Deligne's conjecture on companions in full generality (over normal algebraic varieties). |
