Realization of Voevodsky's Motives Acknowledgments I Would like to Thank B. Kahn and M. Spiess Who Organized an Enlightening Arbeitsgemeinschaft on Voevodsky's Work in Oberwolfach. I Prooted from Discussions With

Introduction The theory of motives has always had two faces. One is the geometric face where a universal cohomology theory for varieties is cooked up from geometric objects like cycles. The other one is the linear algebra face where restricting conditions are put on objects of linear algebra like vector spaces with an operation of the Galois group. The ideal theorem would be an equivalence of these two approaches. For pure motives, Grothendieck proposed a geometric construction. The linear algebra side is covered by l-adic cohomology for all primes l together with singular cohomology equipped with its Hodge structure. The relation between the two sides is made by the Tate or the Hodge conjecture which tell us that geometry and linear algebra should be very close to each other. However, we neither know whether Grothendieck's construction has the required properties nor what the image of the category of motives on the linear algebra side is. I.e., we cannot tell whether a Galois module is motivic just from checking linear algebra conditions (there are conjectures though). For mixed motives, Voevodsky's work goes a long way in constructing the geometric side of the story. The linear algebra side is given by Deligne's absolute Hodge motives, independently considered by Jannsen under the name of mixed realizations. By Beilinson's conjectures the interplay between geometry and linear algebra should be measured by special values of L-functions of motives. However, we are far from proving the ideal theorem. The main aim of the present article is to provide the expected functor between the two sides. More precisely, we construct a realization functor from Voevodsky's triangulated category of geometrical motives (which should be thought of as the derived category of mixed motives) to the \derived cate-gory" of mixed realizations which we constructed in Hu1]. Indeed, most of the present article is a follow-up of loc. cit. where the realization functor was constructed on the category of simplicial varieties. As a direct corollary we also obtain realizations functors to continuous l-adic cohomology and to absolute Hodge cohomology. Their existence is not a surprise (cf. Vo2]) but was not in the literature yet. We want to mention that Levine has a triangulated category of motives ((Le3]). Over a eld of characteristic zero it is equivalent to Voedvodsky's. He also constructs realization functors in his setting starting from a diierent 1 set of axioms. We can show (2.3.6) that the …