says that for a smooth scheme X over a field k and m a positive integer invertible in k, the cycle map is an isomorphism for all j ≥ i and is injective for all j ≥ i − 1. Relation to motives For any field k and commutative ring R, Voevodsky defined an R-linear triangulated category called the derived category of motives over k with coefficients in R, DM(k; R). Each scheme X over k determines two objects in DM called the motive of X, M(X), and the compactly supported motive of X, Mc(X); the two are isomorphic if X is proper over k. One basic point of the derived category of motives is that the four types of motivic homology and motivic cohomology all arise as sets of morphisms in this category. To describe this, first note that there are Tate motives R(j) in DM(k; R) for all integers j, such that the motive of projective space is a direct sum of Tate motives: where M ↦ M[1] denotes the shift or "translation functor" in the triangulated category DM(k; R). In these terms, motivic cohomology (for example) is given by for every scheme X of finite type over k. When the coefficients R are the rational numbers, a modern version of a conjecture by Beilinson predicts that the subcategory of compact objects in DM(k; Q) is equivalent to the bounded derived category of an abelian category MM(k), the category of mixed motives over k. In particular, the conjecture would imply that motivic cohomology groups can be identified with Ext groups in the category of mixed motives. This is far from known. Concretely, Beilinson's conjecture would imply the Beilinson-Soulé conjecture that Hi(X,Q(j)) is zero for i < 0, which is known only in a few cases. Conversely, a variant of the Beilinson-Soulé conjecture, together with Grothendieck's standard conjectures and Murre's conjectures on Chow motives, would imply the existence of an abelian category MM(k) as the heart of a t-structure on DM(k; Q). More would be needed in order to identify Ext groups in MM(k) with motivic cohomology. For k a subfield of the complex numbers, a candidate for the abelian category of mixed motives has been defined by Nori. If a category MM(k) with the expected properties exists (notably that the Betti realization functor from MM(k) to Q-vector spaces is faithful), then it must be equivalent to Nori's category.