M ar 2 00 3 Finite dimensional motives and the Conjectures of Beilinson and Murre Vladimir

Let k be a field of characteristic 0 and let Vk be the category of smooth projective varieties over k. By ∼ we denote an adequate equivalence relation for algebraic cycles on varieties [Ja00]. For every X ∈ Vk let A i ∼(X) = (Z (X)/ ∼)⊗Q be the Chow group of codimension i cycles on X modulo the chosen relation ∼ with coefficients in Q. Let X, Y ∈ Vk, let X = ∪Xi be the connected components of X and let di = dim(Xi). Then Corr r ∼(X, Y ) = ⊕iA di+r ∼ (Xi × Y ) is called a space of correspondences of degree r from X into Y . For any f ∈ Corr(X, Y ) and g ∈ Corr(Y, Z) their composition g ◦ f ∈ Corr(X,Z) is defined by the formula g ◦f = pXZ∗(p ∗ XY (f) ·p ∗ Y Z(g)) where pXZ , pXY and pY Z are the appropriate projections. In particular, we have a linear action of correspondences Corr(Y, Z) × A(Y ) → A(Z) defined by the rule (α, x) 7→ pY ∗(α · p ∗ X(x)), where pX and pY are the projections. The category of pure motives M∼ over k with coefficients in Q with respect to the given equivalence relation ∼ can be defined as follows [Sch94]. Its objects are triples M = (X, p,m), where X ∈ Vk, p ∈ Corr 0 ∼(X,X) is a projector (i.e. p ◦ p = p) and m ∈ Z. Morphisms from M = (X, p,m) into N = (Y, q, n) in M∼ are given by correspondences f ∈ Corr n−m ∼ (X, Y ), such that f ◦ p = q ◦ f = f , and compositions of morphisms are induced by compositions of correspondences.