Regular homomorphisms and mixed motives

Let X be a smooth projective variety of dimension d over an algebraically closed field k. The main goal of this paper is to study, in the context of Voevodsky's triangulated category of motives DMₖ, the group CHⁿ₀₋₆ (X) of codimension n algebraic cycles of X, algebraically equivalent to zero, modulo rational equivalence, 1 n d. Namely, for any regular homomorphism ψ (in the sense of Samuel) defined on CHⁿ₀₋₆ (X), we construct Mⁿ_ψ (X) DMₖ, which is a reasonable approximation, with respect to the slice filtration in DMₖ, of the motive of X, M (X) ; and a map z_ψ: Mⁿ_ψ (X) M (X) in DMₖ, which computes the kernel of ψ. We construct as well a map, z₀₁ⁿ: Mⁿ₀₁ (X) M (X) having analogue properties but which instead computes the subgroup CHⁿ₀₁ (X) CHⁿ₀₋₆ (X) of algebraic cycles abelian equivalent to zero (in the sense of Samuel).