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 CHⁿalg(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 CHⁿalg(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, zabⁿ: Mⁿab(X) → M(X) having analogue properties but which instead computes the subgroup CHⁿab(X)⊆ CHⁿalg(X) of algebraic cycles abelian equivalent to zero (in the sense of Samuel).