| Let X be a quasi-projective complex variety. It follows from the work of Voevodsky that the motivic cohomology of X, denoted as Hp,q(X) where q and p are integers with q nonnegative, can be represented in the triangulated category of motives over the field of complex numbers, denoted as DMeff,-Nis . That is, there exists an object ℘ mot(q) in DMeff,-Nis such that Hp,qX=Hom DMeff,-NisM X,℘motq p-2q where M(X) is the motive of X. We construct objects ℘ mor(q) and ℘ Sing(q) in DMeff,-Nis to represent the morphic cohomology LqH p(X) and the singular cohomology HpSing (Xan) of X. More precisely, LqHpX =HomDMeff,-Nis MX,℘mor qp-2q HpSingXan =HomDMeff,-Nis MX,℘Sing qp-2q where X is smooth. If X is singular, we define the morphic cohomology of X by the above formula. As an application, we show that Friedlander's comparison result LqHp(X) ≅ HpSing (Xan), where X is smooth of pure dimension d and q ≥ d, can be generalized to singular varieties. As a second application, the morphic cohomology operations are considered. |
