Citation/Abstract

Let X be a quasi-projective complex variety. It follows from the work of Voevodsky that the motivic cohomology of X , denoted as H ^p,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 [Special characters omitted.] . That is, there exists an object [Special characters omitted.] _mot ( q ) in [Special characters omitted.] such that [Special characters omitted.] where M ( X ) is the motive of X . We construct objects [Special characters omitted.] _mor ( q ) and [Special characters omitted.] _Sing ( q ) in [Special characters omitted.] to represent the morphic cohomology L ^q H ^p ( X ) and the singular cohomology [Special characters omitted.] ( X ^an ) of X . More precisely, [Special characters omitted.] 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 L ^q H ^p ( X ) [congruent with] [Special characters omitted.] ( X ^an ), 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.