Adequate equivalence relations for non-projective varieties.

Question

When defining adequate equivalence relations on algebraic cycles, I have always seen it to be defined for smooth projective varieties. Is there a problem for defining them for all smooth varieties? For example the Standard Conjecture D expects numerical and homological relations to coincide for smooth projective varieties. Does this fail if remove the projectivity? (for example under quasi-projectivity). The same question can be asked for comparing the smash nilpotence and homological relations (which are expected to coincide for smooth projective varieties).