  Let X be a complex surface with no nontrivial 2-forms. Then we show that
Bloch's conjecture is true (i.e. the Albanese map in this case is injective) if
and only if any homologically trivial idempotent in the ring of correspondences
vanishes. Furthermore the cube of the ideal of homologically trivial
correspondences is zero if these equivalent conditions are satisfied (e.g. if X
is not of general type).
