Let X be a smooth, projective algebraic variety over an algebraically closed field k, and let H(X) denote a Weil cohomology theory. The existence of algebraic cycles on X×X inducing as correspondences the various Künneth projectors π : H(X) → H (X) is one of the standard conjectures of Grothendieck, [8], [9]. It is known in general only for the cases i = 0, 1, 2d − 1, 2d where d = dimX. The pur...