Chow ring
For a smooth complex projective variety, the cycle map from the Chow ring to ordinary cohomology factors through a richer theory, Deligne cohomology. This incorporates the Abel–Jacobi map from cycles homologically equivalent to zero to the intermediate Jacobian. The exponential sequence shows that "CH"1("X") maps isomorphically to Deligne cohomology, but that fails for "CH""j"("X") with "j" &gt; 1.