 David Mumford proved that, on a smooth complete complex algebraic surface "S" with positive geometric genus, the analogous statement for the group formula_1 of rational equivalence classes of codimension two cycles in "S" is false.The hypothesis that the geometric genus is positive essentially means (by the Lefschetz theorem on (1,1)-classes) that the cohomology group formula_2 contains transcendental information, and in effect Mumford's theorem implies that, despite formula_1 having a purely algebraic definition, it shares transcendental information with formula_2.Mumford's theorem has since been greatly generalized.