 External product of Chow classes

For smooth projective varieties $X,Y$, pull back Chow classes from the two factors and take their intersection product on $X\times Y$. This gives the external product and a graded map ${\mathrm{CH}}^{\ast }(X)\otimes {\mathrm{CH}}^{\ast }(Y)\to {\mathrm{CH}}^{\ast }(X\times Y)$. Unlike the cohomological Künneth theorem, it need not generate every class of the product.