Chow ring ($A^{\ast }(X)$)

For a smooth variety, intersection of cycles gives the graded Chow ring $A^{\ast }(X)$.

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.

On $E\times E$ for a complex elliptic curve, the two factor curves and the diagonal have intersection matrix $\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right)$, of determinant two. Their cohomology classes are therefore independent. The degree-one image of the external product of Chow classes has cohomology contained in the span of the two factor classes. The cycle class map thus shows that the diagonal is outside that image, proving failure of surjectivity.