Relation between the Divisor Class Groups and Chow Groups

Question

What if any, is the relation between the Divisor Class Groups and Chow Groups? Can anybody explain with some concrete examples?

Answer 1

If $X$ is irreducible, $A^1(X)$ is actually the divisor class group of $X$. This is proposition 1.9 in Fulton's book on intersection theory. If you compare both definitions it's more or less clear that in both case, this is the quotient of the free abelian group generated by the divisor, quotiented by the ideal generated by $f^{-1}(0) - f^{-1}(\infty)$ for $f \in k(X)$.

In fact, the motivation of the Chow ring is to generalise the construction of the divisor class group to varieties of higher codimension.