 Nagata theorem for divisor class groups

If $A$ is a Noetherian normal domain and $S$ is a multiplicative set, then $\mathrm{Cl}(S^{-1}A)$ is the quotient of $\mathrm{Cl}(A)$ by the classes of height-one primes meeting $S$. Tracking the units of $S^{-1}A$ determines the relations among those prime divisors.