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.

For the three-dimensional affine quadric conethe divisor class group is infinite cyclic. A generator is either ruling plane $V(x,z)$ or $V(x,w)$, and the principal divisor of $x$ gives the relation $[V(x,z)]+[V(x,w)]=0$.