Localization sequence for the divisor class group

Let $X$ be a Noetherian integral scheme that is regular in codimension one, and let $U=X\setminus {\bigcup }_{j=1}^r{Z}_j$, where the $Z_j$ are the codimension-one irreducible components of the complement. Restriction gives an exact sequence

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 $[V(x,z)]+[V(x,w)]=0$.

For a Noetherian integral scheme $X$ that is regular in codimension one,where $H=X\times {\mathbb{P}}^{n-1}$ is a hyperplane divisor. Restriction to $X\times {\mathbb{A}}^n$ gives the first summand, while restriction to the generic projective-space fiber detects the coefficient of $H$.