Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 113 / 3 / b

Restriction sends a Weil divisor on $X$ to the sum of its components meeting $U$. Every prime divisor on $U$ has a codimension-one closure in the Noetherian scheme $X$, so the induced map $\mathrm{Cl}(X)\to \mathrm{Cl}(U)$ is surjective.

If the class of $D$ restricts to zero, then $D{|}_U={\mathrm{div}}_U(f)$ for some $f\in K(X{)}^{\ast }$. The divisor $D-{\mathrm{div}}_X(f)$ is supported on $Z$, hence is an integral combination of $Z_1,\dots ,Z_n$. Conversely, every such combination restricts to zero. Thereforewhere the first map sends the $i$th basis vector to $[Z_i]$, is exact.