Picard-group localization on a smooth variety (${⨁}_j\mathbb{Z}[D_j]\to \mathrm{Pic}(X)\to \mathrm{Pic}(U)\to 0$)

For a smooth integral variety, every Weil divisor is a Cartier divisor, so the Picard group equals the divisor class group. If $U$ is obtained by removing irreducible Weil divisors $D_j$, the localization sequence for the divisor class group gives the displayed exact sequence. A rational trivialization of a line bundle on $U$ has divisor supported on the removed divisors, describing the kernel concretely.