= Picard-group localization on a smooth variety
{c}
{title2=$\bigoplus_j\mathbb Z[D_j]\to\operatorname{Pic}(X)\to\operatorname{Pic}(U)\to0$}
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.
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