Localization sequence for the divisor class group
= 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
$$
\bigoplus_{j=1}^r\mathbb Z[Z_j]\longrightarrow\operatorname{Cl}(X)\longrightarrow\operatorname{Cl}(U)\longrightarrow0.
$$