The Stacks project

Lemma 33.38.11. Let $i : Z \to X$ be a closed immersion of schemes. If the underlying topological space of $X$ is Noetherian and $\dim (X) \leq 1$, then $\mathop{\mathrm{Pic}}\nolimits (X) \to \mathop{\mathrm{Pic}}\nolimits (Z)$ is surjective.

Proof. Consider the short exact sequence

\[ 0 \to (1 + \mathcal{I}) \cap \mathcal{O}_ X^* \to \mathcal{O}^*_ X \to i_*\mathcal{O}^*_ Z \to 0 \]

of sheaves of abelian groups on $X$ where $\mathcal{I}$ is the quasi-coherent sheaf of ideals corresponding to $Z$. Since $\dim (X) \leq 1$ we see that $H^2(X, \mathcal{F}) = 0$ for any abelian sheaf $\mathcal{F}$, see Cohomology, Proposition 20.20.7. Hence the map $H^1(X, \mathcal{O}^*_ X) \to H^1(X, i_*\mathcal{O}_ Z^*)$ is surjective. By Cohomology, Lemma 20.20.1 we have $H^1(X, i_*\mathcal{O}_ Z^*) = H^1(Z, \mathcal{O}_ Z^*)$. This proves the lemma by Cohomology, Lemma 20.6.1. $\square$


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