Lemma 33.38.11 (09NY)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.38: One dimensional Noetherian schemes
Lemma 33.38.11 (cite)

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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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