Lemma 33.30.2. Let $A \to B$ be a faithfully flat ring map. Let $X$ be a scheme over $A$ such that

1. $X$ is quasi-compact and quasi-separated, and

2. $R = H^0(X, \mathcal{O}_ X)$ is a semi-local ring.

Then the pullback map $\mathop{\mathrm{Pic}}\nolimits (X) \to \mathop{\mathrm{Pic}}\nolimits (X_ B)$ is injective.

Proof. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module whose pullback $\mathcal{L}'$ to $X_ B$ is trivial. Set $M = H^0(X, \mathcal{L})$ and $N = H^0(X, \mathcal{L}^{\otimes - 1})$. By Lemma 33.30.1 the $R$-modules $M$ and $N$ are invertible. Since $R$ is semi-local $M \cong R$ and $N \cong R$, see Algebra, Lemma 10.78.7. Choose generators $s \in M$ and $t \in N$. Then $st \in R = H^0(X, \mathcal{O}_ X)$ is a unit by the last part of Lemma 33.30.1. We conclude that $s$ and $t$ define trivializations of $\mathcal{L}$ and $\mathcal{L}^{\otimes -1}$ over $X$. $\square$

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