Lemma 33.30.3. Let $k'/k$ be a field extension. Let $X$ be a scheme over $k$ such that

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

$R = H^0(X, \mathcal{O}_ X)$ is semi-local, e.g., if $\dim _ k R < \infty $.

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

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