Lemma 33.30.3 (0CC5)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.30: Picard groups of varieties
Lemma 33.30.3 (cite)

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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.

Proof. Special case of Lemma 33.30.2. If $\dim _ k R < \infty$ , then $R$ is Artinian and hence semi-local (Algebra, Lemmas 10.53.2 and 10.53.3). $\square$

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