Lemma 33.30.5. Let k be a field. Let X be a normal variety over k. Let U \subset \mathbf{A}^ n_ k be an open subscheme with k-rational points p, q \in U(k). For every invertible module \mathcal{L} on X \times _{\mathop{\mathrm{Spec}}(k)} U the restrictions \mathcal{L}|_{X \times p} and \mathcal{L}|_{X \times q} are isomorphic.
Proof. The fibres of X \times _{\mathop{\mathrm{Spec}}(k)} U \to X are open subschemes of affine n-space over fields. Hence these fibres have trivial Picard groups by Divisors, Lemma 31.28.4. Applying Divisors, Lemma 31.28.1 we see that \mathcal{L} is the pullback of an invertible module \mathcal{N} on X. \square
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