Loading web-font TeX/Math/Italic

The Stacks project

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


Comments (0)


Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.