Lemma 37.48.4. Let $S$ be a quasi-compact and quasi-separated scheme. If $U \subset S$ is a quasi-compact open and $V \to U$ is a surjective finite morphism, then there exists a surjective finite morphism $\overline{V} \to S$ with $\overline{V} \times _ S U$ isomorphic to $V$ as schemes over $U$.
Proof. By Zariski's Main Theorem (Lemma 37.43.3) we can assume $V$ is a quasi-compact open in a scheme $V'$ finite over $S$. After replacing $V'$ by the scheme theoretic image of $V$ we may assume that $V$ is dense in $V'$. It follows that $V' \times _ S U = V$ because $V \to V' \times _ S U$ is closed as $V$ is finite over $U$. Let $Z$ be the reduced induced scheme structure on $S \setminus U$. Then $\overline{V} = V' \amalg Z$ works. $\square$
Post a comment
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)