Lemma 38.31.5 (0CP1)—The Stacks project

Processing math: 100%

Lemma 38.31.5. Let $S$ be a scheme. Let $U \subset W \subset S$ be open subschemes. Let $f : X \to W$ be a morphism and let $s : U \to X$ be a morphism such that $f \circ s = \text{id}_ U$ . Assume

$f$ is proper,
$S$ is quasi-compact and quasi-separated, and
$U$ and $W$ are quasi-compact.

Then there exists a $U$ -admissible blowup $b : S' \to S$ and a morphism $s' : b^{-1}(W) \to X$ extending $s$ with $f \circ s' = b|_{b^{-1}(W)}$ .

Proof. We may and do replace $X$ by the scheme theoretic image of $s$ . Then $X \to W$ is an isomorphism over $U$ , see Morphisms, Lemma 29.6.8. By Lemma 38.31.4 there exists a $U$ -admissible blowup $W' \to W$ and an extension $W' \to X$ of $s$ . We finish the proof by applying Divisors, Lemma 31.34.3 to extend $W' \to W$ to a $U$ -admissible blowup of $S$ . $\square$

Comments (0)

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).

Comments (0)

Post a comment