Lemma 76.39.6. Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $U \subset W \subset Y$ be open subspaces. Let $f : X \to W$ and let $s : U \to X$ be morphisms such that $f \circ s = \text{id}_ U$. Assume
$f$ is proper,
$Y$ is quasi-compact and quasi-separated, and
$U$ and $W$ are quasi-compact.
Then there exists a $U$-admissible blowup $b : Y' \to Y$ 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 of Spaces, Lemma 67.16.7. By Lemma 76.39.5 there exists a $U$-admissible blowup $W' \to W$ and an extension $W' \to X$ of $s$. We finish the proof by applying Divisors on Spaces, Lemma 71.19.3 to extend $W' \to W$ to a $U$-admissible blowup of $Y$. $\square$
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