The Stacks project

Proof. Let $B' \to B$ be a $U$-admissible blowup. Note that the strict transform of $X$ is quasi-compact and quasi-separated over $B'$ as $X$ is quasi-compact and quasi-separated over $B$. Hence we only need to worry about finding a $U$-admissible blowup such that the strict transform becomes flat and locally of finite presentation. We cannot directly apply Theorem 76.38.2 because $X$ is not locally of finite presentation over $B$.

Choose an affine scheme $V$ and a surjective étale morphism $V \to X$. (This is possible as $X$ is quasi-compact as a finite type space over the quasi-compact space $B$.) Then it suffices to show the result for the morphism $V \to B$ (as strict transform commutes with étale localization, see Divisors on Spaces, Lemma 71.18.2). Hence we may assume that $X \to B$ is separated as well as finite type. In this case we can find a closed immersion $i : X \to Y$ with $Y \to B$ separated and of finite presentation, see Limits of Spaces, Proposition 70.11.7.

Apply Theorem 76.38.2 to $\mathcal{F} = i_*\mathcal{O}_ X$ on $Y/B$. We find a $U$-admissible blowup $B' \to B$ such that strict transform of $\mathcal{F}$ is flat over $B'$ and of finite presentation. Let $X'$ be the strict transform of $X$ under the blowup $B' \to B$. Let $i' : X' \to Y \times _ B B'$ be the induced morphism. Since taking strict transform commutes with pushforward along affine morphisms (Divisors on Spaces, Lemma 71.18.5), we see that $i'_*\mathcal{O}_{X'}$ is flat over $B'$ and of finite presentation as a $\mathcal{O}_{Y \times _ B B'}$-module. Thus $X' \to B'$ is flat and locally of finite presentation. This implies the lemma by our earlier remarks. $\square$

Proof. By Lemma 76.39.1 we may assume that $X \to B$ is flat and of finite presentation. After replacing $B$ by a $U$-admissible blowup if necessary, we may assume that $U \subset B$ is scheme theoretically dense. Then $f$ is finite by Lemma 76.37.4. Hence $f$ is finite locally free by Morphisms of Spaces, Lemma 67.46.6. $\square$

Proof. By Lemma 76.39.1 we may assume that $X \to B$ is flat and of finite presentation. After replacing $B$ by a $U$-admissible blowup if necessary, we may assume that $U \subset B$ is scheme theoretically dense. Then $f$ is finite by Lemma 76.37.4 and an open immersion by Lemma 76.37.5. Hence $f$ is an open immersion whose image is closed and contains the dense open $U$, whence $f$ is an isomorphism. $\square$

Proof. This lemma is a generalization of Lemma 76.39.3. As the composition of $U$-admissible blowups is $U$-admissible (Divisors on Spaces, Lemma 71.19.2) we can proceed in stages. Pick a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ B$ with $|B| \setminus |U| = |V(\mathcal{I})|$. Replace $B$ by the blowup of $B$ in $\mathcal{I}$ and $X$ by the strict transform of $X$. After this replacement $B \setminus U$ is the support of an effective Cartier divisor $D$ (Divisors on Spaces, Lemma 71.17.4). In particular $U$ is scheme theoretically dense in $B$ (Divisors on Spaces, Lemma 71.6.4). Next, we do another $U$-admissible blowup to get to the situation where $X \to B$ is flat and of finite presentation, see Lemma 76.39.1. Note that $U$ is still scheme theoretically dense in $B$. Hence $X \to B$ is an open immersion by Lemma 76.37.5. $\square$

Proof. By Lemma 76.39.3 we may find a $U$-admissible blowup $B' \to B$ such that the strict transform $X'$ maps isomorphically to $B'$. Then we can use $B' = X' \to X$ as the factorization. $\square$

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 $Y$. $\square$