The Stacks project

Lemma 76.39.1. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $U \subset B$ be an open subspace. Assume

  1. $B$ is quasi-compact and quasi-separated,

  2. $U$ is quasi-compact,

  3. $f : X \to B$ is of finite type and quasi-separated, and

  4. $f^{-1}(U) \to U$ is flat and locally of finite presentation.

Then there exists a $U$-admissible blowup $B' \to B$ such that the strict transform $X'$ of $X$ is flat and of finite presentation over $B'$.

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 ├ętale 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 ├ętale 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. $\square$


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