The Stacks project

Theorem 76.38.2. Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $X$ be an algebraic space over $B$. Let $\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \subset B$ be a quasi-compact open subspace. Assume

  1. $X$ is quasi-compact,

  2. $X$ is locally of finite presentation over $B$,

  3. $\mathcal{F}$ is a module of finite type,

  4. $\mathcal{F}_ U$ is of finite presentation, and

  5. $\mathcal{F}_ U$ is flat over $U$.

Then there exists a $U$-admissible blowup $B' \to B$ such that the strict transform $\mathcal{F}'$ of $\mathcal{F}$ is an $\mathcal{O}_{X \times _ B B'}$-module of finite presentation and flat over $B'$.

Proof. Choose an affine scheme $V$ and a surjective étale morphism $V \to X$. Because strict transform commutes with étale localization (Divisors on Spaces, Lemma 71.18.2) it suffices to prove the result with $X$ replaced by $V$. Hence we may assume that $X \to B$ is representable (in addition to the hypotheses of the lemma).

Assume that $X \to B$ is representable. Choose an affine scheme $W$ and a surjective étale morphism $\varphi : W \to B$. Note that $X \times _ B W$ is a scheme. By the case of schemes (More on Flatness, Theorem 38.30.7) we can find a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ W$ such that (a) $|V(\mathcal{I})| \cap |\varphi ^{-1}(U)| = \emptyset $ and (b) the strict transform of $\mathcal{F}|_{X \times _ B W}$ with respect to the blowing up $W' \to W$ in $\mathcal{I}$ becomes flat over $W'$ and is a module of finite presentation. Choose a finite type sheaf of ideals $\mathcal{J} \subset \mathcal{O}_ B$ as in Lemma 76.38.1. Let $B' \to B$ be the blowing up of $\mathcal{J}$. We claim that this blowup works. Namely, it is clear that $B' \to B$ is $U$-admissible by our choice of ideal $\mathcal{J}$. Moreover, the base change $B' \times _ B W \to W$ is the blowup of $W$ in $\varphi ^{-1}\mathcal{J} = \mathcal{I}\mathcal{I}'$ (compatibility of blowup with flat base change, see Divisors on Spaces, Lemma 71.17.3). Hence there is a factorization

\[ W \times _ B B' \to W' \to W \]

where the first morphism is a blowup as well, see Divisors on Spaces, Lemma 71.17.10). The restriction of $\mathcal{F}'$ (which lives on $B' \times _ B X$) to $W \times _ B B' \times _ B X$ is the strict transform of $\mathcal{F}|_{X \times _ B W}$ (Divisors on Spaces, Lemma 71.18.2) and hence is the twice repeated strict transform of $\mathcal{F}|_{X \times _ B W}$ by the two blowups displayed above (Divisors on Spaces, Lemma 71.18.7). After the first blowup our sheaf is already flat over the base and of finite presentation (by construction). Whence this holds after the second strict transform as well (since this is a pullback by Divisors on Spaces, Lemma 71.18.4). Thus we see that the restriction of $\mathcal{F}'$ to an étale cover of $B' \times _ B X$ has the desired properties and the theorem is proved. $\square$

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