The Stacks project

Theorem 38.30.7. Let $S$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme over $S$. Let $\mathcal{F}$ be a quasi-coherent module on $X$. Let $U \subset S$ be a quasi-compact open. Assume

  1. $X$ is quasi-compact,

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

  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 $S' \to S$ such that the strict transform $\mathcal{F}'$ of $\mathcal{F}$ is an $\mathcal{O}_{X \times _ S S'}$-module of finite presentation and flat over $S'$.

Proof. We first prove that we can find a $U$-admissible blowup such that the strict transform is flat. The question is ├ętale local on the source and the target, see Lemma 38.30.6 for a precise statement. In particular, we may assume that $S = \mathop{\mathrm{Spec}}(R)$ and $X = \mathop{\mathrm{Spec}}(A)$ are affine. For $s \in S$ write $\mathcal{F}_ s = \mathcal{F}|_{X_ s}$ (pullback of $\mathcal{F}$ to the fibre). As $X \to S$ is of finite type $d = \max _{s \in S} \dim (\text{Supp}(\mathcal{F}_ s))$ is an integer. We will do induction on $d$.

Let $x \in X$ be a point of $X$ lying over $s \in S$ with $\dim _ x(\text{Supp}(\mathcal{F}_ s)) = d$. Apply Lemma 38.3.2 to get $g : X' \to X$, $e : S' \to S$, $i : Z' \to X'$, and $\pi : Z' \to Y'$. Observe that $Y' \to S'$ is a smooth morphism of affines with geometrically irreducible fibres of dimension $d$. Because the problem is ├ętale local it suffices to prove the theorem for $g^*\mathcal{F}/X'/S'$. Because $i : Z' \to X'$ is a closed immersion of finite presentation (and since strict transform commutes with affine pushforward, see Divisors, Lemma 31.33.4) it suffices to prove the flattening result for $\mathcal{G}$. Since $\pi $ is finite (hence also affine) it suffices to prove the flattening result for $\pi _*\mathcal{G}/Y'/S'$. Thus we may assume that $X \to S$ is a smooth morphism of affines with geometrically irreducible fibres of dimension $d$.

Next, we apply a blowup as in Lemma 38.30.3. Doing so we reach the situation where there exists an open $V \subset X$ surjecting onto $S$ such that $\mathcal{F}|_ V$ is finite locally free. Let $\xi \in X$ be the generic point of $X_ s$. Let $r = \dim _{\kappa (\xi )} \mathcal{F}_\xi \otimes \kappa (\xi )$. Choose a map $\alpha : \mathcal{O}_ X^{\oplus r} \to \mathcal{F}$ which induces an isomorphism $\kappa (\xi )^{\oplus r} \to \mathcal{F}_\xi \otimes \kappa (\xi )$. Because $\mathcal{F}$ is locally free over $V$ we find an open neighbourhood $W$ of $\xi $ where $\alpha $ is an isomorphism. Shrink $S$ to an affine open neighbourhood of $s$ such that $W \to S$ is surjective. Say $\mathcal{F}$ is the quasi-coherent module associated to the $A$-module $N$. Since $\mathcal{F}$ is flat over $S$ at all generic points of fibres (in fact at all points of $W$), we see that

\[ \alpha _\mathfrak p : A_\mathfrak p^{\oplus r} \to N_\mathfrak p \]

is universally injective for all primes $\mathfrak p$ of $R$, see Lemma 38.10.1. Hence $\alpha $ is universally injective, see Algebra, Lemma 10.82.12. Set $\mathcal{H} = \mathop{\mathrm{Coker}}(\alpha )$. By Divisors, Lemma 31.33.7 we see that, given a $U$-admissible blowup $S' \to S$ the strict transforms of $\mathcal{F}'$ and $\mathcal{H}'$ fit into an exact sequence

\[ 0 \to \mathcal{O}_{X \times _ S S'}^{\oplus r} \to \mathcal{F}' \to \mathcal{H}' \to 0 \]

Hence Lemma 38.10.1 also shows that $\mathcal{F}'$ is flat at a point $x'$ if and only if $\mathcal{H}'$ is flat at that point. In particular $\mathcal{H}_ U$ is flat over $U$ and $\mathcal{H}_ U$ is a module of finite presentation. We may apply the induction hypothesis to $\mathcal{H}$ to see that there exists a $U$-admissible blowup such that the strict transform $\mathcal{H}'$ is flat as desired.

To finish the proof of the theorem we still have to show that $\mathcal{F}'$ is a module of finite presentation (after possibly another $U$-admissible blowup). This follows from Lemma 38.11.1 as we can assume $U \subset S$ is scheme theoretically dense (see third paragraph of Remark 38.30.1). This finishes the proof of the theorem. $\square$


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