The Stacks project

73.38 Blowing up and flatness

Instead of redoing the work in More on Flatness, Section 38.30 we prove an analogue of More on Flatness, Lemma 38.30.5 which tells us that the problem of finding a suitable blowup is often étale local on the base.

Lemma 73.38.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\varphi : W \to X$ be a quasi-compact separated étale morphism. Let $U \subset X$ be a quasi-compact open subspace. Let $\mathcal{I} \subset \mathcal{O}_ U$ be a finite type quasi-coherent sheaf of ideals such that $V(\mathcal{I}) \cap \varphi ^{-1}(U) = \emptyset $. Then there exists a finite type quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_ X$ such that

  1. $V(\mathcal{J}) \cap U = \emptyset $, and

  2. $\varphi ^{-1}(\mathcal{J})\mathcal{O}_ W = \mathcal{I} \mathcal{I}'$ for some finite type quasi-coherent ideal $\mathcal{I}' \subset \mathcal{O}_ W$.

Proof. Choose a factorization $W \to Y \to X$ where $j : W \to Y$ is a quasi-compact open immersion and $\pi : Y \to X$ is a finite morphism of finite presentation (Lemma 73.34.4). Let $V = j(W) \cup \pi ^{-1}(U) \subset Y$. Note that $\mathcal{I}$ on $W \cong j(W)$ and $\mathcal{O}_{\pi ^{-1}(U)}$ glue to a finite type quasi-coherent sheaf of ideals $\mathcal{I}_1 \subset \mathcal{O}_ V$. By Limits of Spaces, Lemma 67.9.8 there exists a finite type quasi-coherent sheaf of ideals $\mathcal{I}_2 \subset \mathcal{O}_ Y$ such that $\mathcal{I}_2|_ V = \mathcal{I}_1$. In other words, $\mathcal{I}_2 \subset \mathcal{O}_ Y$ is a finite type quasi-coherent sheaf of ideals such that $V(\mathcal{I}_2)$ is disjoint from $\pi ^{-1}(U)$ and $j^{-1}\mathcal{I}_2 = \mathcal{I}$. Denote $i : Z \to Y$ the corresponding closed immersion which is of finite presentation (Morphisms of Spaces, Lemma 64.28.12). In particular the composition $\tau = \pi \circ i : Z \to X$ is finite and of finite presentation (Morphisms of Spaces, Lemmas 64.28.2 and 64.45.4).

Let $\mathcal{F} = \tau _*\mathcal{O}_ Z$ which we think of as a quasi-coherent $\mathcal{O}_ X$-module. By Descent on Spaces, Lemma 71.5.7 we see that $\mathcal{F}$ is a finitely presented $\mathcal{O}_ X$-module. Let $\mathcal{J} = \text{Fit}_0(\mathcal{F})$. (Insert reference to fitting modules on ringed topoi here.) This is a finite type quasi-coherent sheaf of ideals on $X$ (as $\mathcal{F}$ is of finite presentation, see More on Algebra, Lemma 15.8.4). Part (1) of the lemma holds because $|\tau |(|Z|) \cap |U| = \emptyset $ by our choice of $\mathcal{I}_2$ and because the $0$th Fitting ideal of the trivial module equals the structure sheaf. To prove (2) note that $\varphi ^{-1}(\mathcal{J})\mathcal{O}_ W = \text{Fit}_0(\varphi ^*\mathcal{F})$ because taking Fitting ideals commutes with base change. On the other hand, as $\varphi : W \to X$ is separated and étale we see that $(1, j) : W \to W \times _ X Y$ is an open and closed immersion. Hence $W \times _ Y Z = V(\mathcal{I}) \amalg Z'$ for some finite and finitely presented morphism of algebraic spaces $\tau ' : Z' \to W$. Thus we see that

\begin{align*} \text{Fit}_0(\varphi ^*\mathcal{F}) & = \text{Fit}_0((W \times _ Y Z \to W)_*\mathcal{O}_{W \times _ Y Z}) \\ & = \text{Fit}_0(\mathcal{O}_ W/\mathcal{I}) \cdot \text{Fit}_0(\tau '_*\mathcal{O}_{Z'}) \\ & = \mathcal{I} \cdot \text{Fit}_0(\tau '_*\mathcal{O}_{Z'}) \end{align*}

the second equality by More on Algebra, Lemma 15.8.4 translated in sheaves on ringed topoi. Setting $\mathcal{I}' = \text{Fit}_0(\tau '_*\mathcal{O}_{Z'})$ finishes the proof of the lemma. $\square$

Theorem 73.38.2. 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 68.18.2) 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 73.38.1. 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 68.17.3). 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 68.17.10). 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 68.18.2) 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 68.18.7). 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 68.18.4). 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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