The Stacks project

76.39 Applications

In this section we apply the result on flattening by blowing up.

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. $\square$

Lemma 76.39.2. 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 proper, and

  4. $f^{-1}(U) \to U$ is finite locally free.

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

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. $\square$

Lemma 76.39.3. 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 proper, and

  4. $f^{-1}(U) \to U$ is an isomorphism.

Then there exists a $U$-admissible blowup $B' \to B$ such that the strict transform $X'$ of $X$ maps isomorphically to $B'$.

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. $\square$

Lemma 76.39.4. 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. $B$ quasi-compact and quasi-separated,

  2. $U$ is quasi-compact,

  3. $f$ is of finite type

  4. $f^{-1}(U) \to U$ is an isomorphism.

Then there exists a $U$-admissible blowup $B' \to B$ such that $U$ is scheme theoretically dense in $B'$ and such that the strict transform $X'$ of $X$ maps isomorphically to an open subspace of $B'$.

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. $\square$

The following lemma says that a modification can be dominated by a blowup.

Lemma 76.39.5. 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 proper,

  4. $f^{-1}(U) \to U$ us an isomorphism.

Then there exists a $U$-admissible blowup $B' \to B$ which dominates $X$, i.e., such that there exists a factorization $B' \to X \to B$ of the blowup morphism.

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. $\square$

Lemma 76.39.6. Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $U \subset W \subset Y$ be open subspaces. Let $f : X \to W$ and let $s : U \to X$ be morphisms such that $f \circ s = \text{id}_ U$. Assume

  1. $f$ is proper,

  2. $Y$ is quasi-compact and quasi-separated, and

  3. $U$ and $W$ are quasi-compact.

Then there exists a $U$-admissible blowup $b : Y' \to Y$ and a morphism $s' : b^{-1}(W) \to X$ extending $s$ with $f \circ s' = b|_{b^{-1}(W)}$.

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 $Y$. $\square$


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