The Stacks project

68.18 Strict transform

This section is the analogue of Divisors, Section 31.33. Let $S$ be a scheme, let $B$ be an algebraic space over $S$, and let $Z \subset B$ be a closed subspace. Let $b : B' \to B$ be the blowing up of $B$ in $Z$ and denote $E \subset B'$ the exceptional divisor $E = b^{-1}Z$. In the following we will often consider an algebraic space $X$ over $B$ and form the cartesian diagram

\[ \xymatrix{ \text{pr}_{B'}^{-1}E \ar[r] \ar[d] & X \times _ B B' \ar[r]_-{\text{pr}_ X} \ar[d]_{\text{pr}_{B'}} & X \ar[d]^ f \\ E \ar[r] & B' \ar[r] & B } \]

Since $E$ is an effective Cartier divisor (Lemma 68.17.4) we see that $\text{pr}_{B'}^{-1}E \subset X \times _ B B'$ is locally principal (Lemma 68.6.9). Thus the inclusion morphism of the complement of $\text{pr}_{B'}^{-1}E$ in $X \times _ B B'$ is affine and in particular quasi-compact (Lemma 68.6.3). Consequently, for a quasi-coherent $\mathcal{O}_{X \times _ B B'}$-module $\mathcal{G}$ the subsheaf of sections supported on $|\text{pr}_{B'}^{-1}E|$ is a quasi-coherent submodule, see Limits of Spaces, Definition 67.14.6. If $\mathcal{G}$ is a quasi-coherent sheaf of algebras, e.g., $\mathcal{G} = \mathcal{O}_{X \times _ B B'}$, then this subsheaf is an ideal of $\mathcal{G}$.

Definition 68.18.1. With $Z \subset B$ and $f : X \to B$ as above.

  1. Given a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the strict transform of $\mathcal{F}$ with respect to the blowup of $B$ in $Z$ is the quotient $\mathcal{F}'$ of $\text{pr}_ X^*\mathcal{F}$ by the submodule of sections supported on $|\text{pr}_{B'}^{-1}E|$.

  2. The strict transform of $X$ is the closed subspace $X' \subset X \times _ B B'$ cut out by the quasi-coherent ideal of sections of $\mathcal{O}_{X \times _ B B'}$ supported on $|\text{pr}_{B'}^{-1}E|$.

Note that taking the strict transform along a blowup depends on the closed subspace used for the blowup (and not just on the morphism $B' \to B$).

Lemma 68.18.2 (Étale localization and strict transform). In the situation of Definition 68.18.1. Let

\[ \xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ V \ar[r] & B } \]

be a commutative diagram of morphisms with $U$ and $V$ schemes and étale horizontal arrows. Let $V' \to V$ be the blowup of $V$ in $Z \times _ B V$. Then

  1. $V' = V \times _ B B'$ and the maps $V' \to B'$ and $U \times _ V V' \to X \times _ B B'$ are étale,

  2. the strict transform $U'$ of $U$ relative to $V' \to V$ is equal to $X' \times _ X U$ where $X'$ is the strict transform of $X$ relative to $B' \to B$, and

  3. for a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the restriction of the strict transform $\mathcal{F}'$ to $U \times _ V V'$ is the strict transform of $\mathcal{F}|_ U$ relative to $V' \to V$.

Proof. Part (1) follows from the fact that blowup commutes with flat base change (Lemma 68.17.3), the fact that étale morphisms are flat, and that the base change of an étale morphism is étale. Part (3) then follows from the fact that taking the sheaf of sections supported on a closed commutes with pullback by étale morphisms, see Limits of Spaces, Lemma 67.14.5. Part (2) follows from (3) applied to $\mathcal{F} = \mathcal{O}_ X$. $\square$

Lemma 68.18.3. In the situation of Definition 68.18.1.

  1. The strict transform $X'$ of $X$ is the blowup of $X$ in the closed subspace $f^{-1}Z$ of $X$.

  2. For a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the strict transform $\mathcal{F}'$ is canonically isomorphic to the pushforward along $X' \to X \times _ B B'$ of the strict transform of $\mathcal{F}$ relative to the blowing up $X' \to X$.

Proof. Let $X'' \to X$ be the blowup of $X$ in $f^{-1}Z$. By the universal property of blowing up (Lemma 68.17.5) there exists a commutative diagram

\[ \xymatrix{ X'' \ar[r] \ar[d] & X \ar[d] \\ B' \ar[r] & B } \]

whence a morphism $i : X'' \to X \times _ B B'$. The first assertion of the lemma is that $i$ is a closed immersion with image $X'$. The second assertion of the lemma is that $\mathcal{F}' = i_*\mathcal{F}''$ where $\mathcal{F}''$ is the strict transform of $\mathcal{F}$ with respect to the blowing up $X'' \to X$. We can check these assertions étale locally on $X$, hence we reduce to the case of schemes (Divisors, Lemma 31.33.2). Some details omitted. $\square$

Lemma 68.18.4. In the situation of Definition 68.18.1.

  1. If $X$ is flat over $B$ at all points lying over $Z$, then the strict transform of $X$ is equal to the base change $X \times _ B B'$.

  2. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $\mathcal{F}$ is flat over $B$ at all points lying over $Z$, then the strict transform $\mathcal{F}'$ of $\mathcal{F}$ is equal to the pullback $\text{pr}_ X^*\mathcal{F}$.

Proof. Omitted. Hint: Follows from the case of schemes (Divisors, Lemma 31.33.3) by étale localization (Lemma 68.18.2). $\square$

Lemma 68.18.5. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $Z \subset B$ be a closed subspace. Let $b : B' \to B$ be the blowing up of $Z$ in $B$. Let $g : X \to Y$ be an affine morphism of spaces over $B$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $g' : X \times _ B B' \to Y \times _ B B'$ be the base change of $g$. Let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$ relative to $b$. Then $g'_*\mathcal{F}'$ is the strict transform of $g_*\mathcal{F}$.

Proof. Omitted. Hint: Follows from the case of schemes (Divisors, Lemma 31.33.4) by étale localization (Lemma 68.18.2). $\square$

Lemma 68.18.6. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $Z \subset B$ be a closed subspace. Let $D \subset B$ be an effective Cartier divisor. Let $Z' \subset B$ be the closed subspace cut out by the product of the ideal sheaves of $Z$ and $D$. Let $B' \to B$ be the blowup of $B$ in $Z$.

  1. The blowup of $B$ in $Z'$ is isomorphic to $B' \to B$.

  2. Let $f : X \to B$ be a morphism of algebraic spaces and let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If the subsheaf of $\mathcal{F}$ of sections supported on $|f^{-1}D|$ is zero, then the strict transform of $\mathcal{F}$ relative to the blowing up in $Z$ agrees with the strict transform of $\mathcal{F}$ relative to the blowing up of $B$ in $Z'$.

Proof. Omitted. Hint: Follows from the case of schemes (Divisors, Lemma 31.33.5) by étale localization (Lemma 68.18.2). $\square$

Lemma 68.18.7. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $Z \subset B$ be a closed subspace. Let $b : B' \to B$ be the blowing up with center $Z$. Let $Z' \subset B'$ be a closed subspace. Let $B'' \to B'$ be the blowing up with center $Z'$. Let $Y \subset B$ be a closed subscheme such that $|Y| = |Z| \cup |b|(|Z'|)$ and the composition $B'' \to B$ is isomorphic to the blowing up of $B$ in $Y$. In this situation, given any scheme $X$ over $B$ and $\mathcal{F} \in \mathit{QCoh}(\mathcal{O}_ X)$ we have

  1. the strict transform of $\mathcal{F}$ with respect to the blowing up of $B$ in $Y$ is equal to the strict transform with respect to the blowup $B'' \to B'$ in $Z'$ of the strict transform of $\mathcal{F}$ with respect to the blowup $B' \to B$ of $B$ in $Z$, and

  2. the strict transform of $X$ with respect to the blowing up of $B$ in $Y$ is equal to the strict transform with respect to the blowup $B'' \to B'$ in $Z'$ of the strict transform of $X$ with respect to the blowup $B' \to B$ of $B$ in $Z$.

Proof. Omitted. Hint: Follows from the case of schemes (Divisors, Lemma 31.33.6) by étale localization (Lemma 68.18.2). $\square$

Lemma 68.18.8. In the situation of Definition 68.18.1. Suppose that

\[ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 \]

is an exact sequence of quasi-coherent sheaves on $X$ which remains exact after any base change $T \to B$. Then the strict transforms of $\mathcal{F}_ i'$ relative to any blowup $B' \to B$ form a short exact sequence $0 \to \mathcal{F}'_1 \to \mathcal{F}'_2 \to \mathcal{F}'_3 \to 0$ too.

Proof. Omitted. Hint: Follows from the case of schemes (Divisors, Lemma 31.33.7) by étale localization (Lemma 68.18.2). $\square$

Lemma 68.18.9. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ B$-module. Let $Z_ k \subset S$ be the closed subscheme cut out by $\text{Fit}_ k(\mathcal{F})$, see Section 68.5. Let $B' \to B$ be the blowup of $B$ in $Z_ k$ and let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$. Then $\mathcal{F}'$ can locally be generated by $\leq k$ sections.

Proof. Omitted. Follows from the case of schemes (Divisors, Lemma 31.35.1) by étale localization (Lemma 68.18.2). $\square$

Lemma 68.18.10. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ B$-module. Let $Z_ k \subset S$ be the closed subscheme cut out by $\text{Fit}_ k(\mathcal{F})$, see Section 68.5. Assume that $\mathcal{F}$ is locally free of rank $k$ on $B \setminus Z_ k$. Let $B' \to B$ be the blowup of $B$ in $Z_ k$ and let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$. Then $\mathcal{F}'$ is locally free of rank $k$.

Proof. Omitted. Follows from the case of schemes (Divisors, Lemma 31.35.2) by étale localization (Lemma 68.18.2). $\square$


Comments (2)

Comment #2060 by Sasha on

I cannot understand the last two sentences of the first paragraph (just before Def. 59.7.1). For instance, what does it mean "sections supported on " Also in the least sentence what means "this subsheaf is an ideal of "?

Comment #2090 by on

The "subsheaf of sections supported on" is defined in Defn 67.14.6. I have replaced the reference to the lemma to a reference to this definition, see here. In the last sentence of that paragraph we are just saying that the submodule of sections supported on a closed of a (quasi-coherent) sheaf of algebras, is automatically an ideal of that sheaf of algebras. This seems quite obvious to me, but if it isn't then we'll just add a lemma and give a detailed proof. Let me know if you think we should do this.

Also: It may be a good idea to read the version of this section dealing with schemes first...


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0861. Beware of the difference between the letter 'O' and the digit '0'.