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31.33 Strict transform

In this section we briefly discuss strict transform under blowing up. Let $S$ be a scheme and let $Z \subset S$ be a closed subscheme. Let $b : S' \to S$ be the blowing up of $S$ in $Z$ and denote $E \subset S'$ the exceptional divisor $E = b^{-1}Z$. In the following we will often consider a scheme $X$ over $S$ and form the cartesian diagram

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

Since $E$ is an effective Cartier divisor (Lemma 31.32.4) we see that $\text{pr}_{S'}^{-1}E \subset X \times _ S S'$ is locally principal (Lemma 31.13.11). Thus the complement of $\text{pr}_{S'}^{-1}E$ in $X \times _ S S'$ is retrocompact (Lemma 31.13.3). Consequently, for a quasi-coherent $\mathcal{O}_{X \times _ S S'}$-module $\mathcal{G}$ the subsheaf of sections supported on $\text{pr}_{S'}^{-1}E$ is a quasi-coherent submodule, see Properties, Lemma 28.24.5. If $\mathcal{G}$ is a quasi-coherent sheaf of algebras, e.g., $\mathcal{G} = \mathcal{O}_{X \times _ S S'}$, then this subsheaf is an ideal of $\mathcal{G}$.

Definition 31.33.1. With $Z \subset S$ and $f : X \to S$ 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 $S$ in $Z$ is the quotient $\mathcal{F}'$ of $\text{pr}_ X^*\mathcal{F}$ by the submodule of sections supported on $\text{pr}_{S'}^{-1}E$.

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

Note that taking the strict transform along a blowup depends on the closed subscheme used for the blowup (and not just on the morphism $S' \to S$). This notion is often used for closed subschemes of $S$. It turns out that the strict transform of $X$ is a blowup of $X$.

Lemma 31.33.2. In the situation of Definition 31.33.1.

  1. The strict transform $X'$ of $X$ is the blowup of $X$ in the closed subscheme $f^{-1}Z$ 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 $X' \to X \times _ S S'$ 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 31.32.5) there exists a commutative diagram

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

whence a morphism $X'' \to X \times _ S S'$. Thus the first assertion is that this morphism is a closed immersion with image $X'$. The question is local on $X$. Thus we may assume $X$ and $S$ are affine. Say that $S = \mathop{\mathrm{Spec}}(A)$, $X = \mathop{\mathrm{Spec}}(B)$, and $Z$ is cut out by the ideal $I \subset A$. Set $J = IB$. The map $B \otimes _ A \bigoplus _{n \geq 0} I^ n \to \bigoplus _{n \geq 0} J^ n$ defines a closed immersion $X'' \to X \times _ S S'$, see Constructions, Lemmas 27.11.6 and 27.11.5. We omit the verification that this morphism is the same as the one constructed above from the universal property. Pick $a \in I$ corresponding to the affine open $\mathop{\mathrm{Spec}}(A[\frac{I}{a}]) \subset S'$, see Lemma 31.32.2. The inverse image of $\mathop{\mathrm{Spec}}(A[\frac{I}{a}])$ in the strict transform $X'$ of $X$ is the spectrum of

\[ B' = (B \otimes _ A A[\textstyle {\frac{I}{a}}])/a\text{-power-torsion} \]

see Properties, Lemma 28.24.5. On the other hand, letting $b \in J$ be the image of $a$ we see that $\mathop{\mathrm{Spec}}(B[\frac{J}{b}])$ is the inverse image of $\mathop{\mathrm{Spec}}(A[\frac{I}{a}])$ in $X''$. By Algebra, Lemma 10.70.3 the open $\mathop{\mathrm{Spec}}(B[\frac{J}{b}])$ maps isomorphically to the open subscheme $\text{pr}_{S'}^{-1}(\mathop{\mathrm{Spec}}(A[\frac{I}{a}]))$ of $X'$. Thus $X'' \to X'$ is an isomorphism.

In the notation above, let $\mathcal{F}$ correspond to the $B$-module $N$. The strict transform of $\mathcal{F}$ corresponds to the $B \otimes _ A A[\frac{I}{a}]$-module

\[ N' = (N \otimes _ A A[\textstyle {\frac{I}{a}}])/a\text{-power-torsion} \]

see Properties, Lemma 28.24.5. The strict transform of $\mathcal{F}$ relative to the blowup of $X$ in $f^{-1}Z$ corresponds to the $B[\frac{J}{b}]$-module $N \otimes _ B B[\frac{J}{b}]/b\text{-power-torsion}$. In exactly the same way as above one proves that these two modules are isomorphic. Details omitted. $\square$

Lemma 31.33.3. In the situation of Definition 31.33.1.

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

  2. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $\mathcal{F}$ is flat over $S$ 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. We will prove part (2) as it implies part (1) by the definition of the strict transform of a scheme over $S$. The question is local on $X$. Thus we may assume that $S = \mathop{\mathrm{Spec}}(A)$, $X = \mathop{\mathrm{Spec}}(B)$, and that $\mathcal{F}$ corresponds to the $B$-module $N$. Then $\mathcal{F}'$ over the open $\mathop{\mathrm{Spec}}(B \otimes _ A A[\frac{I}{a}])$ of $X \times _ S S'$ corresponds to the module

\[ N' = (N \otimes _ A A[\textstyle {\frac{I}{a}}])/a\text{-power-torsion} \]

see Properties, Lemma 28.24.5. Thus we have to show that the $a$-power-torsion of $N \otimes _ A A[\frac{I}{a}]$ is zero. Let $y \in N \otimes _ A A[\frac{I}{a}]$ with $a^ n y = 0$. If $\mathfrak q \subset B$ is a prime and $a \not\in \mathfrak q$, then $y$ maps to zero in $(N \otimes _ A A[\frac{I}{a}])_\mathfrak q$. on the other hand, if $a \in \mathfrak q$, then $N_\mathfrak q$ is a flat $A$-module and we see that $N_\mathfrak q \otimes _ A A[\frac{I}{a}] =(N \otimes _ A A[\frac{I}{a}])_\mathfrak q$ has no $a$-power torsion (as $A[\frac{I}{a}]$ doesn't). Hence $y$ maps to zero in this localization as well. We conclude that $y$ is zero by Algebra, Lemma 10.23.1. $\square$

Lemma 31.33.4. Let $S$ be a scheme. Let $Z \subset S$ be a closed subscheme. Let $b : S' \to S$ be the blowing up of $Z$ in $S$. Let $g : X \to Y$ be an affine morphism of schemes over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $g' : X \times _ S S' \to Y \times _ S S'$ 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. Observe that $g'_*\text{pr}_ X^*\mathcal{F} = \text{pr}_ Y^*g_*\mathcal{F}$ by Cohomology of Schemes, Lemma 30.5.1. Let $\mathcal{K} \subset \text{pr}_ X^*\mathcal{F}$ be the subsheaf of sections supported in the inverse image of $Z$ in $X \times _ S S'$. By Properties, Lemma 28.24.7 the pushforward $g'_*\mathcal{K}$ is the subsheaf of sections of $\text{pr}_ Y^*g_*\mathcal{F}$ supported in the inverse image of $Z$ in $Y \times _ S S'$. As $g'$ is affine (Morphisms, Lemma 29.11.8) we see that $g'_*$ is exact, hence we conclude. $\square$

Lemma 31.33.5. Let $S$ be a scheme. Let $Z \subset S$ be a closed subscheme. Let $D \subset S$ be an effective Cartier divisor. Let $Z' \subset S$ be the closed subscheme cut out by the product of the ideal sheaves of $Z$ and $D$. Let $S' \to S$ be the blowup of $S$ in $Z$.

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

  2. Let $f : X \to S$ be a morphism of schemes and let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $\mathcal{F}$ has no nonzero local sections supported in $f^{-1}D$, 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 $S$ in $Z'$.

Proof. The first statement follows on combining Lemmas 31.32.12 and 31.32.7. Using Lemma 31.32.2 the second statement translates into the following algebra problem. Let $A$ be a ring, $I \subset A$ an ideal, $x \in A$ a nonzerodivisor, and $a \in I$. Let $M$ be an $A$-module whose $x$-torsion is zero. To show: the $a$-power torsion in $M \otimes _ A A[\frac{I}{a}]$ is equal to the $xa$-power torsion. The reason for this is that the kernel and cokernel of the map $A \to A[\frac{I}{a}]$ is $a$-power torsion, so this map becomes an isomorphism after inverting $a$. Hence the kernel and cokernel of $M \to M \otimes _ A A[\frac{I}{a}]$ are $a$-power torsion too. This implies the result. $\square$

Lemma 31.33.6. Let $S$ be a scheme. Let $Z \subset S$ be a closed subscheme. Let $b : S' \to S$ be the blowing up with center $Z$. Let $Z' \subset S'$ be a closed subscheme. Let $S'' \to S'$ be the blowing up with center $Z'$. Let $Y \subset S$ be a closed subscheme such that $Y = Z \cup b(Z')$ set theoretically and the composition $S'' \to S$ is isomorphic to the blowing up of $S$ in $Y$. In this situation, given any scheme $X$ over $S$ 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 $S$ in $Y$ is equal to the strict transform with respect to the blowup $S'' \to S'$ in $Z'$ of the strict transform of $\mathcal{F}$ with respect to the blowup $S' \to S$ of $S$ in $Z$, and

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

Proof. Let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$ with respect to the blowup $S' \to S$ of $S$ in $Z$. Let $\mathcal{F}''$ be the strict transform of $\mathcal{F}'$ with respect to the blowup $S'' \to S'$ of $S'$ in $Z'$. Let $\mathcal{G}$ be the strict transform of $\mathcal{F}$ with respect to the blowup $S'' \to S$ of $S$ in $Y$. We also label the morphisms

\[ \xymatrix{ X \times _ S S'' \ar[r]_ q \ar[d]^{f''} & X \times _ S S' \ar[r]_ p \ar[d]^{f'} & X \ar[d]^ f \\ S'' \ar[r] & S' \ar[r] & S } \]

By definition there is a surjection $p^*\mathcal{F} \to \mathcal{F}'$ and a surjection $q^*\mathcal{F}' \to \mathcal{F}''$ which combine by right exactness of $q^*$ to a surjection $(p \circ q)^*\mathcal{F} \to \mathcal{F}''$. Also we have the surjection $(p \circ q)^*\mathcal{F} \to \mathcal{G}$. Thus it suffices to prove that these two surjections have the same kernel.

The kernel of the surjection $p^*\mathcal{F} \to \mathcal{F}'$ is supported on $(f \circ p)^{-1}Z$, so this map is an isomorphism at points in the complement. Hence the kernel of $q^*p^*\mathcal{F} \to q^*\mathcal{F}'$ is supported on $(f \circ p \circ q)^{-1}Z$. The kernel of $q^*\mathcal{F}' \to \mathcal{F}''$ is supported on $(f' \circ q)^{-1}Z'$. Combined we see that the kernel of $(p \circ q)^*\mathcal{F} \to \mathcal{F}''$ is supported on $(f \circ p \circ q)^{-1}Z \cup (f' \circ q)^{-1}Z' = (f \circ p \circ q)^{-1}Y$. By construction of $\mathcal{G}$ we see that we obtain a factorization $(p \circ q)^*\mathcal{F} \to \mathcal{F}'' \to \mathcal{G}$. To finish the proof it suffices to show that $\mathcal{F}''$ has no nonzero (local) sections supported on $(f \circ p \circ q)^{-1}(Y) = (f \circ p \circ q)^{-1}Z \cup (f' \circ q)^{-1}Z'$. This follows from Lemma 31.33.5 applied to $\mathcal{F}'$ on $X \times _ S S'$ over $S'$, the closed subscheme $Z'$ and the effective Cartier divisor $b^{-1}Z$. $\square$

Lemma 31.33.7. In the situation of Definition 31.33.1. 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 $T \to S$. Then the strict transforms of $\mathcal{F}_ i'$ relative to any blowup $S' \to S$ form a short exact sequence $0 \to \mathcal{F}'_1 \to \mathcal{F}'_2 \to \mathcal{F}'_3 \to 0$ too.

Proof. We may localize on $S$ and $X$ and assume both are affine. Then we may push $\mathcal{F}_ i$ to $S$, see Lemma 31.33.4. We may assume that our blowup is the morphism $1 : S \to S$ associated to an effective Cartier divisor $D \subset S$. Then the translation into algebra is the following: Suppose that $A$ is a ring and $0 \to M_1 \to M_2 \to M_3 \to 0$ is a universally exact sequence of $A$-modules. Let $a\in A$. Then the sequence

\[ 0 \to M_1/a\text{-power torsion} \to M_2/a\text{-power torsion} \to M_3/a\text{-power torsion} \to 0 \]

is exact too. Namely, surjectivity of the last map and injectivity of the first map are immediate. The problem is exactness in the middle. Suppose that $x \in M_2$ maps to zero in $M_3/a\text{-power torsion}$. Then $y = a^ n x \in M_1$ for some $n$. Then $y$ maps to zero in $M_2/a^ nM_2$. Since $M_1 \to M_2$ is universally injective we see that $y$ maps to zero in $M_1/a^ nM_1$. Thus $y = a^ n z$ for some $z \in M_1$. Thus $a^ n(x - y) = 0$. Hence $y$ maps to the class of $x$ in $M_2/a\text{-power torsion}$ as desired. $\square$

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