The Stacks project

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. $\square$

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