The Stacks project

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. $\square$

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