Lemma 38.41.1. Let $X$ be a scheme. Let $\mathcal{F}$ be a perfect $\mathcal{O}_ X$-module of tor dimension $\leq 1$. For any blowup $b : X' \to X$ we have $Lb^*\mathcal{F} = b^*\mathcal{F}$ and $b^*\mathcal{F}$ is a perfect $\mathcal{O}_ X$-module of tor dimension $\leq 1$.
Proof. We may assume $X = \mathop{\mathrm{Spec}}(A)$ is affine and we may assume the $A$-module $M$ corresponding to $\mathcal{F}$ has a presentation
\[ 0 \to A^{\oplus m} \to A^{\oplus n} \to M \to 0 \]
Suppose $I \subset A$ is an ideal and $a \in I$. Recall that the affine blowup algebra $A[\frac{I}{a}]$ is a subring of $A_ a$. Since localization is exact we see that $A_ a^{\oplus m} \to A_ a^{\oplus n}$ is injective. Hence $A[\frac{I}{a}]^{\oplus m} \to A[\frac{I}{a}]^{\oplus n}$ is injective too. This proves the lemma. $\square$
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