Lemma 38.41.4. Let $X$ be a scheme. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a homorphism of perfect $\mathcal{O}_ X$-modules of tor dimension $\leq 1$. Let $U \subset X$ be a scheme theoretically dense open such that $\mathcal{F}|_ U = 0$ and $\mathcal{G}|_ U = 0$. Then there is a $U$-admissible blowup $b : X' \to X$ such that the kernel, image, and cokernel of $b^*\varphi $ are perfect $\mathcal{O}_{X'}$-modules of tor dimension $\leq 1$.
Proof. The assumptions tell us that the object $(\mathcal{F} \to \mathcal{G})$ of $D(\mathcal{O}_ X)$ is perfect. Thus we get a $U$-admissible blowup that works for the cokernel and kernel by Lemmas 38.40.2 and 38.41.1 (to see what the complex looks like after pullback). The image is the kernel of the cokernel and hence is going to be perfect of tor dimension $\leq 1$ as well. $\square$
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