Lemma 38.41.2. Let $X$ be a scheme. Let $\mathcal{F}$ be a perfect $\mathcal{O}_ X$-module of tor dimension $\leq 1$. Let $U \subset X$ be a scheme theoretically dense open such that $\mathcal{F}|_ U$ is finite locally free of constant rank $r$. Then there exists a $U$-admissible blowup $b : X' \to X$ such that there is a canonical short exact sequence
\[ 0 \to \mathcal{K} \to b^*\mathcal{F} \to \mathcal{Q} \to 0 \]
where $\mathcal{Q}$ is finite locally free of rank $r$ and $\mathcal{K}$ is a perfect $\mathcal{O}_ X$-module of tor dimension $\leq 1$ whose restriction to $U$ is zero.
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