Lemma 31.35.3. Let $X$ be a scheme. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_ X$-module. Let $U \subset X$ be a scheme theoretically dense open such that $\mathcal{F}|_ U$ is finite locally free of constant rank $r$. Then

the blowup $b : X' \to X$ of $X$ in the $r$th Fitting ideal of $\mathcal{F}$ is $U$-admissible,

the strict transform $\mathcal{F}'$ of $\mathcal{F}$ with respect to $b$ is locally free of rank $r$,

the kernel $\mathcal{K}$ of the surjection $b^*\mathcal{F} \to \mathcal{F}'$ is finitely presented and $\mathcal{K}|_ U = 0$,

$b^*\mathcal{F}$ and $\mathcal{K}$ are perfect $\mathcal{O}_{X'}$-modules of tor dimension $\leq 1$.

## Comments (0)