Lemma 38.40.2. Let $X$ be a scheme. Let $E \in D(\mathcal{O}_ X)$ be perfect. Let $U \subset X$ be a scheme theoretically dense open subscheme such that $H^ i(E|_ U)$ is finite locally free of constant rank $r_ i$ for all $i \in \mathbf{Z}$. Then there exists a $U$-admissible blowup $b : X' \to X$ such that $H^ i(Lb^*E)$ is a perfect $\mathcal{O}_{X'}$-module of tor dimension $\leq 1$ for all $i \in \mathbf{Z}$.
Proof. We will construct and study the blowup affine locally. Namely, suppose that $V \subset X$ is an affine open subscheme such that $E|_ V$ can be represented by the complex
\[ \mathcal{O}_ V^{\oplus n_ a} \xrightarrow {d_ a} \ldots \xrightarrow {d_{b - 1}} \mathcal{O}_ V^{\oplus n_ b} \]
Set $k_ i = r_{i + 1} - r_{i + 2} + \ldots + (-1)^{b - i + 1}r_ b$. A computation which we omit show that over $U \cap V$ the rank of $d_ i$ is
\[ \rho _ i = - k_ i + n_{i + 1} - n_{i + 2} + \ldots + (-1)^{b - i + 1}n_ b \]
in the sense that the cokernel of $d_ i$ is finite locally free of rank $n_{i + 1} - \rho _ i$. Let $\mathcal{I}_ i \subset \mathcal{O}_ V$ be the ideal generated by the minors of size $\rho _ i \times \rho _ i$ in the matrix of $d_ i$.
On the one hand, comparing with Lemma 38.40.1 we see the ideal $\mathcal{I}_ i$ corresponds to the global ideal $\text{Fit}_{i, k_ i}(E)$ which was shown to be independent of the choice of the complex representing $E|_ V$. On the other hand, $\mathcal{I}_ i$ is the $(n_{i + 1} - \rho _ i)$th Fitting ideal of $\mathop{\mathrm{Coker}}(d_ i)$. Please keep this in mind.
We let $b : X' \to X$ be the blowing up in the product of the ideals $\text{Fit}_{i, k_ i}(E)$; this makes sense as locally on $X$ almost all of these ideals are equal to the unit ideal (see above). This blowup dominates the blowups $b_ i : X'_ i \to X$ in the ideals $\text{Fit}_{i, k_ i}(E)$, see Divisors, Lemma 31.32.12. By Divisors, Lemma 31.35.3 each $b_ i$ is a $U$-admissible blowup. It follows that $b$ is a $U$-admissible blowup (tiny detail omitted; compare with the proof of Divisors, Lemma 31.34.4). Finally, $U$ is still a scheme theoretically dense open subscheme of $X'$. Thus after replacing $X$ by $X'$ we end up in the situation discussed in the next paragraph.
Assume $\text{Fit}_{i, k_ i}(E)$ is an invertible ideal for all $i$. Choose an affine open $V$ and a complex of finite free modules representing $E|_ V$ as above. It follows from Divisors, Lemma 31.35.3 that $\mathop{\mathrm{Coker}}(d_ i)$ has tor dimension $\leq 1$. Thus $\mathop{\mathrm{Im}}(d_ i)$ is finite locally free as the kernel of a map from a finite locally free module to a finitely presented module of tor dimension $\leq 1$. Hence $\mathop{\mathrm{Ker}}(d_ i)$ is finite locally free as well (same argument). Thus the short exact sequence
\[ 0 \to \mathop{\mathrm{Im}}(d_{i - 1}) \to \mathop{\mathrm{Ker}}(d_ i) \to H^ i(E)|_ V \to 0 \]
shows what we want and the proof is complete. $\square$
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