The Stacks project

Lemma 38.40.3. Let $X$ be an integral scheme. Let $E \in D(\mathcal{O}_ X)$ be perfect. Then there exists a nonempty open $U \subset X$ such that $H^ i(E|_ U)$ is finite locally free of constant rank $r_ i$ for all $i \in \mathbf{Z}$ and 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 strongly urge the reader to find their own proof of the existence of $U$. Let $\eta \in X$ be the generic point. The restriction of $E$ to $\eta $ is isomorphic in $D(\kappa (\eta ))$ to a finite complex $V^\bullet $ of finite dimensional vector spaces with zero differentials. Set $r_ i = \dim _{\kappa (\eta )} V^ i$. Then the perfect object $E'$ in $D(\mathcal{O}_ X)$ represented by the complex with terms $\mathcal{O}_ X^{\oplus r_ i}$ and zero differentials becomes isomorphic to $E$ after pulling back to $\eta $. Hence by Derived Categories of Schemes, Lemma 36.35.9 there is an open neighbourhood $U$ of $\eta $ such that $E|_ U$ and $E'|_ U$ are isomorphic. This proves the first assertion. The second follows from the first and Lemma 38.40.2 as any nonempty open is scheme theoretically dense in the integral scheme $X$. $\square$

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