The Stacks project

Proof. Note that if $V \subset X$ is some open such that $H^ i(E)|_ V$ is finite locally free for all $i \in \mathbf{Z}$ then $V \subset U$. Let $x \in U$. We will show that an open neighbourhood of $x$ is contained in $U$ and that $H^ i(E)$ is finite locally free on this neighbourhood for all $i$. This will finish the proof. During the proof we may (finitely many times) replace $X$ by an open neighbourhood of $x$. Hence we may assume $E$ is represented by a strictly perfect complex $\mathcal{E}^\bullet $. Say $\mathcal{E}^ i = 0$ for $i \not\in [a, b]$. We will prove the result by induction on $b - a$. The module $H^ b(E) = \mathop{\mathrm{Coker}}(d^{b - 1} : \mathcal{E}^{b - 1} \to \mathcal{E}^ b)$ is of finite presentation. Since $H^ b(E)_ x$ is finite free, we conclude $H^ b(E)$ is finite free in an open neighbourhood of $x$ by Modules, Lemma 17.11.6. Thus after replacing $X$ by a (possibly smaller) open neighbourhood we may assume we have a direct sum decomposition $\mathcal{E}^ b = \mathop{\mathrm{Im}}(d^{b - 1}) \oplus H^ b(E)$ and $H^ b(E)$ is finite free, see Lemma 20.46.5. Doing the same argument again, we see that we may assume $\mathcal{E}^{b - 1} = \mathop{\mathrm{Ker}}(d^{b - 1}) \oplus \mathop{\mathrm{Im}}(d^{b - 1})$. The complex $\mathcal{E}^ a \to \ldots \to \mathcal{E}^{b - 2} \to \mathop{\mathrm{Ker}}(d^{b - 1})$ is a strictly perfect complex representing a perfect object $E'$ with $H^ i(E) = H^ i(E')$ for $i \not= b$. Hence we conclude by our induction hypothesis. $\square$