The Stacks project

Lemma 75.26.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \in D(\mathcal{O}_ X)$ be perfect. Given $i, r \in \mathbf{Z}$, there exists an open subspace $U \subset X$ characterized by the following

  1. $E|_ U \cong H^ i(E|_ U)[-i]$ and $H^ i(E|_ U)$ is a locally free $\mathcal{O}_ U$-module of rank $r$,

  2. a morphism $f : Y \to X$ factors through $U$ if and only if $Lf^*E$ is isomorphic to a locally free module of rank $r$ placed in degree $i$.

Proof. Omitted. Hints: Follows from the case of schemes by étale localization. See Derived Categories of Schemes, Lemma 36.31.3. $\square$

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