The Stacks project

Situation 42.50.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $i : Z \to X$ be a closed immersion. Let $E \in D(\mathcal{O}_ X)$ be an object. Let $p \geq 0$. Assume

1. $E$ is a perfect object of $D(\mathcal{O}_ X)$,

2. the restriction $E|_{X \setminus Z}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z}$-module of rank $< p$ sitting in cohomological degree $0$, and

3. at least one¹ of the following is true: (a) $X$ is quasi-compact, (b) $X$ has quasi-compact irreducible components, (c) there exists a locally bounded complex of finite locally free $\mathcal{O}_ X$-modules representing $E$, or (d) there exists a morphism $X \to X'$ of schemes locally of finite type over $S$ such that $E$ is the pullback of a perfect object on $X'$ and the irreducible components of $X'$ are quasi-compact.