The Stacks project

Definition 65.13.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F$ be an algebraic space over $S$. Let $\Delta : F \to F \times F$ be the diagonal morphism.

  1. We say $F$ is separated over $S$ if $\Delta $ is a closed immersion.

  2. We say $F$ is locally separated over $S$1 if $\Delta $ is an immersion.

  3. We say $F$ is quasi-separated over $S$ if $\Delta $ is quasi-compact.

  4. We say $F$ is Zariski locally quasi-separated over $S$2 if there exists a Zariski covering $F = \bigcup _{i \in I} F_ i$ such that each $F_ i$ is quasi-separated.

[1] In the literature this often refers to quasi-separated and locally separated algebraic spaces.
[2] This definition was suggested by B. Conrad.

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