The Stacks project

Definition 64.3.1. (Compare Spaces, Definition 63.13.2.) Consider a big fppf site $\mathit{Sch}_{fppf} = (\mathit{Sch}/\mathop{\mathrm{Spec}}(\mathbf{Z}))_{fppf}$. Let $X$ be an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Let $\Delta : X \to X \times X$ be the diagonal morphism.

  1. We say $X$ is separated if $\Delta $ is a closed immersion.

  2. We say $X$ is locally separated1 if $\Delta $ is an immersion.

  3. We say $X$ is quasi-separated if $\Delta $ is quasi-compact.

  4. We say $X$ is Zariski locally quasi-separated2 if there exists a Zariski covering $X = \bigcup _{i \in I} X_ i$ (see Spaces, Definition 63.12.5) such that each $X_ i$ is quasi-separated.

Let $S$ is a scheme contained in $\mathit{Sch}_{fppf}$, and let $X$ be an algebraic space over $S$. Then we say $X$ is separated, locally separated, quasi-separated, or Zariski locally quasi-separated if $X$ viewed as an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ (see Spaces, Definition 63.16.2) has the corresponding property.

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

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