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Tag 02X5

Chapter 53: Algebraic Spaces > Section 53.13: Separation conditions on algebraic spaces

Definition 53.13.2. Let $S$ be a scheme contained in $\textit{Sch}_{fppf}$. 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.

The code snippet corresponding to this tag is a part of the file spaces.tex and is located in lines 2099–2116 (see updates for more information).

\begin{definition}
\label{definition-separated}
Let $S$ be a scheme contained in $\Sch_{fppf}$.
Let $F$ be an algebraic space over $S$.
Let $\Delta : F \to F \times F$ be the diagonal morphism.
\begin{enumerate}
\item We say $F$ is {\it separated over $S$} if $\Delta$ is a closed immersion.
\item We say $F$ is {\it locally separated over $S$}\footnote{In the
literature this often refers to quasi-separated and
locally separated algebraic spaces.} if $\Delta$ is an
immersion.
\item We say $F$ is {\it quasi-separated over $S$} if $\Delta$ is quasi-compact.
\item We say $F$ is {\it Zariski locally quasi-separated over $S$}\footnote{This
definition was suggested by B.\ Conrad.} if there
exists a Zariski covering $F = \bigcup_{i \in I} F_i$ such that
each $F_i$ is quasi-separated.
\end{enumerate}
\end{definition}

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