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Tag 02X4

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

Lemma 53.13.1. 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. Then

  1. $\Delta$ is locally of finite type,
  2. $\Delta$ is a monomorphism,
  3. $\Delta$ is separated, and
  4. $\Delta$ is locally quasi-finite.

Proof. Let $F = U/R$ be a presentation of $F$. As in the proof of Lemma 53.10.4 the diagram $$ \xymatrix{ R \ar[r] \ar[d]_j & F \ar[d]^\Delta \\ U \times_S U \ar[r] & F \times F } $$ is cartesian. Hence according to Lemma 53.11.4 it suffices to show that $j$ has the properties listed in the lemma. (Note that each of the properties (1) – (4) occur in the lists of Remarks 53.4.1 and 53.4.3.) Since $j$ is an equivalence relation it is a monomorphism. Hence it is separated by Schemes, Lemma 25.23.3. As $R$ is an étale equivalence relation we see that $s, t : R \to U$ are étale. Hence $s, t$ are locally of finite type. Then it follows from Morphisms, Lemma 28.14.8 that $j$ is locally of finite type. Finally, as it is a monomorphism its fibres are finite. Thus we conclude that it is locally quasi-finite by Morphisms, Lemma 28.19.7. $\square$

    The code snippet corresponding to this tag is a part of the file spaces.tex and is located in lines 2049–2061 (see updates for more information).

    \begin{lemma}
    \label{lemma-properties-diagonal}
    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.
    Then
    \begin{enumerate}
    \item $\Delta$ is locally of finite type,
    \item $\Delta$ is a monomorphism,
    \item $\Delta$ is separated, and
    \item $\Delta$ is locally quasi-finite.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Let $F = U/R$ be a presentation of $F$.
    As in the proof of Lemma \ref{lemma-presentation-quasi-compact} the diagram
    $$
    \xymatrix{
    R \ar[r] \ar[d]_j & F \ar[d]^\Delta \\
    U \times_S U \ar[r] & F \times F
    }
    $$
    is cartesian. Hence according to
    Lemma \ref{lemma-representable-morphisms-spaces-property}
    it suffices to show that $j$ has the properties listed in the lemma.
    (Note that each of the properties (1) -- (4) occur in the lists
    of Remarks \ref{remark-list-properties-stable-base-change}
    and \ref{remark-list-properties-fpqc-local-base}.)
    Since $j$ is an equivalence relation it is a monomorphism.
    Hence it is separated by
    Schemes, Lemma \ref{schemes-lemma-monomorphism-separated}.
    As $R$ is an \'etale equivalence relation we see that
    $s, t : R \to U$ are \'etale. Hence $s, t$ are locally of finite
    type. Then it follows from
    Morphisms, Lemma \ref{morphisms-lemma-permanence-finite-type} that
    $j$ is locally of finite type. Finally, as it is a monomorphism
    its fibres are finite. Thus we conclude that it is locally quasi-finite by
    Morphisms, Lemma \ref{morphisms-lemma-finite-fibre}.
    \end{proof}

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