The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 57.13.1. 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. 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 57.10.4 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 57.11.4 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 57.4.1 and 57.4.3.) 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. $\square$

Comments (1)

Comment #3836 by slogan_bot on

Suggested slogan: "The diagonal of any algebraic space is a separated, locally quasi-finite monomorphism"

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