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Tag 0CL0

Chapter 91: Morphisms of Algebraic Stacks > Section 91.6: Higher diagonals

Lemma 91.6.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.

  1. The following are equivalent
    1. $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is separated,
    2. $\Delta_{f, 1} = \Delta_f : \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$ is separated, and
    3. $\Delta_{f, 2} = e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is a closed immersion.
  2. The following are equivalent
    1. $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is quasi-separated,
    2. $\Delta_{f, 1} = \Delta_f : \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$ is quasi-separated, and
    3. $\Delta_{f, 2} = e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is a quasi-compact.
  3. The following are equivalent
    1. $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is locally separated,
    2. $\Delta_{f, 1} = \Delta_f : \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$ is locally separated, and
    3. $\Delta_{f, 2} = e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is an immersion.
  4. The following are equivalent
    1. $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is unramified,
    2. $f$ is DM.
  5. The following are equivalent
    1. $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is locally quasi-finite,
    2. $f$ is quasi-DM.

Proof. Proof of (1), (2), and (3). Choose an algebraic space $U$ and a surjective smooth morphism $U \to \mathcal{X}$. Then $G = U \times_\mathcal{X} \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is an algebraic space over $U$ (Lemma 91.5.1). In fact, $G$ is a group algebraic space over $U$ by the group law on relative inertia constructed in Remark 91.5.2. Moreover, $G \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is surjective and smooth as a base change of $U \to \mathcal{X}$. Finally, the base change of $e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ by $G \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is the identity $U \to G$ of $G/U$. Thus the equivalence of (a) and (c) follows from Groupoids in Spaces, Lemma 69.6.1. Since $\Delta_{f, 2}$ is the diagonal of $\Delta_f$ we have (b) $\Leftrightarrow$ (c) by definition.

Proof of (4) and (5). Recall that (4)(b) means $\Delta_f$ is unramified and (5)(b) means that $\Delta_f$ is locally quasi-finite. Choose a scheme $Z$ and a morphism $a : Z \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$. Then $a = (x_1, x_2, \alpha)$ where $x_i : Z \to \mathcal{X}$ and $\alpha : f \circ x_1 \to f \circ x_2$ is a $2$-morphism. Recall that $$ \vcenter{ \xymatrix{ \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha(x_1, x_2) \ar[d] \ar[r] & Z \ar[d] \\ \mathcal{X} \ar[r]^{\Delta_f} & \mathcal{X} \times_\mathcal{Y} \mathcal{X} } } \quad\text{and}\quad \vcenter{ \xymatrix{ \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x_2, x_2) \ar[d] \ar[r] & Z \ar[d]^{x_2} \\ \mathcal{I}_{\mathcal{X}/\mathcal{Y}} \ar[r] & \mathcal{X} } } $$ are cartesian squares. By Lemma 91.5.4 the algebraic space $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha(x_1, x_2)$ is a pseudo torsor for $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x_2, x_2)$ over $Z$. Thus the equivalences in (4) and (5) follow from Groupoids in Spaces, Lemma 69.9.5. $\square$

    The code snippet corresponding to this tag is a part of the file stacks-morphisms.tex and is located in lines 1350–1405 (see updates for more information).

    \begin{lemma}
    \label{lemma-diagonal-diagonal}
    Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
    \begin{enumerate}
    \item
    The following are equivalent
    \begin{enumerate}
    \item $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$
    is separated,
    \item $\Delta_{f, 1} = \Delta_f :
    \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$
    is separated, and
    \item $\Delta_{f, 2} = e :
    \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$
    is a closed immersion.
    \end{enumerate}
    \item
    The following are equivalent
    \begin{enumerate}
    \item $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$
    is quasi-separated,
    \item $\Delta_{f, 1} = \Delta_f :
    \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$
    is quasi-separated, and
    \item $\Delta_{f, 2} = e :
    \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$
    is a quasi-compact.
    \end{enumerate}
    \item
    The following are equivalent
    \begin{enumerate}
    \item $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$
    is locally separated,
    \item $\Delta_{f, 1} = \Delta_f :
    \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$
    is locally separated, and
    \item $\Delta_{f, 2} = e :
    \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$
    is an immersion.
    \end{enumerate}
    \item
    The following are equivalent
    \begin{enumerate}
    \item $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$
    is unramified,
    \item $f$ is DM.
    \end{enumerate}
    \item
    The following are equivalent
    \begin{enumerate}
    \item $\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$
    is locally quasi-finite,
    \item $f$ is quasi-DM.
    \end{enumerate}
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Proof of (1), (2), and (3).
    Choose an algebraic space $U$ and a surjective smooth morphism
    $U \to \mathcal{X}$. Then
    $G = U \times_\mathcal{X} \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$
    is an algebraic space over $U$ (Lemma \ref{lemma-inertia}).
    In fact, $G$ is a group algebraic space over $U$
    by the group law on relative
    inertia constructed in Remark \ref{remark-inertia-is-group-in-spaces}.
    Moreover, $G \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$
    is surjective and smooth as a base change of $U \to \mathcal{X}$.
    Finally, the base change of
    $e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$
    by $G \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$
    is the identity $U \to G$ of $G/U$.
    Thus the equivalence of (a) and (c) follows from
    Groupoids in Spaces, Lemma
    \ref{spaces-groupoids-lemma-group-scheme-separated}.
    Since $\Delta_{f, 2}$ is the diagonal of $\Delta_f$ we have
    (b) $\Leftrightarrow$ (c) by definition.
    
    \medskip\noindent
    Proof of (4) and (5). Recall that (4)(b) means $\Delta_f$ is
    unramified and (5)(b) means that $\Delta_f$ is locally quasi-finite.
    Choose a scheme $Z$ and a morphism
    $a : Z \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$.
    Then $a = (x_1, x_2, \alpha)$ where $x_i : Z \to \mathcal{X}$
    and $\alpha : f \circ x_1  \to f \circ x_2$ is a $2$-morphism.
    Recall that
    $$
    \vcenter{
    \xymatrix{
    \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha(x_1, x_2)
    \ar[d] \ar[r] &
    Z \ar[d] \\
    \mathcal{X} \ar[r]^{\Delta_f} &
    \mathcal{X} \times_\mathcal{Y} \mathcal{X}
    }
    }
    \quad\text{and}\quad
    \vcenter{
    \xymatrix{
    \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x_2, x_2)
    \ar[d] \ar[r] &
    Z \ar[d]^{x_2} \\
    \mathcal{I}_{\mathcal{X}/\mathcal{Y}} \ar[r] &
    \mathcal{X}
    }
    }
    $$
    are cartesian squares. By Lemma \ref{lemma-isom-pseudo-torsor-aut-over-space}
    the
    algebraic space $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha(x_1, x_2)$
    is a pseudo torsor for $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x_2, x_2)$
    over $Z$. Thus the equivalences in (4) and (5) follow from
    Groupoids in Spaces, Lemma
    \ref{spaces-groupoids-lemma-pseudo-torsor-implications}.
    \end{proof}

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