The Stacks Project


Tag 04XJ

Chapter 77: Properties of Algebraic Stacks > Section 77.4: Points of algebraic stacks

Lemma 77.4.5. Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{X} = [U/R]$ be a presentation of $\mathcal{X}$, see Algebraic Stacks, Definition 71.16.5. Then the image of $|R| \to |U| \times |U|$ is an equivalence relation and $|\mathcal{X}|$ is the quotient of $|U|$ by this equivalence relation.

Proof. The assumption means that we have a smooth groupoid $(U, R, s, t, c)$ in algebraic spaces, and an equivalence $f : [U/R] \to \mathcal{X}$. We may assume $\mathcal{X} = [U/R]$. The induced morphism $p : U \to \mathcal{X}$ is smooth and surjective, see Algebraic Stacks, Lemma 71.17.2. Hence $|U| \to |\mathcal{X}|$ is surjective by Lemma 77.4.4. Note that $R = U \times_\mathcal{X} U$, see Groupoids in Spaces, Lemma 60.21.2. Hence Lemma 77.4.3 implies the map $$ |R| \longrightarrow |U| \times_{|\mathcal{X}|} |U| $$ is surjective. Hence the image of $|R| \to |U| \times |U|$ is exactly the set of pairs $(u_1, u_2) \in |U| \times |U|$ such that $u_1$ and $u_2$ have the same image in $|\mathcal{X}|$. Combining these two statements we get the result of the lemma. $\square$

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    \begin{lemma}
    \label{lemma-points-presentation}
    Let $\mathcal{X}$ be an algebraic stack.
    Let $\mathcal{X} = [U/R]$ be a presentation of $\mathcal{X}$, see
    Algebraic Stacks, Definition \ref{algebraic-definition-presentation}.
    Then the image of $|R| \to |U| \times |U|$ is an equivalence relation
    and $|\mathcal{X}|$ is the quotient of $|U|$ by this equivalence relation.
    \end{lemma}
    
    \begin{proof}
    The assumption means that we have a smooth groupoid $(U, R, s, t, c)$
    in algebraic spaces, and an equivalence $f : [U/R] \to \mathcal{X}$.
    We may assume $\mathcal{X} = [U/R]$.
    The induced morphism $p : U \to \mathcal{X}$ is smooth and surjective, see
    Algebraic Stacks,
    Lemma \ref{algebraic-lemma-smooth-quotient-smooth-presentation}.
    Hence $|U| \to |\mathcal{X}|$ is surjective by
    Lemma \ref{lemma-characterize-surjective}.
    Note that $R = U \times_\mathcal{X} U$, see
    Groupoids in Spaces,
    Lemma \ref{spaces-groupoids-lemma-quotient-stack-2-cartesian}.
    Hence
    Lemma \ref{lemma-points-cartesian}
    implies the map
    $$
    |R| \longrightarrow |U| \times_{|\mathcal{X}|} |U|
    $$
    is surjective. Hence the image of $|R| \to |U| \times |U|$ is
    exactly the set of pairs $(u_1, u_2) \in |U| \times |U|$
    such that $u_1$ and $u_2$ have the same image in $|\mathcal{X}|$.
    Combining these two statements we get the result of the lemma.
    \end{proof}

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