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Tag 0CPW

Chapter 87: Morphisms of Algebraic Stacks > Section 87.37: Scheme theoretic image

Lemma 87.37.7. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces and separated. Let $\mathcal{V} \subset \mathcal{Y}$ be an open substack such that $\mathcal{V} \to \mathcal{Y}$ is quasi-compact. Let $s : \mathcal{V} \to \mathcal{X}$ be a morphism such that $f \circ s = \text{id}_\mathcal{V}$. Let $\mathcal{Y}'$ be the scheme theoretic image of $s$. Then $\mathcal{Y}' \to \mathcal{Y}$ is an isomorphism over $\mathcal{V}$.

Proof. By Lemma 87.7.7 the morphism $s : \mathcal{V} \to \mathcal{Y}$ is quasi-compact. Hence the construction of the scheme theoretic image $\mathcal{Y}'$ of $s$ commutes with flat base change by Lemma 87.37.5. Thus to prove the lemma we may assume $\mathcal{Y}$ is representable by an algebraic space and we reduce to the case of algebraic spaces which is Morphisms of Spaces, Lemma 55.16.7. $\square$

    The code snippet corresponding to this tag is a part of the file stacks-morphisms.tex and is located in lines 8499–8509 (see updates for more information).

    \begin{lemma}
    \label{lemma-scheme-theoretic-image-of-partial-section}
    Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks
    which is representable by algebraic spaces and separated.
    Let $\mathcal{V} \subset \mathcal{Y}$ be an open substack such that
    $\mathcal{V} \to \mathcal{Y}$ is quasi-compact.
    Let $s : \mathcal{V} \to \mathcal{X}$ be a morphism such that
    $f \circ s = \text{id}_\mathcal{V}$.
    Let $\mathcal{Y}'$ be the scheme theoretic image of $s$.
    Then $\mathcal{Y}' \to \mathcal{Y}$ is an isomorphism over $\mathcal{V}$.
    \end{lemma}
    
    \begin{proof}
    By Lemma \ref{lemma-quasi-compact-permanence}
    the morphism $s : \mathcal{V} \to \mathcal{Y}$ is quasi-compact.
    Hence the construction of the scheme theoretic image $\mathcal{Y}'$
    of $s$ commutes with flat base change by
    Lemma \ref{lemma-existence-plus-flat-base-change}.
    Thus to prove the lemma
    we may assume $\mathcal{Y}$ is representable by an algebraic space
    and we reduce to the case of algebraic spaces which is
    Morphisms of Spaces, Lemma
    \ref{spaces-morphisms-lemma-scheme-theoretic-image-of-partial-section}.
    \end{proof}

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