The Stacks project

Lemma 101.38.3. Let $f : \mathcal{Y} \to \mathcal{X}$ be a morphism of algebraic stacks. Then the scheme theoretic image of $f$ exists.

Proof. Choose a scheme $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. By Lemma 101.38.2 we may replace $\mathcal{Y}$ by $V$. Thus it suffices to show that if $X \to \mathcal{X}$ is a morphism from a scheme to an algebraic stack, then the scheme theoretic image exists. Choose a scheme $U$ and a surjective smooth morphism $U \to \mathcal{X}$. Set $R = U \times _\mathcal {X} U$. We have $\mathcal{X} = [U/R]$ by Algebraic Stacks, Lemma 94.16.2. By Properties of Stacks, Lemma 100.9.11 the closed substacks $\mathcal{Z}$ of $\mathcal{X}$ are in $1$-to-$1$ correspondence with $R$-invariant closed subschemes $Z \subset U$. Let $Z_1 \subset U$ be the scheme theoretic image of $X \times _\mathcal {X} U \to U$. Observe that $X \to \mathcal{X}$ factors through $\mathcal{Z}$ if and only if $X \times _\mathcal {X} U \to U$ factors through the corresponding $R$-invariant closed subscheme $Z$ (details omitted; hint: this follows because $X \times _\mathcal {X} U \to X$ is surjective and smooth). Thus we have to show that there exists a smallest $R$-invariant closed subscheme $Z \subset U$ containing $Z_1$.

Let $\mathcal{I}_1 \subset \mathcal{O}_ U$ be the quasi-coherent ideal sheaf corresponding to the closed subscheme $Z_1 \subset U$. Let $Z_\alpha $, $\alpha \in A$ be the set of all $R$-invariant closed subschemes of $U$ containing $Z_1$. For $\alpha \in A$, let $\mathcal{I}_\alpha \subset \mathcal{O}_ U$ be the quasi-coherent ideal sheaf corresponding to the closed subscheme $Z_\alpha \subset U$. The containment $Z_1 \subset Z_\alpha $ means $\mathcal{I}_\alpha \subset \mathcal{I}_1$. The $R$-invariance of $Z_\alpha $ means that

\[ s^{-1}\mathcal{I}_\alpha \cdot \mathcal{O}_ R = t^{-1}\mathcal{I}_\alpha \cdot \mathcal{O}_ R \]

as (quasi-coherent) ideal sheaves on (the algebraic space) $R$. Consider the image

\[ \mathcal{I} = \mathop{\mathrm{Im}}\left( \bigoplus \nolimits _{\alpha \in A} \mathcal{I}_\alpha \to \mathcal{I}_1 \right) = \mathop{\mathrm{Im}}\left( \bigoplus \nolimits _{\alpha \in A} \mathcal{I}_\alpha \to \mathcal{O}_ X \right) \]

Since direct sums of quasi-coherent sheaves are quasi-coherent and since images of maps between quasi-coherent sheaves are quasi-coherent, we find that $\mathcal{I}$ is quasi-coherent. Since pull back is exact and commutes with direct sums we find

\[ s^{-1}\mathcal{I} \cdot \mathcal{O}_ R = t^{-1}\mathcal{I} \cdot \mathcal{O}_ R \]

Hence $\mathcal{I}$ defines an $R$-invariant closed subscheme $Z \subset U$ which is contained in every $Z_\alpha $ and containes $Z_1$ as desired. $\square$

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