The Stacks project

Lemma 67.16.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $Z \subset Y$ be the scheme theoretic image of $f$. If $f$ is quasi-compact then

  1. the sheaf of ideals $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ Y \to f_*\mathcal{O}_ X)$ is quasi-coherent,

  2. the scheme theoretic image $Z$ is the closed subspace corresponding to $\mathcal{I}$,

  3. for any étale morphism $V \to Y$ the scheme theoretic image of $X \times _ Y V \to V$ is equal to $Z \times _ Y V$, and

  4. the image $|f|(|X|) \subset |Z|$ is a dense subset of $|Z|$.

Proof. To prove (3) it suffices to prove (1) and (2) since the formation of $\mathcal{I}$ commutes with étale localization. If (1) holds then in the proof of Lemma 67.16.1 we showed (2). Let us prove that $\mathcal{I}$ is quasi-coherent. Since the property of being quasi-coherent is étale local we may assume $Y$ is an affine scheme. As $f$ is quasi-compact, we can find an affine scheme $U$ and a surjective étale morphism $U \to X$. Denote $f'$ the composition $U \to X \to Y$. Then $f_*\mathcal{O}_ X$ is a subsheaf of $f'_*\mathcal{O}_ U$, and hence $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ Y \to \mathcal{O}_{X'})$. By Lemma 67.11.2 the sheaf $f'_*\mathcal{O}_ U$ is quasi-coherent on $Y$. Hence $\mathcal{I}$ is quasi-coherent as a kernel of a map between coherent modules. Finally, part (4) follows from parts (1), (2), and (3) as the ideal $\mathcal{I}$ will be the unit ideal in any point of $|Y|$ which is not contained in the closure of $|f|(|X|)$. $\square$

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