Lemma 66.16.3. 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

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

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

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

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

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