Properties of the graph of a morphism of algebraic spaces as a consequence of separation properties of the target.

Lemma 66.4.6. Let $S$ be a scheme. Let $T$ be an algebraic space over $S$. Let $g : X \to Y$ be a morphism of algebraic spaces over $T$. Consider the graph $i : X \to X \times _ T Y$ of $g$. Then

1. $i$ is representable, locally of finite type, locally quasi-finite, separated and a monomorphism,

2. if $Y \to T$ is locally separated, then $i$ is an immersion,

3. if $Y \to T$ is separated, then $i$ is a closed immersion, and

4. if $Y \to T$ is quasi-separated, then $i$ is quasi-compact.

Proof. This is a special case of Lemma 66.4.5 applied to the morphism $X = X \times _ Y Y \to X \times _ T Y$. $\square$

Comment #912 by Matthieu Romagny on

Suggested slogan: Graph of a morphism of algebraic spaces vs separation properties of the target.

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