The Stacks project

Lemma 67.4.6. 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.