The top arrow of a “magic diagram” of algebraic spaces has nice immersion-like properties, and under separatedness hypotheses these get stronger.

Lemma 65.4.5. Let $S$ be a scheme. Let $f : X \to Z$, $g : Y \to Z$ and $Z \to T$ be morphisms of algebraic spaces over $S$. Consider the induced morphism $i : X \times _ Z Y \to X \times _ T Y$. Then

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

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

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

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

Proof. By general category theory the following diagram

$\xymatrix{ X \times _ Z Y \ar[r]_ i \ar[d] & X \times _ T Y \ar[d] \\ Z \ar[r]^-{\Delta _{Z/T}} \ar[r] & Z \times _ T Z }$

is a fibre product diagram. Hence $i$ is the base change of the diagonal morphism $\Delta _{Z/T}$. Thus the lemma follows from Lemma 65.4.1, and the material in Section 65.3. $\square$

Comment #1106 by Evan Warner on

Suggested slogan: The top arrow of a "magic diagram" of algebraic spaces has nice immersion-related properties, and under separatedness hypotheses these get stronger.

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