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

Lemma 66.4.5. 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

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

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

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

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 66.4.1, and the material in Section 66.3.
$\square$

## Comments (3)

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