Lemma 87.30.3 (0ARQ)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 87: Formal Algebraic Spaces
Section 87.30: Separation axioms for morphisms
Lemma 87.30.3 (cite)

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Lemma 87.30.3. Let $S$ be a scheme. Let $f : X \to Z$, $g : Y \to Z$ and $Z \to T$ be morphisms of formal algebraic spaces over $S$. Consider the induced morphism $i : X \times _ Z Y \to X \times _ T Y$. Then

$i$ is representable (by schemes), locally of finite type, locally quasi-finite, separated, and a monomorphism,
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 87.15.5. $\square$

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