Lemma 101.4.7 (050H)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 101: Morphisms of Algebraic Stacks
Section 101.4: Separation axioms
Lemma 101.4.7 (cite)

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Lemma 101.4.7. Let $f : \mathcal{X} \to \mathcal{Z}$, $g : \mathcal{Y} \to \mathcal{Z}$ and $\mathcal{Z} \to \mathcal{T}$ be morphisms of algebraic stacks. Consider the induced morphism $i : \mathcal{X} \times _\mathcal {Z} \mathcal{Y} \to \mathcal{X} \times _\mathcal {T} \mathcal{Y}$. Then

$i$ is representable by algebraic spaces and locally of finite type,
if $\Delta _{\mathcal{Z}/\mathcal{T}}$ is quasi-separated, then $i$ is quasi-separated,
if $\Delta _{\mathcal{Z}/\mathcal{T}}$ is separated, then $i$ is separated,
if $\mathcal{Z} \to \mathcal{T}$ is DM, then $i$ is unramified,
if $\mathcal{Z} \to \mathcal{T}$ is quasi-DM, then $i$ is locally quasi-finite,
if $\mathcal{Z} \to \mathcal{T}$ is separated, then $i$ is proper, and
if $\mathcal{Z} \to \mathcal{T}$ is quasi-separated, then $i$ is quasi-compact and quasi-separated.

Proof. The following diagram

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

is a $2$-fibre product diagram, see Categories, Lemma 4.31.13. Hence $i$ is the base change of the diagonal morphism $\Delta _{\mathcal{Z}/\mathcal{T}}$. Thus the lemma follows from Lemma 101.3.3, and the material in Properties of Stacks, Section 100.3. $\square$

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