Lemma 26.21.9 (01KR)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 26: Schemes
Section 26.21: Separation axioms
Lemma 26.21.9 (cite)

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Lemma 26.21.9. Let $f : X \to T$ and $g : Y \to T$ be morphisms of schemes with the same target. Let $h : T \to S$ be a morphism of schemes. Then the induced morphism $i : X \times _ T Y \to X \times _ S Y$ is an immersion. If $T \to S$ is separated, then $i$ is a closed immersion. If $T \to S$ is quasi-separated, then $i$ is a quasi-compact morphism.

Proof. By general category theory the following diagram

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

is a fibre product diagram. The lemma follows from Lemmas 26.21.2, 26.17.6 and 26.19.3. $\square$

Comments (2)

Comment #8468 by Elías Guisado on June 07, 2023 at 09:59

For the sake of having some reference that is actually instructive, here's a neat proof of the cartesianity of the square https://mathoverflow.net/a/80812/101848

Comment #9083 by Stacks project on May 07, 2024 at 10:03

Thanks.

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