Lemma 29.11.11 (01SG)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.11: Affine morphisms
Lemma 29.11.11 (cite)

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Lemma 29.11.11. Suppose $g : X \to Y$ is a morphism of schemes over $S$ .

If $X$ is affine over $S$ and $\Delta : Y \to Y \times _ S Y$ is affine, then $g$ is affine.
If $X$ is affine over $S$ and $Y$ is separated over $S$ , then $g$ is affine.
A morphism from an affine scheme to a scheme with affine diagonal is affine.
A morphism from an affine scheme to a separated scheme is affine.

Proof. Proof of (1). The base change $X \times _ S Y \to Y$ is affine by Lemma 29.11.8. The morphism $(1, g) : X \to X \times _ S Y$ is the base change of $Y \to Y \times _ S Y$ by the morphism $X \times _ S Y \to Y \times _ S Y$ . Hence it is affine by Lemma 29.11.8. The composition of affine morphisms is affine (see Lemma 29.11.7) and (1) follows. Part (2) follows from (1) as a closed immersion is affine (see Lemma 29.11.9) and $Y/S$ separated means $\Delta$ is a closed immersion. Parts (3) and (4) are special cases of (1) and (2). $\square$

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