The Stacks project

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

  1. If $X$ is affine over $S$ and $\Delta : Y \to Y \times _ S Y$ is affine, then $g$ is affine.

  2. If $X$ is affine over $S$ and $Y$ is separated over $S$, then $g$ is affine.

  3. A morphism from an affine scheme to a scheme with affine diagonal is affine.

  4. 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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