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The Stacks project

Lemma 67.20.11. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Suppose $g : X \to Y$ is a morphism of algebraic spaces over $B$.

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

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

  3. A morphism from an affine scheme to an algebraic space with affine diagonal over $\mathbf{Z}$ (as in Properties of Spaces, Definition 66.3.1) is affine.

  4. A morphism from an affine scheme to a separated algebraic space is affine.

Proof. Proof of (1). The base change $X \times _ B Y \to Y$ is affine by Lemma 67.20.5. The morphism $(1, g) : X \to X \times _ B Y$ is the base change of $Y \to Y \times _ B Y$ by the morphism $X \times _ B Y \to Y \times _ B Y$. Hence it is affine by Lemma 67.20.5. The composition of affine morphisms is affine (see Lemma 67.20.4) and (1) follows. Part (2) follows from (1) as a closed immersion is affine (see Lemma 67.20.6) and $Y/B$ separated means $\Delta $ is a closed immersion. Parts (3) and (4) are special cases of (1) and (2). $\square$

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