Lemma 66.20.11. 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 $B$.

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

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

A morphism from an affine scheme to an algebraic space with affine diagonal over $\mathbf{Z}$ (as in Properties of Spaces, Definition 65.3.1) is affine.

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

## Comments (0)