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.

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