Lemma 29.11.2. An affine morphism is separated and quasi-compact.

**Proof.**
Let $f : X \to S$ be affine. Quasi-compactness is immediate from Schemes, Lemma 26.19.2. We will show $f$ is separated using Schemes, Lemma 26.21.7. Let $x_1, x_2 \in X$ be points of $X$ which map to the same point $s \in S$. Choose any affine open $W \subset S$ containing $s$. By assumption $f^{-1}(W)$ is affine. Apply the lemma cited with $U = V = f^{-1}(W)$.
$\square$

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