II.4.4

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This has H = 4 (can use II.3.11d, II.3.13, II.2.7), V = ....

HAPPY

• First show $f(Z)$ is separated and of finite type:
• Recall cor. II.4.8, show the composition $Z \to X \to Y$ is proper.
• Conclude the natural inclusion $f(Z) \to Y$ is a closed immersion, hence of finite type and separated.
• Conclude that $f(Z) \to Y \to S$ is separated.
• Properness:
• For any $Y' \to Y$ argue $Y' \times Z \to Y' \times f(Z)$ is surjective. Can apply ex II.2.7
• For any closed set $C \subset Y' \times f(Z)$, have $n(C) = (n \circ m)(m^{-1}(C))$, where $Y' \times Z \xrightarrow{m} Y' \times f(Z) \xrightarrow{n} Y'$