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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'

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