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II.4.6

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This has H = 2, V = ...; ex. II.4.1 can be applied.

HAPPY

Say \mbox{Spec }B = X \xrightarrow{f} Y =\mbox{Spec }A is determined by  \theta \colon A \to B.

  • Replace  A with A/\ker \theta and use II.4.1 to reduce to the case  \theta injective.
  • Show there are injections  A \subset B \subset \overline{A} where the last is the integral closure (in the fraction field of B).
  • use 4.11A to reduce this to  B \subset R_v for all valuation rings (of K(B)/k) containing A
  • use the valuation ring criterion of properness.

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