# 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.