# II.4.5

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This H = 2, V = ....

### HAPPY

• parts a,b follows from results II.4.3, II.4.7 and application of ex II.2.7
• part c is missing
• part d, assume there is a nonconstant global section, use it to show there is a valuation of $K(X)/k$ over the ground field k with no center in X.
• it will be helpful to prove for an integral scheme all the restriction maps are injective (what is useful in this problem is to show any nonzero global section remains nonzero when restricted to the stalk at any point).
• Let $a$ be the nonconstant global section, set $b = 1/a$consider the ring $k[b]_{(b)}$; it must be dominated by some valuation ring $R$
• this means $m_x = \mathcal{O}_{X,x} \cap m_R$; and $b \in m_R$
• check $a_x \in m_x$ and $a_x \not \in m_x$ both give contradictions.