# I.3.20

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This has H = 1

### HAPPY

$f\in A(Y-P)$

• Consider $V(f) \subset Y-p$. If $p$ is in the clsoure (taken in $Y$), then extend by zero.
• Otherwise $p$ is not in the closure, argue $1/f \in A(U - p)$ for a sufficiently small $U \ni p$ disjoint from $V(f)$
• Argue in fact $1/f \in \mathcal{O}_p$; for all $q \in U-p$, $1/f = g/h$ near $q$
• If $h(p) = 0$ use $\dim Y \ge 2$ to show $V(f) \cap U \ne \emptyset$; contradiction.
• Consider $J \subset \{g \in \mathcal{O}_p| fg \in \mathcal{O}_p\}$
• By the above, conclude $f^nJ \subset J, \forall n$.
• Use the determinant trick to show $f$ is integral over $\mathcal{O}_p$.

### Commutative Algebra

Determinant Trick: Let $M$ be a finitely generated (say by $n$ elements) $A$ module and $\phi \colon M \to M$ be endomorphism such that there is an ideal $J \subset A$ such that $\phi(M) \subset JM$, then $\phi$ satifies a monic polynomial relation of degree $n$.