Lemma 31.26.4. Let $X$ be a locally Noetherian integral scheme. Let $f \in R(X)^*$. Then the collections

and

are locally finite in $X$.

Lemma 31.26.4. Let $X$ be a locally Noetherian integral scheme. Let $f \in R(X)^*$. Then the collections

\[ \{ Z \subset X \mid Z\text{ a prime divisor with generic point }\xi \text{ and }f\text{ not in }\mathcal{O}_{X, \xi }\} \]

and

\[ \{ Z \subset X \mid Z \text{ a prime divisor and }\text{ord}_ Z(f) \not= 0\} \]

are locally finite in $X$.

**Proof.**
There exists a nonempty open subscheme $U \subset X$ such that $f$ corresponds to a section of $\Gamma (U, \mathcal{O}_ X^*)$. Hence the prime divisors which can occur in the sets of the lemma are all irreducible components of $X \setminus U$. Hence Lemma 31.26.1 gives the desired result.
$\square$

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