Lemma 54.7.3. In Situation 54.7.1 there exists a nonzero $f \in \mathfrak m$ such that for every $i = 1, \ldots , r$ there exist
a closed point $x_ i \in C_ i$ with $x_ i \not\in C_ j$ for $j \not= i$,
a factorization $f = g_ i f_ i$ of $f$ in $\mathcal{O}_{X, x_ i}$ such that $g_ i \in \mathfrak m_{x_ i}$ maps to a nonzero element of $\mathcal{O}_{C_ i, x_ i}$.
Proof. We will use the observations made following Situation 54.7.1 without further mention. Pick a closed point $x_ i \in C_ i$ which is not in $C_ j$ for $j \not= i$. Pick $g_ i \in \mathfrak m_{x_ i}$ which maps to a nonzero element of $\mathcal{O}_{C_ i, x_ i}$. Since the fraction field of $A$ is the fraction field of $\mathcal{O}_{X_ i, x_ i}$ we can write $g_ i = a_ i/b_ i$ for some $a_ i, b_ i \in A$. Take $f = \prod a_ i$. $\square$
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