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Let $\mathcal{O}$ be a Noetherian, one dimensional domain and $P\subset \mathcal{O}$ be a prime ideal. I want to prove that if $0\neq I\subseteq P$ we have $\mathcal{O}/(\mathcal{O}\cap I\mathcal{O}_P) \cong \mathcal{O}_P/ I\mathcal{O}_P$.

It's clearly enough to prove that the map $\mathcal{O}\rightarrow \mathcal{O}_P/ I\mathcal{O}_P$ is surjective, but I can't prove that.

This is used in the proof of propostion 12.3 in Neukirch Algebraic Number Theory.

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Since $\mathcal{O}$ is a $1$-dimensional domain and $I\neq 0$, $\mathcal{O}/I$ is $0$-dimensional. Since it is Noetherian, that means $\mathcal{O}/I$ is the product of its localizations at each of its prime ideals. In particular, the map from $\mathcal{O}/I$ to any localization of $\mathcal{O}/I$ is surjective (since it is just the projection onto a factor of a product). Thus the map $\mathcal{O}/I\to \mathcal{O}_P/I\mathcal{O}_P$ is surjective, which gives the isomorphism you want.

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  • $\begingroup$ Hello Eric, can you please explain why $O/I$ being Noetherian implies that $O/I$ is the product of its localizations at each of its prime ideals? Thanks! $\endgroup$
    – kindasorta
    Jan 17, 2019 at 12:19
  • $\begingroup$ That is a general theorem about Noetherian 0-dimensional rings. See for instance stacks.math.columbia.edu/tag/00JB and stacks.math.columbia.edu/tag/00KH. $\endgroup$ Jan 17, 2019 at 16:41
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Here is the argument that I suspect Neukirch intended. In the first part of the proof of Prop 12.3 in Neukirch's book, he states that if $P \supseteq I$, then $P$ is the only prime ideal containing $\widetilde{I}_P$ where $\widetilde{I}_P = (\mathcal{O} \cap I\mathcal{O}_P)$. He doesn't offer a proof, but a simple one is given in Proof of a stronger form of Chinese remainder theorem (12.3) in Neukirch

That means that $\mathcal{O}/\widetilde{I}_P$ has a single maximal ideal, $P/\widetilde{I}_P$ and so everything not in that ideal is a unit. Therefore, if $x/s \in \mathcal{O}_P$ with $s \notin P$, and writing modulo $\widetilde{I}_P$ with an overbar, $\overline{s}$ has an inverse $\overline{s}' \in \mathcal{O}/\widetilde{I}_P$ so that $xs' \in \mathcal{O}$ maps to $\overline{x}/\overline{s}$, showing surjection.

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