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Let $(R,\frak m)$ be a Noetherian local ring. I am trying to prove that if the natural map $\operatorname{Spec}(\widehat R)\rightarrow \operatorname{Spec}(R)$ is bijective then $\mathfrak{p}\widehat R $ is irreducible for all $\mathfrak p\in \operatorname{Spec}(R)$.

Thank you for your proof or comment.

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  • $\begingroup$ What's an example of such an $R$? $\endgroup$
    – Angelo
    Commented Dec 20, 2019 at 20:54
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    $\begingroup$ A DVR (or any local domain of dimension 1 whose completion is still a domain) would be an example. I can't think of anything in higher dimension though. $\endgroup$
    – Hailong Dao
    Commented Dec 21, 2019 at 16:48
  • $\begingroup$ Thank you for your comment Hailong Dao! $\endgroup$
    – TNAn
    Commented Jun 22, 2020 at 3:31

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