Lemma 10.162.11. Let $R$ be a Noetherian local ring. Let $\mathfrak p \subset R$ be a prime. Assume

$R_{\mathfrak p}$ is a discrete valuation ring, and

$\mathfrak p$ is analytically unramified.

Then for any associated prime $\mathfrak q$ of $R^\wedge /\mathfrak pR^\wedge $ the local ring $(R^\wedge )_{\mathfrak q}$ is a discrete valuation ring.

**Proof.**
Assumption (2) says that $R^\wedge /\mathfrak pR^\wedge $ is a reduced ring. Hence an associated prime $\mathfrak q \subset R^\wedge $ of $R^\wedge /\mathfrak pR^\wedge $ is the same thing as a minimal prime over $\mathfrak pR^\wedge $. In particular we see that the maximal ideal of $(R^\wedge )_{\mathfrak q}$ is $\mathfrak p(R^\wedge )_{\mathfrak q}$. Choose $x \in R$ such that $xR_{\mathfrak p} = \mathfrak pR_{\mathfrak p}$. By the above we see that $x \in (R^\wedge )_{\mathfrak q}$ generates the maximal ideal. As $R \to R^\wedge $ is faithfully flat we see that $x$ is a nonzerodivisor in $(R^\wedge )_{\mathfrak q}$. Hence we win.
$\square$

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