TITLE: Minimal prime ideals of $\hat R$ contract to minimal primes of $R$?
QUESTION [5 upvotes]: In what conditions from $P\in\operatorname{Min}\widehat R$ can we deduce that $P\cap R\in \operatorname{Min} R$ ?

Here $(R,\mathfrak m)$ is a commutative Noetherian local ring, $\widehat R$ its $\mathfrak m$-adic completion and  $\operatorname{Min} R$ the minimal primes of $R$.

REPLY [3 votes]: As I've already told you, the going-down theorem holds for flat $R$-algebras (see, for instance, Matsumura, Commutative Ring Theory, Theorem 9.5). Since $R\to\hat{R}$ is (faithfully) flat and $P$ a minimal prime in $\hat R$, then $\mathfrak p=P\cap R$ is also minimal. Otherwise let $\mathfrak q$ be a prime ideal of $R$ with $\mathfrak q\subset \mathfrak p$. Going-down theorem says that there exists a prime $Q$ of $\hat R$ with $Q\subset P$ such that $Q\cap R=\mathfrak q$ (obviously $Q\neq P$) and this contradicts the minimality of $P$.