The Stacks project

Lemma 110.12.7. There exists a ring $A$ complete with respect to a principal ideal $I$ and an element $f \in A$ such that the $I$-adic completion $A_ f^\wedge $ of $A_ f$ is not flat over $A$.

Proof. Set $A = R[[x]]$ and $I = (x)$ and observe that $R_ f[[x]]$ is the completion of $R[[x]]_ f$. $\square$

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