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Let $A$ be a noetherian local ring with maximal ideal $m$ and let $I\subset m$. Is the $I$-adic completion of $A$ necessarily local?

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Yes, and the proof goes like I did here.

But a more general result holds: the $I$-adic completion $\widehat R$ is quasi-local iff $R/I$ is quasi-local. (Quasi-local means local, but not necessarily noetherian.) See here on page $6$.

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