Lemma 109.17.1. There exists a local Noetherian $2$-dimensional domain $(B, \mathfrak m)$ complete with respect to a principal ideal $I = (b)$ and an element $f \in \mathfrak m$, $f \not\in I$ such that the $I$-adic completion $C = (B_ f)^\wedge $ of the principal localization $B_ f$ is nonreduced and even such that $C_ b = C[1/b] = (B_ f)^\wedge [1/b]$ is nonreduced.

**Proof.**
See discussion above.
$\square$

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