The Stacks project

Example 10.119.5. Let $k$ be a field of characteristic $p > 0$ such that $k$ has infinite degree over its subfield $k^ p$ of $p$th powers. For example $k = \mathbf{F}_ p(t_1, t_2, t_3, \ldots )$. Consider the ring

\[ A = \left\{ \sum a_ i x^ i \in k[[x]] \text{ such that } [k^ p(a_0, a_1, a_2, \ldots ) : k^ p] < \infty \right\} \]

Then $A$ is a discrete valuation ring and its completion is $A^\wedge = k[[x]]$. Note that the induced extension of fraction fields of $A \subset k[[x]]$ is infinite purely inseparable. Choose any $f \in k[[x]]$, $f \not\in A$. Let $R = A[f] \subset k[[x]]$. Then $R$ is a Noetherian local domain of dimension $1$ whose completion $R^\wedge $ is nonreduced (think!).

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