Example of a reduced ring whose completion is not reduced Let $(R,\mathfrak{m})$ be a local Noetherian ring. Give an example such that R is reduced but $\mathfrak{m}$-adic completion of R is not reduced.
 A: Perhaps the "simplest" kind of example would be a local noetherian domain that is finite flat over a discrete valuation ring, though any such example has to be in equicharacteristic $p > 0$ since otherwise excellence considerations for the discrete valuation ring of generic characteristic 0 would rule it out. 
Such an example is lurking in Example 11 in section 3.6 of the book "Neron Models". That example  (which omits a few details in its justification) provides a discrete valuation ring $A$ over $k = \mathbf{F}_p$ with residue field $k$ such that there is $a \in A$ with no $p$th root in $A$ but a $p$th root in $\widehat{A}$.  This ring $A$ is built between $k(T)$ and $k(T,U)$ (and it is not essentially of finite type over $k$, as otherwise it would be excellent, contradicting what is below).
The ring $R = A[x]/(x^p-a)$ is a domain since it is $A$-flat and its localization to the fraction field $K$ of $A$ is the ring $K[x]/(x^p-a)$ that is a field (as $x^p - a$ is irreducible over $K$, due to the normality of $A$). The ring $R$ is local noetherian of dimension 1 since it is finite flat over $A$ with special fiber $k[x]/(x^p- \overline{a})$ that is artin local.  But $A$-finiteness also implies that $\widehat{R} = \widehat{A} \otimes_A R = \widehat{A}[x]/(x^p-a)$, and by inspection this is non-reduced (since $a$ admits a $p$th root in $\widehat{A}$ by design).
A: In the Stacks project you can find the standard Example Tag 00PB as well as the more interesting Example Tag 02JD due to Ferrand-Raynaud.
