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.

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 $R = A[x]/(x^pa)$ is a domain since it is $A$flat and its localization to the fraction field $K$ of $A$ is the ring $K[x]/(x^pa)$ 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 $\widehat{R} = \widehat{A} \otimes_A R = \widehat{A}[x]/(x^pa)$, and by inspection this is nonreduced (since $a$ admits a $p$th root in $\widehat{A}$ by design). 


In the Stacks project you can find the standard Example Tag 00PB as well as the more interesting Example Tag 02JD due to FerrandRaynaud. 

