# Example of the completion of a noetherian domain at a prime that is not a domain

Let $R$ be a Noetherian domain, and let $\mathfrak{p}$ be a prime ideal; consider the completion $\hat R_{\mathfrak{p}}$ of $R$ at $\mathfrak{p}$ (the inverse limit of the system of quotients $R/\mathfrak{p}^n$). If $R$ is a PID, it is easy to see that $\hat R_{\mathfrak{p}}$ is a domain.

Someone asked in sci.math if $\hat R_{\mathfrak{p}}$ would always be a domain. I thought it would, but looking at Eisenbud's "Commutative Algebra", I found a reference to a theorem of Larfeldt and Lech that says that if $A$ is any finite-dimensional algebra over a field $k$, then there is a Noetherian local integral domain $R$ with maximal ideal $\mathfrak{m}$ such that $\hat{R_{\mathfrak{M}}}\cong A[[x_1,\ldots,x_n]]$ for some $n$; and so this completion will not be a domain if $A$ is not a domain. I would like to know an example directly, if possible.

Does someone know an easy example of a noetherian domain $R$ and a prime ideal $\mathfrak{p}$ such that $\hat R_{\mathfrak{p}}$ is not a domain? Thanks in advance.

• Serre's homological conditions $R_i$ and $S_i$ are well-behaved under completion for excellent local rings, so properties characterized in this way such as reducedness ($R_0$ + $S_1$) and normality ($R_1$ + $S_2$) are preserved under completion for such rings. So those notions are well-behaved under analytification of algebraic schemes (over complex numbers, as well as over non-archimedean fields) since algebraic and analytic local rings are excellent and have the "same" completion. – BCnrd Mar 17 '10 at 16:12
• By the way, a local ring whose completion is a domain is called sometimes analytically irreducible, for obvious reasons. – Karl Schwede Jan 27 '11 at 19:44
• Hm, perhaps I should also mention that a curve singularity whose completion is a domain is called unibranch. – Karl Schwede Mar 20 '13 at 12:51

Let $R=\mathbb{C}[x,y]/(y^2-x^2(x-1))$. This is the nodal cubic in the plane. Look at the prime $\mathfrak{p}=(x,y)$, corresponding to the nodal point. The completion here is isomorphic to $\mathbb{C}[[x,y]]/(xy)$.
At least for algebraic local domains R, we have 1-1 correspondence between the minimal primes in the completion $\cap{R}$ and the maximal ideals in the integral closure of R in its quotient field. This will produce a lot of examples including curves near a point whose neighbourhood in the curve can't be covered by a single parameterization. In fact, various connections come to the fore here. For several ways to characterize the number of minimal primes in the completion of a local ring of an irreducible plane curve, one can look at S. S. Abhyankar's Chavounet prize winning paper "Historical Ramblings in Algebraic Geometry and related algebra".