Example of the completion of a noetherian domain at a prime that is not a domain - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T18:33:58Z http://mathoverflow.net/feeds/question/18496 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18496/example-of-the-completion-of-a-noetherian-domain-at-a-prime-that-is-not-a-domain Example of the completion of a noetherian domain at a prime that is not a domain Arturo Magidin 2010-03-17T15:26:34Z 2013-05-13T07:30:26Z <p>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. </p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/18496/example-of-the-completion-of-a-noetherian-domain-at-a-prime-that-is-not-a-domain/18498#18498 Answer by Charles Siegel for Example of the completion of a noetherian domain at a prime that is not a domain Charles Siegel 2010-03-17T15:31:42Z 2010-03-17T15:31:42Z <p>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 $\mathbb{C}[[x,y]]/(xy)$.</p> http://mathoverflow.net/questions/18496/example-of-the-completion-of-a-noetherian-domain-at-a-prime-that-is-not-a-domain/18499#18499 Answer by Franz Lemmermeyer for Example of the completion of a noetherian domain at a prime that is not a domain Franz Lemmermeyer 2010-03-17T15:39:11Z 2010-03-17T15:39:11Z <p>There's another example (going back to Nagat) in <a href="http://www.math.purdue.edu/~heinzer/preprints/bou16.pdf" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/18496/example-of-the-completion-of-a-noetherian-domain-at-a-prime-that-is-not-a-domain/40978#40978 Answer by Sándor Kovács for Example of the completion of a noetherian domain at a prime that is not a domain Sándor Kovács 2010-10-04T03:19:27Z 2013-05-13T07:30:26Z <p>It might be worth pointing out that you get an example of this by localizing at any point of a variety (scheme) which is irreducible but not (analytically/formally/étale) locally irreducible. In particular, any self-intersecting curve will give an example, just like the one above by Charles.</p> http://mathoverflow.net/questions/18496/example-of-the-completion-of-a-noetherian-domain-at-a-prime-that-is-not-a-domain/58540#58540 Answer by pradip Keskar for Example of the completion of a noetherian domain at a prime that is not a domain pradip Keskar 2011-03-15T15:20:52Z 2011-03-15T15:20:52Z <p>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".</p> <p>Several connections are worth mentioning, eg. concepts such as Henselization and Hensel's lemma, Zariski's Main theorem (as a special case, it mentions that the completion of a normal algebraic local domain is again a normal domain), links associated to the singularities of algebraic curves, various reciprocity laws from Kummer to Artin in Algebraic Number Theory etc.</p>