Is the normalisation of an integral noetherien dimension one ring a finite morphism? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T19:26:10Z http://mathoverflow.net/feeds/question/83626 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83626/is-the-normalisation-of-an-integral-noetherien-dimension-one-ring-a-finite-morphi Is the normalisation of an integral noetherien dimension one ring a finite morphism? name 2011-12-16T15:03:21Z 2011-12-16T19:33:10Z <p>This feels like something I should know but I can't find an answer in Liu or in Atiyah-MacDonald, or a counter-example.</p> <p>To state the question again: let $A$ be an integral Noetherien ring of Krull dimension one and $K$ its field of fractions. Let $B$ be the set of elements of $K$ that are integral over $A$ i.e. $B$ is the normalisation of $A$ in $K$.</p> <p>Is the morphism $A \to B$ finite?</p> <p>Note that this is true if $A$ is excellent (or even just Nagata), and its rather difficult to construct examples of non-excellent rings.</p> http://mathoverflow.net/questions/83626/is-the-normalisation-of-an-integral-noetherien-dimension-one-ring-a-finite-morphi/83647#83647 Answer by Qing Liu for Is the normalisation of an integral noetherien dimension one ring a finite morphism? Qing Liu 2011-12-16T18:08:23Z 2011-12-16T18:08:23Z <p>Let $R$ be any non-excellent DVR with field of fractions $K$, let $L/K$ be a finite extension such that the normalization $B$ of $R$ in $L$ is not finite over $R$. We have $L=K[a_1,...,a_n]$ for some $a_i\in B$. Consider $A=R[a_1,...,a_n]\subseteq B$. Then $B$ in the integral closure of $A$, but is not finite over $A$ (because $A$ is finite over $R$). </p> http://mathoverflow.net/questions/83626/is-the-normalisation-of-an-integral-noetherien-dimension-one-ring-a-finite-morphi/83651#83651 Answer by Mahdi Majidi-Zolbanin for Is the normalisation of an integral noetherien dimension one ring a finite morphism? Mahdi Majidi-Zolbanin 2011-12-16T18:27:17Z 2011-12-16T19:33:10Z <p>For a brief history of this question you can look at Matsumura's <em>Commutative Ring Theory</em>, page 264. In <em>Ein Satz über primäre Integritätsbereiche</em>, Math. Ann. vol. 103 (1930), p.p. 450-465 Krull proved that the integral closure of a one-dimensional Noetherian local domain $A$ is finite over $A$ if and only if the completion of $A$ is reduced. Akizuki constructed the first example of a one-dimensional Noetherian local integral domain with non-reduced completion in 1935. For another counterexample you can see <a href="http://arxiv.org/pdf/alg-geom/9503017" rel="nofollow">Akizuki's counterexample</a></p>