This feels like something I should know but I can't find an answer in Liu or in Atiyah-MacDonald, or a counter-example.


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 $K$.


Is the morphism $A \to B$ finite?


Note that this is true if $A$ is excellent (or even just Nagata), and its rather difficult to construct examples of non-excellent rings.



Is the normalisation of an integral noetherien dimension one ring a finite morphism?
For a brief history of this question you can look at Matsumura's *Commutative Ring Theory*, page 264. In *Ein Satz über primäre Integritätsbereiche*, 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 [Akizuki's counterexample](http://arxiv.org/pdf/alg-geom/9503017)


