A non-catenary, normal, local domain

Ever since Nagata constructed his celebrated example of a non-catenary Noetherian domain, it has been an open question whether or not such an example could be integrally closed. If the "chain conjecture" were valid, it could not be. However, in [3], T. Ogoma showed such an example existed. Precisely, he constructed a Noetherian domain R and showed that the integral closure of R was non-catenary and a finite i^-module (thus Noetherian). The present article is an alternate presentation of Ogoma's example. It is intended to serve two purposes. First, the construction itself has been simplified. It is shorter, requires less machinery, and should be more accessible than the original. Secondly, some new properties of R are observed. Most significantly, R is in fact integrally closed already (Theorem 4). This also simplifies matters. It should be noted that [3] contains other examples, numerous positive results of interest, and a great deal of creativity which are omitted here. We begin the construction. Let F be a countable field and {ai9 bi9 c{\i e Z } be indeterminates. Set Km = F{{ai9 bi9 c{ | / ^ m}) and K = [JKm. Let x9 y, z, w be additional indeterminates. Select a set 0» ( a K[x, y, z, w]) of prime elements, exactly one for each height one prime of S = K[x9 y9 z, w]ix>ytZtw), such that W G ^ and 0* contains infinitely many elements from F[x9 y9 z, w]. Noting &> is countable, these assumptions allow a numbering 0> = {Pi\i e Z} with/?! = w and Pi e Kt_2 [*, y, z, H>] for every i ;> 2. Set qn = nj=i Pm fn = * + HUml gn = y + EU hql hn = z + Eu oat, and Pn = (/„, gn9 hn) S for n ^ 0. Observe that for each n > 0, we have (modulo ((*, y9 z, w)S) ) fn = x + axw9 gn = y + bxw9 hn = z + c\w, and so fn9 gn9 hn9 w is a regular system of parameters. Therefore Pn is a height three prime ideal.