Intermediate rings between a local domain and its completion. II (Q5954294)

Let \(R\) be an excellent normal domain, \(\widehat R\) its completion, \(K\) [respectively \(\widehat K]\) the fraction field of \(R\) [respectively \(\widehat R]\) and \(L/K\) a pure transcendental field subextension of \(\widehat K/K\). The rings of type \(A:= L\cap\widehat R\) provide many interesting examples, as Nagata already noticed in the 1950's. The present authors studied conditions when such a ring \(A\) can be expressed as a direct union of localized polynomial extension rings of \(R\) [see \textit{W.Heinzer, C. Rotthaus} and \textit{S. Wiegand}, Ill. J. Math. 43, No. 1, 19-46 (1999; Zbl 0916.13002) and Ill. J. Math. 44, No. 4, 927-928 (2000; Zbl 0965.13007)]. NEWLINENEWLINENEWLINEThe present paper studies the results connecting flatness of extension rings between an excellent normal local domain and its completion. The authors consider conditions of these rings to have Cohen-Macaulay formal fibers. The article includes several concrete examples illustrating these results.