Corrigendum to our paper ``Intermediate rings between a local domain and its completion''
As the title indicates, this short note is a corrigendum to our earlier paper [Ill. J. Math. 43, No. 1, 19-46 (1999; Zbl 0916.13002)]. The topic of that paper is the structure of an intermediate ring \(A\) between a normal excellent local domain \(R\) and its \({\mathfrak m}\)-adic completion \(\widehat R\) of the form \(A:= K(\tau_1, \tau_2, \dots, \tau_s)\cap \widehat R\), where \(s\in \mathbb{N}\) and \(\tau_1,\tau_2, \dots,\tau_s \in\widehat {\mathfrak m}\) are certain algebraically independent elements over the fraction field of \(R\). We now realize that theorem 5.8 of that paper contains an error; an incorrect equivalence is given for the flatness property to hold for a certain extension involved in this construction. Theorem 5.8 is restated in our new paper ``Intermediate rings between a local domain and its completion. II'' (preprint); the new theorem states: With the notation above, suppose that \(y\in{\mathfrak m}\) and that \(R^*\) is the \(y\)-adic completion of \(R\). Then the following statements are equivalent: (1) \(S:=R[\tau_1, \tau_2,\dots, \tau_s]_{({\mathfrak m},\tau_1, \tau_2,\dots, \tau_s)} \hookrightarrow\widehat R[1/y]\) is flat. (2) If \(P\) is a prime ideal of \(S\) and \(\widehat Q\) is a prime ideal of \(\widehat R\) minimal over \(P\widehat R\) such that \(y\notin\widehat Q\), then \(\text{ht}(\widehat Q)=\text{ht}(P)\). (3) If \(\widehat Q\) is a prime ideal of \(\widehat R\) with \(y\notin \widehat Q\), then \(\text{ht}(\widehat Q)\geq \text{ht}(\widehat Q\cap S)\).
scientific article; zbMATH DE number 1553611