Idealwise algebraic independence for elements of the completion of a local domain (Q1358735)

Suppose \((R,{\mathfrak m})\) is an excellent normal local domain with field of fractions \(K\) and completion \((\widehat R,\widehat {\mathfrak m})\). In this paper we consider three concepts of independence over \(R\) for elements \(\tau_1, \dots, \tau_n\in \widehat {\mathfrak m}\) which are algebraically independent over \(K\). We relate these three concepts of independence to flatness conditions of extensions of Krull domains, establish implications among them, and draw some conclusions concerning their existence and equivalence in special situations. We also investigate their stability under change of base ring. We begin our analysis in \S 2: The elements \(\tau_1, \dots, \tau_n\) are idealwise independent if \(K(\tau_1, \dots, \tau_n) \cap\widehat R\) equals the localized polynomial ring \(R[\tau_1, \dots, \tau_n]_{({ \mathfrak m}, \tau_1, \dots, \tau_n)}\). -- In \S 3 and \S 4 we present two methods for obtaining idealwise independent elements over a countable ring \(R\). The method in \S 3 is to find elements \(\tau_1, \dots, \tau_n\in \widehat {\mathfrak m}\), the maximal ideal of \(\widehat R\), so that (1) \(\tau_1, \dots, \tau_n\) are algebraically independent over the fraction field of \(R\), and (2) for every prime ideal \(P\) of \(S=R[\tau_1, \dots, \tau_n]_{({\mathfrak m}, \tau_1, \dots, \tau_n)}\) with \(\dim(S/P)=n\), the ideal \(P\widehat R\) is \(\widehat {\mathfrak m}\)-primary. If (1) and (2) hold, we say that \(\tau_1,\dots,\tau_n\) are primarily independent over \(R\); we show that primarily independent elements are idealwise independent. If \(R\) is countable and \(\dim(R) >2\), we show the existence over \(R\) of idealwise independent elements that fail to be primarily independent. -- In \S 4 we define \(\tau\in {\mathfrak m} \widehat R\) to be residually algebraically independent over \(R\) if \(\tau\) is algebraically independent over the fraction field of \(R\) and for each height-one prime ideal \(P\) of \(\widehat R\) such that \(P\cap R\neq 0\), the image of \(\tau\) in \(\widehat R/P\) is algebraically independent over \(R/(P\cap R)\). We show: primary independence \(\Rightarrow\) residual algebraic independence \(\Rightarrow\) idealwise independence. In \S 5 we describe the three concepts of idealwise independence, residual algebraic independence, and primary independence in terms of certain flatness conditions on the embedding \(\varphi:R [\tau_1, \dots, \tau_n]_{({\mathfrak m}, \tau_1, \dots, \tau_n)} \hookrightarrow \widehat R\). In \S 6 we investigate the stability of these independence concepts under base change, composition and polynomial extension. We show in \S 7 that both residual algebraic independence and primary independence hold for elements over the original ring \(R\) exactly when they hold over the Henselization \(R^h\) of \(R\). Also idealwise independence descends from the Henselization to the ring \(R\). If \(R\) is Henselian of dimension two, then all three concepts of independence are equivalent for one element \(\tau\in \widehat {\mathfrak m}\).