Presentation depth and the Lipman-Sathaye Jacobian theorem (Q1605643)

A theorem of \textit{J. Lipman} and \textit{A. Sathaye} [Michigan Math. J. 28, 199-222 (1981; Zbl 0438.13019)] is generalized by weakening the hypotheses substantially: Let \(R\) be a Noetherian domain with fraction field \(K\). Assume that the \(S_2\)-locus is open in algebras essentially of finite type over \(R\) (which is always true if \(R\) is excellent). Let \(S\) be an extension algebra, essentially of finite type over \(R\) being torsion free and generically étale. Suppose that for every maximal ideal \(\mathfrak{m} \subset S\), \(\mathfrak{m} \cap R = \mathfrak p\), \(R_{\mathfrak p}\) is normal and \(S_{\mathfrak{m}}\) has a relatively \(S_2\) presentation over \(R_{\mathfrak p}\) (which is always true if \(R\) is normal and Cohen-Macaulay). Let \(L = K \otimes_R S\) and \(S^\prime\) be the integral closure of \(S\) in \(L\) and assume that \(S^\prime\) is a finite \(S\)-module (which is always true if \(R\) is excellent). Suppose that, for every height one prime ideal \(\mathfrak{q}\) of \(S^\prime\) the ring \(R_{\mathfrak{q} \cap R}\) is regular. Then \[ S^\prime\colon{_L J}_{S^\prime|R} \subseteq S\colon{_L J}_{S|R}. \] Here \(J_{S|R}\) is the Jacobian ideal of \(S\) over \(R\).