Geometric Auslander criterion for openness of an algebraic morphism (Q2854234)

Let \(k\) be a field, \(X\) and \(Y\) schemes of finite type over \(k\) and \(\phi:X\to Y\) a morphism locally of finite type. If \(Y\) is normal and of dimension \(n\) it is shown that \(\phi\) is open iff there are no vertical irreducible components in the \(n\)-fold fibred product \(\phi^{\{n\}}:X^{\{n\}}\to Y.\) Algebraically, this translates to the following: let \(R\) be a \(k\)-algebra of finite type and \(A\) an \(R\)-algebra of finite type. Assume that \(R\) is normal of dimension \(n.\) Then the induced morphism \(\mathrm{Spec}(A)\to \mathrm{Spec}(R)\) is open iff for every minimal prime \(P\) of \(A^{\otimes^n_R}\) we have \(P\cap R=(0)\), or equivalently \(A^{\otimes^n_R}\) is a torsion-free \(R\)-module. The inspiration of this result comes from a classical flatness criterion [\textit{M. Auslander}, Ill. J. Math. 5, No. 4, 631--647 (1961; Zbl 0104.26202)].