On the transitivity of regular differential forms (Q916723)

Let R/k and S/R be generically smooth algebras of finite type which are equidimensional of relative dimensions d, resp. n. The author proves that the canonical isomorphism of the meromorphic differential forms \({\mathcal M}(\Omega^ d_{R/k})\otimes_ R{\mathcal M}(\Omega^ n_{S/R})\to {\mathcal M}(\Omega^{d+n}_{S/k})\) always induces a morphism of the regular differential forms in the sense of Kunz and Waldi \(\phi:\;\omega^ d_{R/k}\otimes_ R\omega^ n_{S/R}\to \omega^{d+n}_{S/k},\) and that \(\phi\) is an isomorphism in the following two cases: (1) R/k and S/R are both Cohen-Macaulay algebras (2) R/k is a Gorenstein algebra. (A short description of the construction of the regular differential forms of the highest degree is given in a preliminary paragraph.) An example is given, showing that \(\phi\) need not always be an isomorphism.