Modules of differentials of symmetric algebras (Q1262905)

Symmetric algebras of modules are among the mildest of extensions of a commutative \(ring\quad R\). The aim is to exhibit some simplifying features of their modules of differentials. Two results are proved here. First, if R is a smooth algebra over a field k of characteristic zero and E is a finitely generated R-module, we examine the module \({\mathcal D}\) of k-derivations of the symmetric algebra S(E). The Zariski-Lipman conjecture is shown to hold for these algebras. That is, if \({\mathcal D}\) is S(E)-projective, then S(E) is R-smooth. The other result concerns the homological rigidity of the module of differentials \(\Omega_{S(E)/k}\). The expectation is that if the projective dimension of \(\Omega_{S(E)/k}\), as an S(E)-module, is finite, then S(E) must be a complete intersection. This would imply that E has projective dimension at most 1. Machinery that may be useful to deal with the general case is developed and used to rule out certain cases, e.g. the projective dimension of E cannot be 2.