Syzygies of differentials of forms (Q372656)

From the authors' abstract: ``Given a standard graded polynomial ring \(R=k[x_1,\dots,x_n]\) over a field \(k\) of characteristic zero and a graded \(k\)-subalgebra \(A=k[f_1,\dots,f_m]\subset R\), one relates the module \(\Omega_{_A/k}\) of Kähler \(k\)-differentials of \(A\) to the transposed Jacobian module \(\mathcal D\subset\sum^n_{i=1}Rdx_i\) of the forms \(f_1,\ldots,f_m\) by means of a Leibniz map \(\Omega_{A/k}\to\mathcal D\) whose kernel is the torsion of \(\Omega_{A/k}\). Letting \(\mathfrak D\) denote the \(R\)-submodule generated by the (image of the) syzygy module of \(\Omega_{A/k}\) and \(\mathfrak Z\) the syzygy module of \(\mathcal D\), there is a natural inclusion \(\mathfrak D\subset \mathfrak Z\) coming from the chain rule for composite derivatives. The main goal is to give means to test when this inclusion is an equality -- in which case one says that the forms \(f_1,\dots,f_m\) are polarizable.''