Syzygies of differentials of forms

Abstract

Given a standard graded polynomial ring $R=k[x_1,...,x_n]$ over a field $k$ of characteristic zero and a graded $k$-subalgebra $A=k[f_1,...,f_m]\subset R$, one relates the module $Ω_{A/k}$ of Kähler $k$-differentials of $A$ to the transposed Jacobian module $\mathcal{D}\subset \sum_{i=1}^n R dx_i$ of the forms $f_1,...,f_m$ by means of a {\em Leibniz map} $Ω_{A/k}\rar \mathcal{D}$ whose kernel is the torsion of $Ω_{A/k}$. Letting $\fp$ denote the $R$-submodule generated by the (image of the) syzygy module of $Ω_{A/k}$ and $\fz$ the syzygy module of $\mathcal{D}$, there is a natural inclusion $\fp\subset \fz$ 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 $f_1,...,f_m$ are {\em polarizable}. One surveys some classes of subalgebras that are generated by polarizable forms. The problem has some curious connections with constructs of commutative algebra, such as the Jacobian ideal, the conormal module and its torsion, homological dimension in $R$ and syzygies, complete intersections and Koszul algebras. Some of these connections trigger questions which have interest in their own.