Syzygies of differentials of forms

TL;DR

This paper explores the relationships between differentials of forms, syzygies, and Jacobian modules in polynomial rings, introducing the concept of polarizability and examining its algebraic implications and connections.

Contribution

It introduces the notion of polarizability for forms and provides criteria to determine when certain syzygy modules coincide, linking differential modules with algebraic structures.

Findings

Characterization of polarizable forms

Connections between syzygies and Jacobian modules

Insights into algebraic structures like complete intersections

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 $\Omega_{A/k}$ of K\"ahler $k$-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} $\Omega_{A/k}\rar \mathcal{D}$ whose kernel is the torsion of $\Omega_{A/k}$. Letting $\fp$ denote the $R$-submodule generated by the (image of the) syzygy module of $\Omega_{A/k}$ 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.