On the module of differentials of order $n$ of hypersurfaces

TL;DR

This paper provides an explicit description of the module of differentials of order n for hypersurfaces using a higher-order Jacobian, and explores its properties related to regularity, normality, and projective dimension.

Contribution

It introduces a higher-order Jacobian matrix to explicitly present the module of differentials of order n and extends known results to higher-order cases for hypersurfaces.

Findings

Higher-order Jacobian matrix explicitly describes the module of differentials

Freeness and torsion-freeness relate to hypersurface regularity and normality

Analysis of the module's projective dimension for hypersurfaces

Abstract

We give an explicit presentation of the module of differentials of order $n$ of a finitely generated algebra via a higher-order Jacobian matrix. We use the presentation to study some aspects of this module in the case of hypersurfaces. More precisely, we prove higher-order versions of known results relating freness and torsion-freness of the module of differentials with the regularity and normality of the hypersurface. We also study its projective dimension.