The Canonical Exact Sequence of Differential Modules for 0-Dimensional Schemes

TL;DR

This paper investigates the structure of Kähler differential modules associated with 0-dimensional schemes in projective space, providing a canonical exact sequence and formulas for their Hilbert polynomials in specific cases.

Contribution

It introduces a canonical exact sequence for differential modules of 0-dimensional schemes and derives Hilbert polynomial formulas for certain classes of schemes.

Findings

Established the canonical exact sequence for differential modules.

Derived Hilbert polynomial formulas for fat point schemes.

Analyzed differential powers of ideals in 0-dimensional schemes.

Abstract

Given a 0-dimensional scheme $\X$ in $\mathbb{P}^n_K$ over a perfect field $K$, we examine the second differential power of its homogeneous vanishing ideal. This enables us to establish the canonical exact sequence for the associated K\"ahler differential module. We also provide a formula for the Hilbert polynomial of K\"ahler differential modules when $\X$ is either a fat point scheme or a 0-dimensional locally monomial Gorenstein scheme.