The canonical exact sequence of differential modules for 0-dimensional schemes

Tran N. K. Linh and Le Ngoc Long

Vol. 344 (2026), No. 1, 47–68

DOI: 10.2140/pjm.2026.344.47

Abstract

Given a 0-dimensional scheme $\mathbb{𝕏}$ in ${\mathbb{P}}_K^n$ 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 Kähler differential module. We also provide a formula for the Hilbert polynomial of Kähler differential modules when $\mathbb{𝕏}$ is either a fat point scheme or a 0-dimensional locally monomial Gorenstein scheme.

Keywords

Kähler differential modules, differential powers, Hilbert functions, zero-dimensional schemes, locally monomial Gorenstein schemes

Mathematical Subject Classification

Primary: 13N05

Secondary: 13D40, 13N10, 14N05

Milestones

Received: 25 June 2025

Revised: 16 April 2026

Accepted: 17 May 2026

Published: 10 August 2026