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

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.