Kähler differential algebras for 0-dimensional schemes

Abstract

Given a 0-dimensional scheme in a projective space ${\mathbb{P}}^n$ over a field $K$, we study the Kähler differential algebra ${\Omega }_{R/K}$ of its homogeneous coordinate ring $R$. Using explicit presentations of the modules ${\Omega }_{R/K}^m$ of Kähler differential $m$-forms, we determine many values of their Hilbert functions explicitly and bound their Hilbert polynomials and regularity indices. Detailed results are obtained for subschemes of ${\mathbb{P}}^1$, fat point schemes, and subschemes of ${\mathbb{P}}^2$ supported on a conic.