  Given a 0-dimensional scheme in a projective space $\mathbb{P}^n$ over a
field $K$, we study the K\"ahler differential algebra $\Omega_{R/K}$ of its
homogeneous coordinate ring $R$. Using explicit presentations of the modules
$\Omega^m_{R/K}$ of K\"ahler 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.
