On the De Rham Cohomology of Zero-Dimensional Schemes

Abstract

The (naive) de Rham cohomology of a zero-dimensional scheme is the homology of the Kähler differential algebra of its coordinate ring, viewed as a complex. Since it is well-known to vanish in higher degrees if the ring is quasi-homogeneous, we concentrate on the affine case. For a general affine $K$-algebra $R=P/I$, where ${\rm char}(K)=0$ and $P=K[x_1,\dots,x_n]$, we prove a vanishing theorem for $H_{\rm dR}^m(R)$ based on the shape of a Macaulay basis of $dI\wedge Ω^m_{P/K}$. For an Artinian local algebra $A=P/\langle f_1,\dots,f_n\rangle$, where $\{f_1,\dots,f_n\}$ is a super regular sequence, we show that $H_{\rm dR}^{\bullet}(A)$ is non-trivial in general, but trivial when the natural system of generators of the relation module of $Ω^m_{A/K}$ is a standard basis. Moreover, we provide a detailed study of the dimension of $H_{\rm dR}^0(A)=\ker(d_A)$ for Artinian local rings $A=P/I$. The case of arbitrary affine zero-dimensional schemes is reduced to this case using a Galois splitting, and as a result we obtain the de Rham cohomology of a fat point scheme. Many explicitly computed examples and counterexamples support the results and indicate how subtle the de Rham cohomology of a zero-dimensional scheme is in general.