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 char(K)=0 and P=K[x1,…,xn], we prove a vanishing theorem for HdRm(R) based on the shape of a Macaulay basis of dI∧ ΩmP/K. For an Artinian local algebra A=P/⟨ f1,…,fn⟩, where {f1,…,fn} is a super regular sequence, we show that HdR•(A) is non-trivial in general, but trivial when the natural system of generators of the relation module of ΩmA/K is a standard basis. Moreover, we provide a detailed study of the dimension of HdR0(A)=ker(dA) 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.
Related
- Discrete geometry and de-Rham cohomology of graphs
- On Logarithmic Poisson and De Rham Cohomology Groups of a Class of Inhomogeneous Divisors
- The stacky approach to de Rham-Witt cohomology
- Hochschild cohomology of Hilbert schemes of points on surfaces
- Tropical cohomology for band schemes over the tropical hyperfield
BibTeX
@misc{kreuzer2026rham,
title = {{On the De Rham Cohomology of Zero-Dimensional Schemes}},
author = {Martin Kreuzer and Le Ngoc Long},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34819},
eprint = {2609.34819},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34819},
}
RIS
TY - UNPB
TI - On the De Rham Cohomology of Zero-Dimensional Schemes
AU - Martin Kreuzer
AU - Le Ngoc Long
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34819
UR - https://arxiv.org/abs/2609.34819
AB - 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 char(K)=0 and P=K[x_1,…,x_n], we prove a vanishing theorem for H_dR^m(R) based on the shape of a Macaulay basis of dI∧ Ω^m_P/K. For an Artinian local algebra A=P/⟨ f_1,…,f_n⟩, where {f_1,…,f_n} is a super regular sequence, we show that H_dR^•(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_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.
ER -