发表机构
Universität Passau; University of Education - Hue University（帕绍大学; 顺化教育大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究零维概形的de Rham上同调，证明了基于Macaulay基的消没定理，并分析了Artinian局部环的上同调维数，通过Galois分裂计算了fat point概形的上同调。
AI 中文摘要
零维概形的（朴素）de Rham上同调是其坐标环的Kähler微分代数作为复形的同调。由于众所周知，若环是拟齐次的，则高次上同调消失，我们专注于仿射情形。对于一般的仿射$K$-代数$R=P/I$，其中${\rm char}(K)=0$且$P=K[x_1,\dots,x_n]$，我们基于$dI\wedge \Omega^m_{P/K}$的Macaulay基的形状证明了$H_{\rm dR}^m(R)$的消没定理。对于Artinian局部代数$A=P/\langle f_1,\dots,f_n\rangle$，其中$\{f_1,\dots,f_n\}$是超正则序列，我们证明$H_{\rm dR}^{\bullet}(A)$一般非平凡，但当$\Omega^m_{A/K}$的关系模的自然生成元系是标准基时则为平凡。此外，我们详细研究了Artinian局部环$A=P/I$的$H_{\rm dR}^0(A)=\ker(d_A)$的维数。任意仿射零维概形的情形通过Galois分裂归约到该情形，并由此获得了fat point概形的de Rham上同调。许多显式计算的例子和反例支持了这些结果，并表明零维概形的de Rham上同调通常是非常微妙的。
英文摘要
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.
Comments20 pages
