On the de rham cohomology of algebraic varieties
On the de rham cohomology of algebraic varieties
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DOI：
10.1007/bf02684807
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发表时间：
1966
期刊：
影响因子：
--
通讯作者：
A. Grothendieck
中科院分区：
文献类型：
--
作者：
A. Grothendieck
..。关于哈特肖恩关于对偶的研讨会，我最近看了你和霍奇关于“第二类积分”的联合论文(2)。由于Hironaka已经证明了奇点的分解(3)，那篇论文(第81页)的“猜想C”是正确的，因此该论文的结果依赖于它。现在我想到，在这篇文章中，“猜想C”的全部力量还没有被充分利用，即“第二类积分”理论实质上包含在以下非常简单的定理1中：设X是复数域C上的仿射代数方案；假设X是正则的(即“非奇异”)。然后，复上同调H‘(X，C)可以被计算为代数de Rham复形(即X上“有理且处处定义”的微分形式的复形)的上同调。
... In connection with Hartshorne's seminar on duality, I had a look recently at your joint paper with Hodge on" Integrals of the second kind"(2). As Hironaka has proved the resolution of singularities (3), the" Conjecture C" of that paper (p. 81) holds true, and hence the results of that paper which depend on it. Now it occurred to me that in this paper, the whole strength of the" Conjecture C" has not been fully exploited, namely that the theory of" integrals of second kind" is essentially contained in the following very simpleTheorem 1.--Let X be an affine algebraic scheme over the field C of complex numbers; assume X regular (ie" non singular"). Then the complex cohomology H'(X, C) can be calculated as the cohomology of the algebraic De Rham complex (ie the complex of differential forms on X which are" rational and everywhere defined").