Integration of differential forms on schemes.
Integration of differential forms on schemes.
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方案上微分形式的整合。
DOI：
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发表时间：
1990
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通讯作者：
E. Kunz
中科院分区：
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作者：
R. Hübl;E. Kunz
在过去的二十年里，有大量关于留数代数理论的论文。在这些处理中，该理论已通过各种方法和不同程度的通用性得到发展，并且已应用于代数和几何中的许多问题（参见[RD]，[Ho]，[Ke]，[Kul]，[Ll]，[L2]，[SS 2]，[V]；对于复解析理论，我们在绝对情况下参考[GH]，第五章，在相对情况下参考[B]）。在本文中，我们想介绍一种留数理论的另一种方法。它与 [Kul] 和 [SS 2] 中提出的概念密切相关，但它比这些理论更加普遍。我们对诺特方案的分离态射 / : X — » Y 感兴趣，其中 / 是 d 维的等维且一般是平滑的。 （使用 [KW] 的技术可以弱化后一个条件。为了简单起见，我们将把自己限制在上述情况，并将简单但技术性的概括留给读者。）在上述情况中，X/Y 的 d 次微分形式的正则 r 的层 α)χ/γ 被定义（[KW]，§3，§4）。它是 ^χ(Ωχ/γ) 的子束，即 X/Y 的亚纯 d 型束（参见 [EGA] IV, (20. 1. 5)）。我们感兴趣的是正则 r d 形式的留数的积分和计算，或者更准确地说，它们的局部上同调类的积分和计算，以及 X/Y 的适当有限子方案 Z/Y 的支持。在本文中，我们将通常所说的“余数”称为“积分”，并且我们将保留“余数”一词来表示 X 的闭合点处的准时构造。
In the last two decades numerous papers have been written about the algebraic theory of residues. In these treatments the theory has been developed with various methods and in various degrees of generality, and it has been applied to many problems in algebra and geometry (cf. [RD], [Ho], [Ke], [Kul], [Ll], [L2], [SS 2], [V]; for the complex analytic theory we refer to [GH], chap. V in the absolute case, resp. [B] in the relative Situation). In this paper we would like to introduce one more approach to the theory of residues. It is closely related to the concepts developed in [Kul] and [SS 2], however it is considerably more general than these theories. We are interested in separated morphisms / : X — » Υ of noetherian schemes, with / being equidimensional of dimension d and generically smooth. (The latter condition may be weakened, using the techniques of [KW]. For simplicity we will restrict ourselves to the above case, and we leave the easy but technical generalizations to the reader.) In the Situation described above the sheaf α)χ/γ of regul r differential forms of degree d of X/Y is defined ([KW], §3, §4). It is a subsheaf of ^χ(Ωχ/γ), the sheaf of meromorphic d-forms of X/Y (cf. [EGA] IV, (20. 1. 5)). We are interested in the Integration and the calculation of residues of regul r d-forms, or — to be precise — of their local cohomology classes with supports in suitable finite subschemes Z/Y of X/Y. In this paper we shall call "integral" what is often called "residue", and we shall reserve the word "residue" for a punctual formation at the closed points of X.