Noether-Lasker decomposition of coherent analytic subsheaves
Noether-Lasker decomposition of coherent analytic subsheaves
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相干解析副滑轮的 Noether-Lasker 分解
DOI：
10.1090/s0002-9947-1969-0234019-8
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发表时间：
1969
影响因子：
1.3
通讯作者：
Y. Siu
中科院分区：
文献类型：
--
作者：
Y. Siu
在本文中，我们发展的理论的Noether-Lasker分解的相干解析子层作为一个类似的代数Noether-Lasker分解的理想在Noetherian环。分解可以描述如下：假设θ Y是复空间（X，d '）上的凝聚解析层Y的凝聚解析子层，在Grauert意义下。对于X的每一点x，Y作为J的一个（0，-）子模，有一个Noether-Lasker分解为9 x的准素（0-）子模。这些准素子模的根是（9）的素理想，它定义了X在x处的子簇芽。这些子簇芽被拼接在一起形成X的整体不可约子簇，我们称之为Y的伴随子簇。Y的一个凝聚子层，如果只有一个相关联的子簇，则称为准素层。我们证明了每一个凝聚解析子层都可以表示为“局部有限”的准素子层的交。这个表示就是我们所说的相干解析子层的诺特-拉斯克分解。如果（X，（））是Stein，则凝聚解析层SY的凝聚解析真子层是准素的当且仅当F（X，Y）是F（X，（））-模r（X，Y）的准素子模.子层的Noether-Lasker分解是从Thimm的间隙层理论导出的。在本文的第一部分，我们用层理论的方法阐述了Thimm的间隙层理论。本文第二部分建立了相干解析子层的Noether-Lasker分解。
In this paper we develop the theory of Noether-Lasker decomposition of coherent analytic subsheaves as an analogue of the algebraic Noether-Lasker decomposition of ideals in Noetherian rings. The decomposition can be described as follows: Suppose 9Y is a coherent analytic subsheaf of a coherent analytic sheaf Y on a complex space (X, d') in the sense of Grauert. For every point x of X, Y, as an (9,-submodule of J, has a Noether-Lasker decomposition into primary (0-submodules of 9x. The radicals of these primary submodules are prime ideals of (9, which define subvariety-germs of X at x. These subvariety germs are pieced together to form global irreducible subvarieties of X which we call associated subvarieties of Y. A coherent subsheaf of Y which has only one associated subvariety is called primary. We prove that every coherent analytic subsheaf can be represented as the intersection of "locally finite" primary subsheaves. This representation is what we call the Noether-Lasker decomposition of the coherent analytic subsheaf. If (X, () is Stein, then a coherent analytic proper subsheaf .9 of a coherent analytic sheaf SY is primary if and only if F(X, Y) is a primary submodule of the F(X, ()-module r(X, Y). The Noether-Lasker decomposition of subsheaves is derived from the gapsheaf theory of Thimm [4]. In part I of this paper we give an exposition of Thimm's theory of gap-sheaves by sheaf-theoretical methods. In part II of this paper we establish the Noether-Lasker decomposition of coherent analytic subsheaves.