A generalized Koszul complex. I
A generalized Koszul complex. I
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广义的科祖尔复合体。
DOI：
10.1090/s0002-9947-1964-0159859-0
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发表时间：
1964
影响因子：
1.3
通讯作者：
D. Buchsbaum
中科院分区：
文献类型：
--
作者：
D. Buchsbaum
导论.在文献[1]中，Koszul复形被用来研究余维与重数之间的关系。它还帮助我们研究麦考利模和环，并提供了一个上下文，以证明科恩-麦考利定理有关的unmixness完全交叉。现在有一个推广的科恩-麦考利定理（已知的，我们相信，作为广义科恩-麦考利定理），这与unmixedness的理想所产生的未成年人的矩阵。由于Koszul复形可以被认为是与R”-* R的映射相关联的复形，其中R是交换环，因此似乎应该存在与Rm-> R”的映射相关联的复形。事实上，只要愿意走那么远，为什么不寻找与任何交换环上的任意模映射相关联的复形呢？在本文中，我们定义了一个复杂的映射的模块。在§1中，我们讨论了这个复杂的完整的一般性。在§2中，我们几乎将注意力限制在Rm -* R”的映射上，并建立了附在这个映射上的复形的许多形式性质，这将在本文的其余部分以及随后的部分中需要。§§3和4在这里被包括以给出如何可以推广E-序列的概念的指示，并且表明在局部环上，E-序列的这种推广的概念（如在通常情况下）与阶无关。在随后的论文中，我们将研究这些更一般的思想，将它们相互联系起来（如[1]），并将它们应用于由雅可比矩阵的未成年人产生的理想定义的簇的奇异簇的情况。还可以提到的是，通常的Koszul复形在研究某些特殊模的不变因子[2]中起着作用，我们将随后展示与映射/：Rm-> R”相关联的复形的同调群与/的上核的不变因子之间的联系。若R是Noether映射，且Coker f有有限长，则Coker Sp（f）对每个p都有有限长，其中Sp（f）：SP（R“）-* Sp（R”）表示诱导映射
Introduction. In [1], the Koszul complex was used to study the relationship between codimension and multiplicity. It also helped us investigate Macaulay modules and rings, and provided a context in which to prove the Cohen-Macaulay Theorem concerning the unmixedness of complete intersections. Now there is a generalization of the Cohen-Macaulay Theorem (known, we believe, as the generalized Cohen-Macaulay Theorem) which has to do with the unmixedness of an ideal generated by the minors of a matrix. Since the Koszul complex can be thought of as a complex associated with a map of R" -* R, where R is a commutative ring, it seemed likely that there should be a complex associated with a map of Rm-> R". In fact, as long as one is willing to go that far, why not look for some complex associated with an arbitrary map of modules over any commutative ring. In this paper, we define a complex associated with a map of modules. In §1, we discuss this complex in complete generality. In §2, we pretty much restrict our attention to maps of Rm -* R", and establish many of the formal properties of the complexes attached to this map which will be needed in the rest of this paper as well as in subsequent ones. §§3 and 4 are included here to give an indication of how the notion of E-sequence may be generalized, and to show that over local rings this generalized notion of £-sequence is (as in the usual case) independent of order. In subsequent papers we shall investigate these more general ideas, relating them to each other (as in [1]) and also applying them to the case of the singular variety of a variety which is defined by the ideal generated by the minors of the Jacobian matrix. It might also be mentioned that the usual Koszul complex plays a role in studying the invariant factors [2] of certain special modules, and we shall subsequently show the connection between the homology groups of the complex associated with a map/:Rm-> R", and the invariant factors of the cokernel of/ Another reason for looking at complexes associated with maps/:Rm-> R" is the following. If R is noetherian, and coker / has finite length, then coker Sp(f) has finite length for every p, where Sp(f) :SP(R'") -* Sp(R") denotes the induced map