The Quillen - Suslin theorem and the structure of n-dimensional elementary polynomial matrices
The Quillen - Suslin theorem and the structure of n-dimensional elementary polynomial matrices
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DOI：
10.1109/tcs.1984.1085545
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发表时间：
1984-06
期刊：
影响因子：
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通讯作者：
D. Youla;P. Pickel
中科院分区：
文献类型：
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作者：
D. Youla;P. Pickel
显然，在任何一般的n维线性系统理论中，利用初等多项式矩阵的性质是必要的。在最近的一篇论文[5]中，有人指出，这种矩阵的内部结构可以用三种不同但同样有意义的方式来概念化。设A（z）\equiv A（z_{1}，z_{2}，\cdots，z_{n}）表示n个变量z_{i}的m \times r多项式矩阵，i = 1 \rightarrow n，其中m \leq r .我们说A（z）是射影自由的（PJF），如果它可以被包含为某个r \times r初等多项式矩阵的前m行;说它是幺模的（UM），如果存在r \times m多项式矩阵B（z）使得A（z）B（z）= l_{m};说它是零素的（ZP），如果它的^{r}C_{m} m \times m子式没有公共零点。虽然很容易证明PJF \rightarrow UM \rightarrow ZP和ZP \rightarrow UM [5]，但直到1976年，Quillen [7]和Suslin [10]才（独立地）建立了Serre在1957年提出的猜想UM \rightarrow PJF。在他们的证明中包含的两个主要思想是相当显着的，有很强的综合理论的味道。我们在本文中的目的是解释这些想法，给出一个基本的教程帐户的奎伦-苏斯林定理，是完全用多项式的语言，只使用最少的现代抽象代数。这里提出的发展适用于具有真实的或复系数的多项式。那些对更一般的系数字段感兴趣的人（例如，[11]《礼记》云：“礼者，礼也。在整个过程中，我们一直试图提出明确的建设，在所有的证据。附录涉及的结果UM \rightarrow PJF的标准形式的塞尔猜想投射模。
Apparently, in any general theory of linear n -dimensional of an systems, it is necessary to exploit the properties of elementary polynomial matrices. In a recent paper [5], it was shown that the internal structure of such matrices could be conceptualized in three distinct but equally meaningful ways. Let A(z) \equiv A(z_{1}, z_{2}, \cdots ,z_{n}) denote an m \times r polynomial matrix in the n variables z_{i}, i = 1 \rightarrow n , where m \leq r . We say that A(z) is projectively free (PJF), if it can be included as the first m rows of some r \times r elementary polynomial matrix, that it is unimodular (UM), if there exists an r \times m polynomial matrix B(z) such that A(z)B(z) = l_{m} , and that it is zero-prime (ZP), if its ^{r}C_{m} m \times m minors are devoid of common zeros. Although it is easily shown that PJF \rightarrow UM \rightarrow ZP and that ZP \rightarrow UM [5] , it was not until recently, in 1976, that the conjecture UM \rightarrow PJF made by Serre in 1957 was established (independently) by Quillen [7] and Suslin [10]. The two major ideas contained in their proofs are quite remarkable and have a strong synthesis-theoretic flavor. Our purpose in this paper is to explain these ideas by giving an elementary tutorial account of the Quillen-Suslin theorem that is couched completely in the language of polynomials and uses only a minimum of modern abstract algebra. The development presented here applies to polynomials with real or complex coefficients. Those interested in more general coefficient fields (e.g., finite fields) are encouraged to go on to [11]. Throughout we have attempted to present explicit constructions in all proofs. An appendix relates the result UM \rightarrow PJF to the standard form of the Serre conjecture for projective modules.