The Quillen - Suslin theorem and the structure of n-dimensional elementary polynomial matrices

Abstract

Apparently, in any general theory of linear <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(z) \equiv A(z_{1}, z_{2}, \cdots ,z_{n})</tex> denote an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m \times r</tex> polynomial matrix in the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> variables <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_{i}, i = 1 \rightarrow n</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m \leq r</tex> . We say that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(z)</tex> is projectively free (PJF), if it can be included as the first <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> rows of some <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r \times r</tex> elementary polynomial matrix, that it is unimodular (UM), if there exists an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r \times m</tex> polynomial matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B(z)</tex> such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(z)B(z) = l_{m}</tex> , and that it is zero-prime (ZP), if its <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^{r}C_{m} m \times m</tex> minors are devoid of common zeros. Although it is easily shown that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PJF \rightarrow UM \rightarrow ZP</tex> and that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ZP \rightarrow UM [5]</tex> , it was not until recently, in 1976, that the conjecture <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">UM \rightarrow PJF</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">UM \rightarrow PJF</tex> to the standard form of the Serre conjecture for projective modules.