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A generalization of Serre’s conjecture and some related issues. (English) Zbl 1017.13006

Let \(R=[z_1,z_2, \dots,z_n]\) be the ring of polynomials over a field \(K\) and \(F\) be a \(m\times l\) matrix with coefficients from \(R\) \((m>l)\). The authors prove the equivalence of several conjectures, generalizing the well-known Serre conjecture. In order to state the authors’ main formulation (conjecture 1), it is convenient to use notations other than in the reviewed article [see I. Z. Rozenknop, “On submodules of free modules over the ring of polynomials”, Deposited in VINITI, No. 3143-74 (Moscow 1974) (Russian)]. The columns \(\lambda_1,\lambda_2, \dots,\lambda_l\) of the matrix \(F\) can be considered as elements (vectors) of a free \(R\)-module with generators, say, \(w_1,w_2, \dots, w_m\). Suppose that \(\text{rank} F =l\) and write in the exterior algebra \(R(w_1,w_2, \dots,w_m)\): \(\lambda_1\wedge \lambda_2\wedge \cdots \wedge\lambda_l =dS\), where \(d\) is the greatest common divisor of all the \(l\times l\) minors of \(F\). Then the conjecture in question can be written in the following: \(S\) is complementable, in the sense that \[ (\exists T)\;S\wedge T=w_1 \wedge w_2\wedge \cdots\wedge w_m)\Rightarrow S\text{ is linearly complementable} \] (i.e., \(T\) can be found in the form \(T=\mu_1\wedge \mu_2\wedge \cdots\wedge \mu_{m-l}\) with certain vectors \(\mu_1,\mu_2, \dots, \mu_{m-l})\). If \(d=1\), this is the original Serre conjecture [for matrices proved by D. Quillen, Invent. Math. 36, 167-171 (1976; Zbl 0337.13011) and A. A. Suslin, Sov. Math., Dokl. 17, (1976), 1160-1164 (1977); translation from Dokl. Akad. Nauk SSSR 229, 1063-1066 (1976; Zbl 0354.13010)]. In the general case for \(n=2\), see: J. P. Guiver and N. K. Bose, IEEE Trans. Cirquits Syst. CAS. 29, 649-657 (1982; Zbl 0504.65020). For \(n>2\) the question remains open so far (except of some special cases). It is necessary to emphasize the following conjecture 2: \(S\) is complementable \(\Rightarrow\) there exists a factorization \(S=\nu_1 \wedge\nu_2 \wedge\cdots \wedge\nu_l\) such that \(F=F_0 G_0\), where \(F_0\) consists of columns corresponding to \(\nu_1,\nu_2, \dots, \nu_l\) and \(G_0\) is a square \(l\times l\) matrix with \(\det G_0=d\). If \(\text{rank} F<l\), a suitable \(S=S(F)\) was indicated by the reviewer [I. Z. Rozenknop (loc. cit.)] and obviously rediscovered in other notations by Z. Lin [IEEE Trans. Circuits Syst. 35, 1317-1322 (1988; Zbl 0662.93036)]. Some relevant formulations of the generalized Serre conjecture in this case are presented in the article under review as well.
The field \(K\) is assumed algebraically closed in this work.

MSC:

13C10 Projective and free modules and ideals in commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
15A54 Matrices over function rings in one or more variables
15A23 Factorization of matrices
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References:

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