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A note on unimodular rows. (English) Zbl 0901.13009

The aim of the paper is to show the following: Let \(k\) be an algebraically closed field and let \(R=k [X_0, \dots, X_n]\). Suppose we are given the unimodular row \((f_0, \dots, f_n)\) with homogeneous polynomials. Assume \(n!\mid \ell(R/(f_0, \dots, f_n))\), where \(\ell\) means length. Then the unimodular row \((f_0, \dots, f_n)\) is completable, i.e. there is \(A\in\text{GL}(n+1,R)\) such that \(A(f_0, \dots, f_n)= (1,0, \dots,0)\). This gives an answer in a special case to a question of Nori about unimodular rows, a question generalizing a theorem of Suslin.

MSC:

13C10 Projective and free modules and ideals in commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
19A13 Stability for projective modules
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References:

[1] Mumford, D., Algebraic Geometry I, Complex Projective Varieties, Grundlehren der mathematischen Wissenschaften (1976), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0356.14002
[2] Quillen, D., Projective modules over polynomial rings, Invent. Math., 36, 166-172 (1976)
[3] Suslin, A. A., Stably free modules, Mat. Sb., 102, 537-550 (1977) · Zbl 0389.13002
[4] Swan, R. G., Vector bundles, projective modules and the \(K\), John Moore Conference. John Moore Conference, Annals of Mathematics Studies, 113 (1987), Princeton Univ. Press: Princeton Univ. Press Princeton, p. 432-522
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