Algebraic vector bundles on spheres (Q2921102)

This paper is part of a collection by the authors including [Duke Math. J. 163, No. 14, 2561--2601 (2014; Zbl 1314.14044); J. Am. Math. Soc. 28, No. 4, 1031--1062 (2015; Zbl 1329.14045)], where they effectively use the theory of \(\mathbb{A}^1\)-homotopy developed by \textit{F. Morel} and \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)] to settle various questions about vector bundles on affine varieties, among other interesting results.NEWLINENEWLINEThe sphere here refers to smooth quadric hypersurface of the form, \(Q_{2n-1}\subset \mathbb{A}^{2n}\), defined by the equation \(\sum_{i=1}^n x_iy_i=1\) which is a smooth affine model of an \(\mathbb{A}^1\)-homotopy sphere as well as an universal example for unimodular rows.NEWLINENEWLINELet me describe the application (Section 5) of their results for unimodular rows here. If \((a_0,a_1,\dots,a_n)\) is a unimodular row over a commutative ring \(R\), \textit{A. A. Suslin} had shown that the row \((a_0^{r_0},\dots,a_n^{r_n})\) can always be completed to a non-singular \((n+1)\times(n+1)\) matrix over \(R\) if \(n!\) divides \(\prod r_i\) [Mat. Sb., Nov. Ser. 102(144), 537--550 (1977; Zbl 0354.13005)].NEWLINENEWLINEM.~V.~Nori had asked a more general question inspired by Suslin's theorem, which is as follows. Let \(\phi:R=k[x_0,x_1,\dots,x_n]\to A\) be a \(k\)-algebra homomorphism and let \(f_0,\dots,f_n\in R\) be such that they vanish only at the origin and the length of \(R/(f_0,\dots,f_n)\) is a multiple of \(n!\). Assume that the ideal generated by \(\phi(x_i)\) is the whole ring \(A\), so that \((\phi(f_0),\dots,\phi(f_n))\) is a unimodular row over \(A\). Can this row be completed? The reviewer showed that this is indeed true if we assume that \(k\) ia algebraically closed and the \(f_i\)s are homogeneous [J. Algebra 191, No. 1, 228--234, Art. No. JA966923 (1997; Zbl 0901.13009)]. The second author showed that in general, Nori's question has a negative answer [J. Ramanujan Math. Soc. 27, No. 1, 21--40 (2012; Zbl 1257.19001)] wherein he proposed a refinement of Nori's question. In this paper, the authors prove the refined version.