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Let $A$ be a polynomial algebra in $n$ variables over field $\mathbb{F}$ of characteristic zero which is algebraically closed. Assume that $a_1,\ldots, a_n, b_1,\ldots, b_n\in A$ are such that $a_1b_1+\ldots+ a_nb_n=1$. Consider the free left $A$-module $_{A}M$ with basis $r_1,\ldots, r_n$.

From Suslin-Quillen theorem we know that $_{A}M/(A(a_1r_1+\ldots+a_nr_n))$ is free $A$-module of rank $n-1$.

$\bf{Question.}$ Is there any formula of a basis of this module in terms of $a_i, b_i$?

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    $\begingroup$ No, it is what is known as a stably free module, but in general not free. The standard example is the tangent bundle of the real sphere. Let $A=\mathbb{R}[x,y,z]/(x^2+y^2+z^2-1)$ and take $a_1=x, a_2=y, a_3=z$. $\endgroup$
    – Mohan
    Commented Jul 7, 2020 at 17:00
  • $\begingroup$ Is there any such examples if $\mathbb{F}$ is algebraically closed. And $A$ is polynomial algebra? $\endgroup$
    – solver6
    Commented Jul 7, 2020 at 17:22
  • $\begingroup$ If $A$ is the polynomial ring, then no such exists, so called Quillen-Suslin theorem, answering a question of Serre. If $A$ is not a polynomial ring, there are many such, even for algebras over algebraically closed fields. $\endgroup$
    – Mohan
    Commented Jul 7, 2020 at 17:41
  • $\begingroup$ I edited the question. $\endgroup$
    – solver6
    Commented Jul 7, 2020 at 17:43
  • $\begingroup$ As I said, this is true, and the characteristic of the field is irrelevant. $\endgroup$
    – Mohan
    Commented Jul 7, 2020 at 17:48

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