Geodesic
Unimodular rows over Laurent polynomial rings
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 4, pp. 927-934
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We prove that for any ring ${\bf R}$ of Krull dimension not greater than 1 and $n\geq 3$, the group ${\rm E}_{n}({\bf R}[X, X^{-1}])$ acts transitively on ${\rm Um}_{n}({\bf R} [X, X^{-1}])$. In particular, we obtain that for any ring ${\bf R}$ with Krull dimension not greater than 1, all finitely generated stably free modules over ${\bf R} [X, X^{-1}]$ are free. All the obtained results are proved constructively.
We prove that for any ring ${\bf R}$ of Krull dimension not greater than 1 and $n\geq 3$, the group ${\rm E}_{n}({\bf R}[X, X^{-1}])$ acts transitively on ${\rm Um}_{n}({\bf R} [X, X^{-1}])$. In particular, we obtain that for any ring ${\bf R}$ with Krull dimension not greater than 1, all finitely generated stably free modules over ${\bf R} [X, X^{-1}]$ are free. All the obtained results are proved constructively.
DOI : 10.21136/CMJ.2022.0002-20
Classification : 03F65, 13C10, 14Q20, 19A13
Keywords: Quillen-Suslin theorem; stably free module; Hermite ring conjecture; Laurent polynomial ring; constructive mathematics 
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Mnif, Abdessalem; Amidou, Morou. Unimodular rows over Laurent polynomial rings. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 4, pp. 927-934. doi: 10.21136/CMJ.2022.0002-20

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