Unimodular rows over Laurent polynomial rings

Abdessalem Mnif; Morou Amidou

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 4, page 927-934
  • ISSN: 0011-4642

Abstract

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We prove that for any ring 𝐑 of Krull dimension not greater than 1 and n ≥ 3 , the group E n ( 𝐑 [ X , X - 1 ] ) acts transitively on Um n ( 𝐑 [ X , X - 1 ] ) . In particular, we obtain that for any ring 𝐑 with Krull dimension not greater than 1, all finitely generated stably free modules over 𝐑 [ X , X - 1 ] are free. All the obtained results are proved constructively.

How to cite

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Mnif, Abdessalem, and Amidou, Morou. "Unimodular rows over Laurent polynomial rings." Czechoslovak Mathematical Journal 72.4 (2022): 927-934. <http://eudml.org/doc/298940>.

@article{Mnif2022,
abstract = {We prove that for any ring $\{\bf R\}$ of Krull dimension not greater than 1 and $n\ge 3$, 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.},
author = {Mnif, Abdessalem, Amidou, Morou},
journal = {Czechoslovak Mathematical Journal},
keywords = {Quillen-Suslin theorem; stably free module; Hermite ring conjecture; Laurent polynomial ring; constructive mathematics},
language = {eng},
number = {4},
pages = {927-934},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Unimodular rows over Laurent polynomial rings},
url = {http://eudml.org/doc/298940},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Mnif, Abdessalem
AU - Amidou, Morou
TI - Unimodular rows over Laurent polynomial rings
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 4
SP - 927
EP - 934
AB - We prove that for any ring ${\bf R}$ of Krull dimension not greater than 1 and $n\ge 3$, 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.
LA - eng
KW - Quillen-Suslin theorem; stably free module; Hermite ring conjecture; Laurent polynomial ring; constructive mathematics
UR - http://eudml.org/doc/298940
ER -

References

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