Stably free modules over 𝐑[𝐗] of rank >dim𝐑 are free

Yengui, Ihsen

doi:10.1090/S0025-5718-2010-02427-5

Stably free modules over $\mathbf {R}{[X]}$ of rank $> \dim \mathbf {R}$ are free
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Math. Comp. 80 (2011), 1093-1098 Request permission

Abstract:

We prove that for any finite-dimensional ring $\mathbf {R}$ and $n\geq \dim \mathbf {R} +2$, the group $\textrm {E}_{n}(\textbf {R}[X])$ acts transitively on $\textrm {Um}_{n}(\mathbf {R}[X])$. In particular, we obtain that for any finite-dimensional ring $\mathbf {R}$, all finitely generated stably free modules over $\mathbf {R}[X]$ of rank $> \dim \mathbf {R}$ are free. This result was only known for Noetherian rings. The proof we give is short, simple, and constructive.

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