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Abstract: For a smooth affine algebra of dimension $d$ over an
algebraically closed field $k$ with $d!\in k^{\times}$, it is known that
stably isomorphic projective modules of rank at least $d$ are isomorphic.
Also, this is known not to be true in general when the modules have rank
less than $d-1$.
In this paper (https://arxiv.org/abs/2111.13088) by Fasel, the above is
extended to modules of rank $d-1$ using the \mathbb{A}^1-$homotopy theory.
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