Let $R$ be a discrete valuation ring (DVR) and let $M$ be a projective module of finite type over the polynomial ring $R[t]$. Is $M$ free over $R[t]$?


As far as I understand, this should be a consequence of the Bass-Quillen conjecture for $R$. Is it proven in this particular case?



Are finite projective modules over $R[t]$ free when $R$ is DVR?
The Bass-Quillen conjecture is known to be true for principal ideal domains (that is, if $A$ is a PID, all finitely generated projective modules over $A[T\_1,\dots,T\_n]$ are free). This was proven in theorem 4 of the paper



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> *Quillen, Daniel*, [**Projective modules over polynomial rings**](http://dx.doi.org/10.1007/BF01390008), Invent. Math. 36, 167-171 (1976). [ZBL0337.13011](https://zbmath.org/?q=an:0337.13011).
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