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This is more-or-less question (3) on page 170 of Quillen's "Projective Modules over Polynomial Rings" (link):

Let $A$ be a regular Noetherian ring and let $f \in A$ be an element of $A$ such that $A/(f)A$ is a regular ring. Let $N$ be a finitely generated projective $A[f^{-1}]$-module. Does there necessarily exist a finitely generated projective $A$-module $M$ such that $M \otimes_{A} A[f^{-1}] \simeq N$ as $A[f^{-1}]$-modules?

Keywords: vector bundle, complement, divisor, smooth

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    $\begingroup$ This is known to be true for such local rings containing a field. The affine algebra case was known by a result of Bhatwadekar and Rao (Transactions of the AMS, 1983). Then by a yoga invented by D. Popescu, Popescu deduces it for rings containing a field. $\endgroup$
    – Mohan
    Commented May 21, 2018 at 22:06
  • $\begingroup$ @Mohan Thanks. I remembered just now that Gabber proves the case when $A$ is regular local of dimension 3, in his thesis "Some theorems on Azumaya algebras", Chapter 1, Theorem 1. $\endgroup$
    – Minseon Shin
    Commented May 22, 2018 at 2:19
  • $\begingroup$ See also Rao, "On Projective $R_{f_{1} \dotsb f_{t}}$-Modules", American Journal of Mathematics, 1985. $\endgroup$
    – Minseon Shin
    Commented Oct 3, 2018 at 20:53

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