Bass–Quillen conjecture for vector bundles on affine schemes

Let $X$ be a regular noetherian affine scheme of finite Krull dimension, and let $\mathrm{Vect}_r(X)$ denote the set of isomorphism classes of rank $r$ vector bundles on $X$. For each $n\geq 1$, consider the pullback map induced by the projection $X\times\mathds{A}^n\to X$.

Bass–Quillen conjecture. The pullback map

$$\mathrm{Vect}_r(X)\rightarrow\mathrm{Vect}_r(X\times\mathds{A}^n)$$

is a bijection for all $r\geq 1$ and all $n\geq 1$.

This generalizes the Quillen–Suslin theorem from affine space over a field to regular noetherian affine schemes. It is a fundamental problem about homotopy invariance of vector bundles and is closely related to the classification of finitely generated projective modules over polynomial rings.