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Serre's theorem (one of them) states that for a quasi-coherent sheaf $\mathscr F$ on an affine noetherian scheme $H^i(X,\mathscr{F})$ vanish for $i >0$. I used to think that this would imply that on an affine variety there cannot be non-trivial vector bundles, because such a bundle would define a non-trivial cocycle in $\check{H}^1(X, GL_n)$, and this is isomorphic to the derived functor cohomology $H^1(X, GL_n)$ for "good enough" schemes, and the latter vanishes by Serre's theorem.

But here they give examples of non-trivial line bundles on affine varieties.

Where is my reasoning fallacious? Is it because the trivialising opens for the vector bundle can be non-affine?

Does it mean that in general there is no simple way to classify algebraic vector bundles even on an affine variety?

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$GL_n$ is not a coherent sheaf, because it is not a sheaf of $\mathcal{O}_X$-modules.

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    I don't know, but I imagine this should be really hard. Remember that the Quillen-Suslin theorem -- every algebraic vector bundle on $\mathbb{C}^n$ is trivial -- took 20 years to prove.2011-04-23