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