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?


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

  • $\begingroup$ What if I take the sheaf $F$ given by $F(U)=Hom(U,GL_n)$? Oh, I see, since $GL_n$ is not Abelian for $n > 1$, it is not a module. I see that what I've written barely makes sense. Does it make sense at least for n=1, when $GL_1 = \mathcal{O}_X^*$? $\endgroup$ – Dima Sustretov Apr 21 '11 at 19:00
  • $\begingroup$ For $GL_1$, you have a sheaf of abelian groups, but not an $\mathcal{O}_X$-module. How do you take a section of $GL_1$ and multiply it by a section of $\mathcal{O}_X$ to get another section of $GL_1$? $\endgroup$ – David E Speyer Apr 21 '11 at 20:13
  • $\begingroup$ oh, I am sorry, you are right, of course. I am being dense. $\endgroup$ – Dima Sustretov Apr 21 '11 at 20:21
  • $\begingroup$ Although it goes a bit out of the scope of the question, I hope you don't mind me asking if there is some characterisation of affine schemes that can only have trivial vector bundles over them? $\endgroup$ – Dima Sustretov Apr 21 '11 at 20:30
  • 4
    $\begingroup$ 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. $\endgroup$ – David E Speyer Apr 23 '11 at 4:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.