1
$\begingroup$

I would like to know :

  • why is the category $ \mathfrak{Coh}_X $ of coherent scheaves the smallest abelian category containing line bundles ?
  • Why is the category $ \mathfrak{QCoh}_X $ of quasi coherent scheaves the smallest abelian category containing locally free scheaves ?

Thanks a lot for your help.

$\endgroup$
  • 4
    $\begingroup$ This more-or-less amounts to the observation that quasicoherent (resp. coherent) sheaves arise as the cokernels of locally free sheaves (resp. locally free sheaves of finite rank). $\endgroup$ – Zhen Lin Dec 8 '15 at 19:39

Your Answer

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

Browse other questions tagged or ask your own question.