4
$\begingroup$

In EGAIV$_3$ 8.9.1 it is written: "Si $\mathcal F$ est un $O_X$-Module quasi-cohérent de présentation finie[...]"

Is "$\mathcal F$ de présentation finie" not the same as "$\mathcal F$ admet une présentation finie" of EGAI 5.2.5., like the first one is global(can't really imagine that) and the second one local or something like that?

If it the same then de présentation finie imply quasi-cohérent. I'm confused!

Improve this question
$\endgroup$
1
  • 8
    $\begingroup$ Logically it is redundant, but philosophically it is not: faithfully flat descent theory concerns quasi-coherent sheaves of modules, and so in any questions of descent with sheaves of modules (even without flatness) it is natural to always first say that one's sheaf of modules is quasi-coherent and to then impose whatever extra conditions one wishes on top of that (e.g., to characterize among the quasi-coherent sheaves which ones arise by base change from coherent sheaves on a descent of the scheme to a noetherian subring of the base ring). $\endgroup$
    – BCnrd
    Sep 21, 2010 at 17:05

1 Answer 1

Reset to default
4
$\begingroup$

Well, a finitely presented sheaf of $\mathcal{O}_X$-modules is a cokernel of a map between quasi-coherent $\mathcal{O}_X$-modules, so it is itself quasi-coherent. So yes, the "quasi-coherent" is redundant in your sentence.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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