4
$\begingroup$

Exercise 14.1.I of Ravi Vakil's notes is the following:

Show that locally free sheaves on Noetherian normal schemes satisfy "Hartog's Lemma": sections defined away from a set of codimension at least 2 extend over the set.

With his "Algebraic Hartog's Lemma" (For a Noetherian normal ring R, the intersection (in the field of fractions) of $R_p$ where $p$ ranges over all codimension one primes is R), I can show the statement for free sheaves. But how does one extend this to locally free sheaves?

  • 2
    I am not an algebraic geometer, but if you have a locally free sheaf, why can't you just restrict to a cover of open sets where you are free, extend in each of them using what you've proved, and then glue the sections back together?2012-05-13
  • 0
    @froggie: That works so long as you know the extensions are unique – otherwise it won't satisfy the matching conditions needed to glue sections together. This tends to be a problem when the sheaf isn't sufficiently "separated" as a topological space...2012-05-13
  • 4
    @ZhenLin: I don't think there is a problem. As the scheme (say $X$) is locally integral, the restriction map on $O_X$ to a dense open subset is always injective. Sections of $O_X$ can be viewed as morphisms to the affine line and the latter is separated.2012-05-13
  • 0
    @froggie: It appears I made the mistake of confusing "sections defined away from a set of codimension at least 2" with "sheaves defined away from a set of codimension at least 2." With that mistake cleared up, I think what you're saying works...2012-05-13
  • 0
    @only For future reference, its **Hartogs**, not Hartog. See [Wikipedia](http://enwp.org/wiki/Friedrich_Hartogs).2013-06-06

0 Answers 0