# “Hartog's Lemma” for locally free sheaves on a Noetherian normal scheme

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?

-
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? – froggie May 13 '12 at 18:59
@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... – Zhen Lin May 13 '12 at 19:22
@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. – user18119 May 13 '12 at 20:11
@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... – only May 13 '12 at 21:35
@only For future reference, its Hartogs, not Hartog. See Wikipedia. – Lord_Farin Jun 6 '13 at 15:19