The category of $\mathcal{O}\_X$ modules on a scheme $X$ has enough injectives, every sheaf can be inbedded in an injective sheaf. Now if I take a quasi-coherent sheaf, is this hull again quasi-coherent, and how does one go about proving this? I did not find this fact in Hartshorne.



The injective hull of a quasi-coherent sheaf.
It is an exercise in Hartshorne that every quasi-coherent sheaf in a noetherian scheme can be embedded in an injective quasi-coherent sheaf (see Hartshorne, Chapter III, exercise 3.6). 


EDIT As Brian points out below, this doesn't answer the question since the author is looking for an injective object in the category of $O\_X$-modules that is quasi-coherent. END-EDIT


See Chapter II.7 in Residues and Duality for more details (including ``general nonsense'' about injective hulls).


EDIT2 In particular, Theorem II.7.18 seems to be very close to what you are looking for. END-EDIT2


