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 quasicoherent sheaf, is this hull again quasicoherent, and how does one go about proving this? I did not find this fact in Hartshorne.

It is an exercise in Hartshorne that every quasicoherent sheaf in a noetherian scheme can be embedded in an injective quasicoherent 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 quasicoherent. ENDEDIT 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. ENDEDIT2 

