# The injective hull of a quasi-coherent sheaf.

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.

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See Hartshorne's book "Residues & Duality" for a nice discussion of the structure of injectives in QCoh($X$) for any (locally?) noetherian scheme $X$. His methods there show that on such a scheme every injective in the category of qcoh sheaves is injective in the category of all sheaves of modules. That sounds like it may be an affirmative answer to your question (but hard to tell, since not sure what you mean by "this hull" and by "Hartshorne"). –  BCnrd Jul 12 '10 at 17:05
I meant the hull defined in Hartshorne's book Algebraic Geometry for any module. I'd like to know if this hull is still a quasi-coherent sheaf. –  jef Jul 12 '10 at 18:17
jef, can you point me to that reference for "hull" in Hartshorne's book "Algebraic Geometry". I didn't know he defined that there. –  Karl Schwede Jul 12 '10 at 18:50
It's proposition 2.2 in III.2. He inbeds every stalk in p of a sheaf in an injective module and considers this module as a sheaf on the singleton p, the he pushes those sheaves forward to the whole space and takes the product of all these sheaves. –  jef Jul 12 '10 at 18:59
@jef, thanks. The reason I was confused is because the word "hull" sometimes has a special meaning in this context (see for example Residues and Duality that Brian mentioned above). Basically an injective hull of a module $N$ is an injective object $I$ containing $N$ that is also in some sense "minimal" (meaning, that any non-zero subobject $J \subseteq I$ satisfies $J \cap N \neq 0$) An extension satisfying this type of property is often called "essential". –  Karl Schwede Jul 12 '10 at 19:12

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).