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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3$\begingroup$ 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"). $\endgroup$– BCnrdJul 12, 2010 at 17:05
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$\begingroup$ 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. $\endgroup$– faridrbJul 12, 2010 at 18:17
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$\begingroup$ jef, can you point me to that reference for "hull" in Hartshorne's book "Algebraic Geometry". I didn't know he defined that there. $\endgroup$– Karl SchwedeJul 12, 2010 at 18:50
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$\begingroup$ 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. $\endgroup$– faridrbJul 12, 2010 at 18:59
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1$\begingroup$ @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". $\endgroup$– Karl SchwedeJul 12, 2010 at 19:12
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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
answered Jul 12, 2010 at 17:04
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$\begingroup$ Karl, in that textbook exercise 3.6, "injective quasi-coherent sheaf" means "injective object in the category of qcoh sheaves" rather than "injective sheaf of modules which is also qcoh". I just mention this to clarify what you're saying (since it seems the question may be about injectives in the entire category of sheaves of modules, but hard to tell). $\endgroup$– BCnrdJul 12, 2010 at 18:14
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$\begingroup$ If I prove this exercise, doesn't it follow that this sheaf is also injective in the category of sheaves of modules? (I need this fact). Since this hull is the hull in the category of sheaves of modules, if the hull of a quasi-coherent is still quasi-coherent, then this hull is a quasi-coherent sheaf injective in the category of modules. $\endgroup$– faridrbJul 12, 2010 at 18:22
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1$\begingroup$ @jef, that is what Brian indicated in his first comment. The quote being, "on such a scheme every injective in the category of qcoh sheaves is injective in the category of all sheaves of modules. " $\endgroup$ Jul 12, 2010 at 22:14