TITLE: Why is an injective quasi-coherent sheaf's restriction to an open subset still an injective object?
QUESTION [10 upvotes]: X is a Noetherian scheme, F is an injective object in the category of quasi-coherent sheaves on X. U is an open subset of X. Why F's restriction on U is still an injective object in the category of quasi-coherent sheaves on U?

REPLY [23 votes]: The restriction-by-zero type arguments can actually be made to work, with some effort and an extra hypothesis.  Suppose $X$ is locally Noetherian, $j: U \to X$ the inclusion of an open subscheme.
Let $Mod(X)$ and $QCoh(X)$ be the categories of $O_X$-modules, and quasi-coherent $O_X$-modules, respectively.
The "some effort" is the following Lemma
Lemma If $X$ is locally Noetherian, then the injective objects in $QCoh(X)$ are precisely the injective objects of $Mod(X)$ which are quasi-coherent as sheaves of modules.
Pf: Any injective object of $Mod(X)$ which is quasi-coherent must certainly be injective in the smaller category $QCoh(X)$.  For the converse, it suffices to show that any injective object $I$ of $QCoh(X)$ injects into some $I'$ which is a quasi-coherent injective object of $Mod(X)$, for then $I$ will be a retract of $I'$ and so injective in $Mod(X)$.  This seems tricky, but is proved in Theorem 7.18 of Hartshorne's "Residues and duality".

Now, let's prove the result using the Lemma:  If $J$ is an injective object in $QCoh(X)$, then the hard direction of the Lemma implies that it is injective in $Mod(X)$.  The restriction-by-zero argument applies in this category, allowing us to conclude that $j^* J$ is injective in $Mod(U)$.  It's clearly quasi-coherent, so applying the easy direction of the Lemma we see that it is injective in $QCoh(U)$ as desired.
[Aside: On a Noetherian scheme, any quasi-coherent sheaf is a union of its coherent subsheaves and one can "extend" coherent sheaves on U to coherent sheaves on X (see e.g., Hartshorne Ex. II.5.15).  Using these facts, one should be able to give a more direct argument in the Noetherian case.]