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?



Why is an injective quasi-coherent sheaf's restriction to an open subset still an injective object?
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".




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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.]


