Question

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?

Answer 1

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.]

Answer 2

Restriction to an open subset has an exact left adjoint (extension by zero).

Answer 3

This is false is general. In particular, if $X = SpecA$ is affine, this would imply that given an injective $A$-module $M$ and $f \in A$, one would have $M_f$ is injective over $A_f$; this is FALSE in general (see, for example, "Localization of Injective Modules" by Everett C. Dade (it's in Journal of Algebra ~ April 1981)).

Maybe you need to assume that $X$ is locally Noetherian, or even Noetherian?

Answer 4

Hey, I am the asker for the question, I want to say two points:

1. I forgot the Noetherian condition; X should be Noetherian;

2. I lose the cookie so I can't log in that account any more; I don't know how to add comment or reply others... Why I don't have that button...