Question

Let $X$ be an arbitrary scheme. A quasi coherent sheaf $\cal F$ is said to be injective if $Hom_{ O_X}(-, \cal F)$ is exact. We can also regard a quasi coherent sheaf $\cal G$ on $X$ such that for all open subset $U$ of $X$, $\cal G(U)$ is an injective $\cal O_X$-module. So we can ask a question that

1) Is there any relation between these two kind of sheaves?

2) Which conditions on $X$ (or on $\cal F$) are needed to regard the first kind of these sheaves ($\cal F$) equivalent to the second one?

Answer 1

The condition you want is $X$ a locally noetherian scheme. Then by Hartshorne's "Residues & Dualities," Proposition 7.17, $\cal{F}$ is an injective ${\cal O}_X$-module if and only if for each $x \in X$, the stalks ${\cal F}_x$ are injective ${\cal O}_x$-modules. If the sections are injective ${\cal O}_X(U)$-modules, that should give injectivity on the stalks (abelian groups form a locally noetherian Grothendieck category, so use e.g., Henning Krause's "The Spectrum of a Module Category" Proposition A.11, which says direct limits of injective objects are injective). For the reverse question, I think you need $X$ to be noetherian.