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Let $X$ be a topological space, and let $\mathscr{O}$ be a sheaf of rings on $X$. It is easy to verify that the functor $\Gamma (U, -) : \textbf{Mod}(\mathscr{O}) \to \textbf{Ab}$ is representable, and that the representation is moreover natural in $U$. An easy argument [Hartshorne, Algebraic Geometry, Ch. III, Lem. 2.4] then shows that any injective $\mathscr{O}$-module is flasque in the sense of Godement, hence acyclic for $\Gamma(X, -) : \textbf{Ab}(\textbf{Sh}(X)) \to \textbf{Ab}$ (note the change of category!).

This seems to be a bizarre proof, and not easily generalised to an arbitrary (Grothendieck) topos $\mathcal{E}$. Yet it is true: this is part of what Lemma 03FD in [Stacks] asserts. An even stronger result is true: for all objects $E$ in $\mathcal{E}$, any injective $\mathscr{O}$-module is acyclic for $\mathcal{E}(E, -) : \textbf{Ab}(\mathcal{E}) \to \textbf{Ab}$, i.e. flasque in the sense of Verdier. This is Lemma 072Z in [Stacks]. The proofs, however, go via Čech cohomology, and seem less straightforward than the proof in the special case outlined in the previous paragraph.

Question 1. What is the intuitive meaning of flasqueness in the sense of Verdier? Can it be characterised as a property of sheaves of sets?

Question 2. What is the moral reason for the flasqueness of injective $\mathscr{O}$-modules?

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Linked up the tags from the Stacks project. Hope I got it right. – Dylan Moreland Mar 31 '12 at 16:55
@Dylan: Thanks! I got one of the citations wrong, but I've replaced it with the right one now. – Zhen Lin Mar 31 '12 at 17:14

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