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Let $X$ be a scheme and $F$ be an injective object of $\mathrm{Qcoh}(X)$. Is it true that $F$ is acyclic with respect to the usual sheaf cohomology?

For noetherian schemes $X$ this is well-known; then $F$ even turns out to be flasque. I don't really care for pathological schemes, but I would like to know if it's true for quasi-compact quasi-separated schemes.

If $X$ is affine, then it is also well-known (no matter if $X$ is noetherian or not), because then actually every quasi-coherent module is acyclic. Remark that even on an affine scheme, whose underlying topological space is noetherian, there are injective quasi-coherent modules which are not flasque (SGA 6, Exp. II, App. I); but of course this does not influence the answer.

The background is that I would like to define cohomology within $\mathrm{Qcoh}(X)$, without using the category of (not necessarily quasi-coherent) $\mathcal{O}_X$-modules or even all sheaves on $X$. This works because $\mathrm{Qcoh}(X)$ is a Grothendieck category (without any assumptions on $X$), thus has enough injective objects. This cohomology would turn out to be the usual sheaf cohomology (i.e. computed in $\mathrm{Sh}(X)$) if and only if injective objects are acyclic with respect to the usual sheaf cohomology.

EDIT: a-fortiori answers the question affirmatively if $X$ is quasi-compact and semi-separated. Is there any chance to get the result also when $X$ is just assumed to be quasi-compact and quasi-separated? I've already convinced myself that the proof cannot be translated verbatim.

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Proposition B.8 in Thomason-Trobaugh, Higher Algebraic K-Theory of Schemes (in the Grothendieck Festschrift) has the case $X$ quasi-compact and semi-separated. –  user2035 Feb 24 '12 at 14:25
    
I don't see why the proof that every injective sheaf is flabby (Hartshorne p. 207) should not work for $F$? –  doug Feb 24 '12 at 14:29
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@a-fortiori: Great! Please post this as an answer (it is not just a comment). @Xogn: Extension by zero kills quasi-coherence. –  Martin Brandenburg Feb 24 '12 at 14:30
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The quasi-compact semi-separated case can also be found in Daniel Murfet's notes. See Section 6 of therisingsea.org/notes/… –  Philipp Hartwig Feb 24 '12 at 20:40
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@Zhen Lin: I regard $\mathrm{Qcoh}(X)$ as a cocomplete tensor category; its unit is $\mathcal{O}_X$. This is justified by the fully faithfulness of $X \mapsto \mathrm{Qcoh}(X)$, which I have recently proven with Alexandru Chirvasitu (arxiv.org/abs/1202.5147). Currently I think about cohomology theory internal to an abelian tensor category (and wonder if anyone has done this so far ...). –  Martin Brandenburg Feb 25 '12 at 22:47

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The case of quasi-compact semi-separated schemes is treated in the references given in the comments above.

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