Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
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
@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
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
@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
show 2 more comments

1 Answer 1

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

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.