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

Hi everybody!! I am studing cohomology of sheaves on schemes. I have a question for you. Let $X$ be a noetherian scheme. Do you know if the cathegory QCoh(X) has enough injectives? I know that it has "enough flasque" and that Mod(X) has enough injective but what about QCoh(X)? Thank you

share|improve this question

3 Answers 3

Yes. If $X$ is a Noetherian scheme, then the category of quasi-coherent sheaves on $X$ has enough injectives. See Hartshorne's Algebraic Geometry, exercise III.3.6(a).

share|improve this answer

Yes, it has enough injectives. As explained in Hartshorne's "Residues and Duality", the injective hull of a quasi-coherent sheaf in Mod(X) is again quasi-coherent. This is even more than you asked for.

Less explicit, you can show that Qcoh(X) is a Grothendieck category (see Daniel Murfet's Notes) provided X is concentrated, and as such has enough injectives.

share|improve this answer

First of all, let me quote the famous paper by Grothendieck "Sur quelques points d'algebre homologique" (Tohoku Math part 1 & part 2). According to Theorem 1.10.1 there, every abelian category satisfying (AB5) (equivalent to exactness of filtered direct limits) and with a generator has enough injectives, For every scheme $X$ it is well known that $Qco(X)$ is abelian and satisfies (AB5), see, for instance, EGA I, new edition, Corollaire (2.2.2)(iv) where it is proved that such a limit preserves quasi-coherence.

So, the only issue is the existence of a generator. For some time it was known that over a quasi-compact quasi-separated scheme, $Qco(X)$ has a generator. For a nice geometric argument, consult the proof of Theorem (4) in Kleiman's "Relative duality for quasi-coherent sheaves". Note that any noetherian scheme $X$ is quasi-compact and quasi-separated, EGA I, (6.1.1) and (6.1.13).

Surprisingly, using techniques from relative homological algebra, Enochs and Estrada proved in 2005 the existence of a generator for any scheme. See their paper "Relative homological algebra in the category of quasi-coherent sheaves". (There was a previous unpublished proof by O. Gabber).

Summing up, for any scheme $X$, the category $Qco(X)$ has enough injectives.

share|improve this answer
    
For an english translation of Grothendieck's Tohoku paper by Michael Barr, look at ftp.math.mcgill.ca/pub/barr/pdffiles/gk.pdf –  Leo Alonso Sep 22 '10 at 15:19
    
Is it possible to give a better link to the english translation of Tohoku? When I click on your link, it doesn't work. (Thanks to you and to Michael Barr!) –  Ravi Vakil May 31 '12 at 16:14

Your Answer

 
discard

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.