# injective objects

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

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 (There was a previous unpublished proof by O. Gabber).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.

• For an english translation of Grothendieck's Tohoku paper by Michael Barr, look at ftp.math.mcgill.ca/pub/barr/pdffiles/gk.pdf Sep 22, 2010 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!) May 31, 2012 at 16:14
• @Ravil Vakil Here is a working link: math.mcgill.ca/barr/papers/gk.pdf Oct 15, 2015 at 9:51

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).

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.