I have a short question:
Does the category of quasi-coherent sheaves on a scheme have arbitrary products? I know that it does if the scheme is affine and I know that they will not be isomorphic to the product as $\mathcal{O}_{X}$-modules, but I haven't been able to find a counterexample.
Any help is appreciated!
Best Answer
I don't know about the answer in complete generality, but it is true for quasi-compact and quasi-separated schemes (but see the added bits and the update below).
The main point here is that under these assumptions every quasi-coherent sheaf is the (filtered) colimit of its $\text{sub-}\mathcal{O}_{X}\text{-modules}$ of finite type. One can then show that the finitely presented $\mathcal{O}_{X}\text{-modules}$ are a set of generators. Using this, and the general fact that the inclusion functor of quasi-coherent sheaves in all sheaves preserves and reflects exactness and infinite direct sums (hence all colimits), a rather straightforward application of the special adjoint functor theorem exhibits the quasi-coherent sheaves as a reflective subcategory of all sheaves. This implies that the quasi-coherent sheaves over a quasi-compact quasi-separated scheme are a Grothendieck abelian category (complete, cocomplete, filtered colimits are exact, and there is a set of generators). In particular, there are enough injectives.
For details, I refer you to Appendix B: Modules vs. Quasi-coherent modules (in particular B.2, B.3 and B.12) on pages 409ff of Thomason-Trobaugh, Higher Algebraic $K$-theory of Schemes and of Derived Categories, The Grothendieck Festschrift, Vol. III, 247–435, Progress in Mathematics, 88, Birkhäuser Boston, Boston, MA, 1990, MR 1106918. You'll find precise references to EGA and SGA in there, so that should be sufficient.
In B.2 Thomason remarks that it seemed to be unknown at that time whether the category of quasi-coherent sheaves over a general scheme has all limits (or a set of generators, or enough injectives). Since the existence of arbitrary products would give limits for free, I doubt that you can push the above much further.
Added: The reflector from sheaves to coherent sheaves (i.e. the right adjoint to the inclusion) has many very nice formal properties and is called coherator. It plays a central rôle in the paper of Thomason-Trobaugh.
Towards the end of appendix B Thomason makes several observations how one can use or modify the above arguments to recover related results in Hartshorne's Residues and Duality from the above using suitable assumptions like local noetherian, finite Krull dimension and cohomological finiteness assumptions on the right derived functor of the coherator.
Added: As Akhil points out in a comment below, according to Brian Conrad, Grothendieck duality and base change, Lemma 2.1.7 on p. 28 (in the version available on B.C.'s homepage), Ofer Gabber proved that in fact there is a set of generators on any scheme. This gives that the category of quasi-coherent schemes is Grothendieck abelian without any restrictions. Unfortunately, there is not much information how this is proved.
Update: I just found out about Akhil Mathew's blog post in which Akhil gives his own account of the arguments outlined here. While you're there, please have a look around and enjoy some of the other beautiful posts he collects there: "Climbing Mount Bourbaki, Thoughts on mathematics". Highly recommended reading.