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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!

2 Answers 2

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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.

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    This is nice. Thanks a lot!!2011-07-29
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    The argument that it is a Grothendieck abelian category is very nice: that wasn't the approach in Hartshorne ("Residues and Duality") at all.2011-07-29
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    @Akhil: The article of Thomason-Trobaugh is full of such treasures, definitely worth looking at. I don't remember what exactly Hartshorne (or rather Grothendieck) does in R&D.2011-07-29
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    @Theo: Thanks, I will have a look at it. By the way, according to Brian Conrad's book "Grothendieck duality and base change," (available on his webpage) a result of Gabber (Lemma 2.1.7) shows that there is indeed a set of generators on *any* scheme (and in particular, one gets a Grothendieck abelian category).2011-07-29
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    @Akhil: Thanks a lot for that. Good to know!2011-07-29
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    @user8249: Akhil pointed out that the result seems to be true without any restrictions on the scheme due to a result due to Gabber (see edit).2011-07-29
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    @Theo: Thanks for the link to my blog and the kind words!2011-08-04
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    @Akhil: It was a pleasure to find it! The first comment to your post puzzled me a bit, and I'm still not sure I understand it. However, it may be referring to what Theo J.-F. mentions [here](http://mathoverflow.net/questions/49175/is-every-saft-category-cocomplete) (a cat. is "saft" if "cocontinuous = has right adj."): "It is an easy exercise that a category $C$ is saft if and only if every continuous functor $C^{\rm op} \to Set$ is representable. In particular, any saft category is complete, because any putative limit corresponds to a continuous functor, and hence necessarily representable."2011-08-04
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    @Theo: Thanks for the reference to the MO post. I'm still unsure about this (I never really understood adjoint functor theorems particularly well anyway), but probably will end up learning more category theory next year anyway, so might eventually!2011-08-07
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This was already discussed here:

https://mathoverflow.net/questions/39941/does-qcohx-admit-a-generating-set

https://mathoverflow.net/questions/40587/quasi-coherent-envelope-of-a-module

It should be remarked that infinite products in categories of quasi-coherent sheaves are not well-behaved at all. For example, they are not compatible with restriction maps of open sets. While it is true that this category is presentable, I don't think that this will be really useful for schemes which are not concentrated.