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See this question: Link

I have exactly the same question, but I feel none of the questions explain why the proof fails in the infinite case. I am not looking for a counterexample. I have two related questions too:

1) Is an arbitrary product(respectively, coproduct) of quasicoherent sheaves on a scheme quasicoherent?

2) Is a finite product of quasicoherent sheaves on a scheme quasicoherent?

The products and coproducts are in the category of $\mathcal{O}_X$ modules, NOT that of quasicoherent sheaves. I think I have proven 2), but I'm not sure.

Jehu314
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For the questions listed in the body of your post:

In the infinite case: for infinite coproducts over reasonably nice spaces, this is easy. Over a noetherian scheme (or more generally a noetherian topological space) one may check that the direct sum of the local presentations for each quasi-coherent sheaf in the coproduct add together to form local presentations for the direct sum. (Over a non-noetherian topological space, one must take the presheaf given by $U\mapsto \bigoplus_{i\in I}\mathcal{F}_i(U)$ and sheafify.) For products, things are more complicated. Quasi-coherent sheaves on a scheme is a Grothendieck Abelian Category, and in particular has arbitrary products. See here for references to Gabber's proof of this fact.

For the finite case: Yes, and this may be deduced from the fact that finite products and coproducts are equivalent in abelian categories like the category of quasi-coherent sheaves on a scheme (and then one applies the argument involving coproducts from the first paragraph).

To address the issue that comes up in the linked proof, the problem is that the listed exact sequence with direct sums should really be a sequence of direct products, which are not necessarily equivalent to direct sums except when the index set is finite.

KReiser
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  • Thanks, but I was asking about products in the category of Sheaves over a scheme, not the category of QUASI-COHERENT sheaves. – Jehu314 Jan 08 '19 at 08:05
  • Also, I don't quite understand your statement about infinite coproducts. Each point has an open neighbourhood on which the sheaf has a presentation, but if there are infinitely many sheaves in the coproduct, why is there necessarily an open neighbourhood on which ALL the sheaves have a presentation? – Jehu314 Jan 08 '19 at 08:15
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    I had hastily omitted a condition: one needs a noetherian topological space for infinite coproducts to avoid sheafifying. And as long as one is working in sheaves of $\mathcal{O}_X$ modules, there is no need to adjust anything here - quasicoherent sheaves are a full subcategory of $\mathcal{O}_X$ modules so any categorical construction made in the smaller category is automatically the correct construction in the larger. – KReiser Jan 08 '19 at 08:35
  • So, if the underlying space of a scheme is noetherian, can you define coproducts as $U \rightarrow \bigoplus \mathcal{F}_{i}(U) $ without sheafifying? – Jehu314 Jan 08 '19 at 08:38
  • And are you sure that limits in a full subcategory are limits in the bigger category? – Jehu314 Jan 08 '19 at 08:42
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    Yes to the first. After doing a bit of reading, it appears that the product side is perhaps more complicated than I originally thought (and one should entirely ignore the last sentence of my previous comment, it is incorrect). You'll want to go and have a look at Thomason and Trobaugh "Higher algebraic K-theory of schemes and of derived categories" in The Grothendieck Festschrift, Vol. III, 247--435, Progr. Math., 88, Birkhäuser, Boston, 1990. (MR11069118) for probably the best reference of figuring this out. Either way, it is surprising to me that you are worrying so much about... – KReiser Jan 08 '19 at 10:16
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    ... this interaction between qc sheaves and arbitrary sheaves this (apparently) early in your algebraic geometry career. The reason qc sheaves of modules are worked with is that they behave reasonably nicely - until you have a specific research problem involving some sheaf which is not qc, it will save you some amount of headaches if you do not worry to hard about them (similar to the way in which one does not worry too much about very large groups until one has to). – KReiser Jan 08 '19 at 10:20
  • Thanks, Have you got a reference for the coproduct thing?(I presume it's true for all colimits) – Jehu314 Jan 08 '19 at 12:19