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Here's something I've been wondering about for a few weeks:

Consider a topological space $X$ and a sheaf of rings $\mathscr O_X$ on $X$. Suppose $\mathscr{F}$ and $\mathscr{G}$ are $\mathscr O_X$ modules.

We can define a presheaf on $X$ via $U\mapsto \mathscr{F}(U)\otimes_{\mathscr O(U)}\mathscr{G}(U)$. This is usually not a sheaf. (For instance one can take $X = \mathbb{P}^1_k$ over some algebraically closed $k$ and ${\mathscr O}_X$ the structure sheaf, $\mathscr{F} = \mathscr O_X(n)$ and $\mathscr{G} = \mathscr O_X(m)$, where these are the $\mathscr O_{X}$ modules of invertible sheaves attached to a divisor of degree $-n,-m$ respectively, and $n,m > 0$.) We write $\mathscr{F}\otimes_{\mathscr O_X}\mathscr{G}$ for the sheaf associated to this presheaf.


Here is the general question:

Is there some kind of "obstruction theory" or homotopy theoretic-gadget that will explain when various colimit constructions of sheaves in the category of presheaves fail to be sheaves?

However, this being rather vague, here are two examples of specific questions that I would consider special cases of the above:

Let $X$ be a topological space (perhaps with some "nice" properties, like locally compact Hausdorff), and $\mathscr O_X$ be the ring of complex-valued continuous functions on $X$. Are there homotopy-theoretic conditions we can place on $X$ to ensure that for any two "nice" (perhaps finite-type or coherent) sheaves $\mathscr{F},\mathscr{G}$ of $\mathscr O_X$-modules, the tensor presheaf is a sheaf? (One could also ask this question for arbitrary direct sum presheaves, etc.)

And:

Let $X$ be a scheme, $\mathscr{F},\mathscr{G}$ two quasicoherent $\mathscr O_X$ modules. Are there conditions on $X$ such that $U\mapsto \mathscr{F}(U)\otimes_{\mathscr O_X(U)}\mathscr{G}(U)$ is already a sheaf? (I'd be interested to see an example also where $X$ is not affine and yet $U\mapsto \mathscr{F}(U)\otimes_{\mathscr{O}(U)}\mathscr{G}(U)$ is also a sheaf, if this is possible at all).

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side remark: when U is affine in X, $(F \otimes G)(U) = F(U) \otimes G(U)$ (at least when $F$ and $G$ are quasi-coherent). I guess abstractly this is just stating the existence of a basis for the topology of $X$ which plays well with the given colimit construction. (I repeat: side remark) –  Jacob Bell Apr 25 '13 at 18:09
    
@Jacob: Yes for qc, this is essentially the third paragraph. –  Jason Polak Apr 25 '13 at 18:11
    
"On the other hand, if X=Spec(A) is an affine scheme and F,G are quasicoherent then the above presheaf is a sheaf." Are you sure? The associated sheaf has the same sections as the presheaf on every open affine subset, and then also on every quasicompact open subset, but I think that for general open subsets it will be wrong. You should give $A=k[x_1,x_2,\dotsc]$ and $U=\cup_i D(x_i)$ a try. –  Martin Brandenburg Jun 17 '13 at 15:45
    
@Martin: Ah OK, you are right, I should have been more careful. I guess that presheaf is a sheaf whenever every open sub scheme is affine but this almost never happens. I wonder if there is a reasonable weaker hypothesis. –  Jason Polak Jun 17 '13 at 17:39

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