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Let $(\mathscr{F}_\alpha)_\alpha$ be a family of sheaves on $X$, and $\prod_\alpha\mathscr{F}_\alpha$ the product sheaf. If $x\in X$, is it true that $$\left(\prod_\alpha\mathscr{F}_\alpha\right)_x\simeq\prod_\alpha(\mathscr{F}_\alpha)_x \ ?$$ I think $(\oplus_\alpha\mathscr{F}_\alpha)_x\simeq\oplus_\alpha(\mathscr{F}_\alpha)_x$ may be true, but not the product sheaf.

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up vote 12 down vote accepted

You are right. It is true for direct sums, even arbitrary colimits (since colimits commute with colimits, and remember the description of the stalk as a colimit; or represent it as a pullback functor to the point), and also for finite products, even more generally for finite limits (since these commute with filtered colimits in, say, algebraic categories, in which the sheaves should live). But it is not true for infinite products.

There is always a canonical map $(\prod_{\alpha} F_{\alpha})_x \to \prod_{\alpha} (F_{\alpha})_x$. But it doesn't have to be injective, even for very nice spaces $X$ and sheaves $F_{\alpha}$. Take $X=\mathbb{R}$ and $F_{\alpha}$ the sheaf of continuous function for $\alpha \in \mathbb{N}$, and $x=0$. Let $f_{\alpha} : \mathbb{R} \to \mathbb{R}$ be a continuous function which vanishes on $]-1/(\alpha+1),+1/(\alpha+1)[$, but does not vanish at $1/{\alpha}$. Then $(f_{\alpha})_{\alpha}$ represents an element in the kernel of the canonical map, which is not trivial.

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