# Taking stalk of a product of sheaves

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.

-

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.