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Let $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$ be an inverse system of sheaves of abelian groups on a space $X$. Then for any $x\in X$ we have a natural map $$\left(\lim_i \mathcal{F}_i\right)_x\rightarrow \lim_i \mathcal{F}_{i,x}.$$ However, as a was pointed out on https://mathoverflow.net/q/80322, this morphism is neither injective or surjective in general. Is there a way to gain any sort of information about the stalks of $\lim_i \mathcal{F}_i$ from the stalks of the $\mathcal{F}_i$? For instance, if the $\mathcal{F}_{i+1}\rightarrow \mathcal{F}_i$ are surjective, does that help?

As far as I know, the category $\mathbb{N}$ is not $L$-finite, as in that case the filtered colimit and $\lim_{\mathbb{N}}$ would commute by https://mathoverflow.net/q/140164.

Other Edit: Is the situation better if we assume that the limit is exact, by which I mean that the higher limits $R^i\lim_{j\in I} \mathcal{F}_j=0$ for $i>0$ (where by better I mean there we can get some information from $\lim_i \mathcal{F}_{i,x}$ about $\left(\lim_i \mathcal{F}_i\right)_x$)?

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  • $\begingroup$ I've added a setup where I think taking stalks commutes with limits, but I'm not sure tbh $\endgroup$ Dec 6, 2020 at 1:11
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    $\begingroup$ I don't think you'll get a nice answer at this level of generality, but I might have missed something. Your edit isn't quite right though. What you need is the colimits in the definition of stalks to be $\kappa$-directed. In terms of topology it means for example that open subsets are closed un $\kappa$-small intersection. In that case stalk presrve $\kappa$-small limit, but that's not going to help you much for the étale site. $\endgroup$ Dec 16, 2020 at 19:51
  • $\begingroup$ Is there something at a lower level of generality? For instance assuming that the limit is exact? $\endgroup$ Dec 18, 2020 at 19:12
  • $\begingroup$ What I mean is that I don't know any criterion general enough to be of interest for this kind of commutation (except the one with k-small limits and k-directed colimits of course, but it is unlikely to apply to your situation). But it happens that (in very concrete situation) such limits commute, so, if you wonder about this because of a concrete limits you want to be preserved by the stalk functor, then my intuition would be to focus on your example rather than looking for general result that might apply. $\endgroup$ Dec 18, 2020 at 19:26
  • $\begingroup$ @SimonHenry To be honest, I don't think that in my case the stalk commutes, but my gut tells me that the map ought to be surjective. I think my uneasyness comes from the fact that in some sense the left hand side is completely determined by the $\mathcal{F}_i$'s, and so its stalks ought to be understandable purely in terms of $\mathcal{F}_i$ (of course not by its stalks, but by the whole "global" datum). Not having any control of the left-hand side seems therefore "wrong" to me? $\endgroup$ Dec 18, 2020 at 19:30

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