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Let $k$ be an algebraically closed field of characteristic $p\geq 0$ and $\ell$ a prime different from $p$. For a connected scheme of finite type over $k$ with geometric point $x$, and a lisse $\overline{\mathbb{Q}}_{\ell}$-sheaf $F$ on $X$ one can compute the global sections as follows $$H^0(X,F)=F_x^{\pi_1(X,x)}$$ I don't know where a proof of this is written down, but it seems to me that it works like this: Consider $F$ as projective system $(F_n)$ of $\mathcal{O}_E/\mathbf{m}^n$-sheaves, for $E$ a finite extension of $\mathbb{Q}_{\ell}$, and $\mathbf{m}$ the maximal ideal of $\mathcal{O}_E$. Each $F_n$ corresponds to a étale covering $X_n$ of $X$, and $(F_n)_x$ is the set of geometric points of $X_n$ over $x$. Now the sections of $X_n\rightarrow X$ correspond to points in $(F_n)_x$ which are fixed by the $\pi_1(X,x)$-action. Passing to the limit proves the formula for $H^0$.

My question is: Does this also hold for $k$ not algebraically closed?

I've never seen it stated like this, but my argument doesn't seem to use the fact that $k=\bar{k}$.

I think it is true for smooth curves, because a smooth curve $X\neq \mathbb{P}^1$ over any field of characteristic $p\geq 0$ is even an "étale $K(\pi_1(X,x),1)$ space", i.e. $$H^n(X,\mathbb{Z}_{\ell})=H^n(\pi_1(X,x), \mathbb{Z}_\ell)$$ for all $n\geq 0$.

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You appear to be using the assumption that all the sheaves in your projective system trivialize over a finite etale cover. With this assumption, your argument is correct (without any hypotheses at all on $X$, I should imagine), essentially because you're considering lisse l-adic sheaves over the finite etale site, and the latter is equivalent to the site of finite $\pi_1(X,x)$-sets. – Keerthi Madapusi Pera Oct 11 '11 at 14:07
Hi Keerthi, thanks for your comment. I don't understand where I use that all $F_n$ trivialize over a finite covering: By the definition of "lisse" each $F_n$ is represented by an étale covering, and $\pi_1(X,x)$ acts on the finite set $F_{n,x}$ for each $n$. – Lars Oct 11 '11 at 19:56
Keerthi, I also don't understand your objection. Lars doesn't assume that there's a finite étale cover that trivializes all the sheaves in the projective system (he takes a different one for each sheaf). Lars, I think that your argument is valid for any connected scheme X. (I've seen that kind of result referred to as "a well-known fact" but I can't find a reference. Maybe it's considered too easy to be stated explicitely.) – Alex Oct 11 '11 at 22:36
Thanks Alex! It seems that it's just unraveling of definitions, but I assumed that Deligne would have written it like this in Weil II if it was true. He explicitly writes the formula for $X\times_k \bar{k}$... – Lars Oct 12 '11 at 9:53
Well, he only needs the formula for $X\times_k\overline{k}$, and Deligne doesn't usually write what he doesn't need. – Alex Oct 12 '11 at 13:29

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