Let $X$ be a smooth geometrically connected scheme over a field $k$ of characteristic 0 (but not necessarily algebraically closed, I can take it to be a number field). Let $F$ be a finite algebraic group over $k$.

Is the following statement true: $H_{et}^1(X,F) = H^1(\pi_1^{et}(X), F(\bar{k}))$?

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    $\begingroup$ There is a natural morphism $\pi$ of sites from the etale site of $X$ to its finite etale site. Using the Leray spectral sequence, your question basically reduces to asking: Is $R^1\pi_*F = 0$? Since $R^1\pi_*F$ is the sheafication of $U\mapsto H^1_{et}(U,F)$ over the finite etale site, another way of saying this is: If $U$ is a $k$-scheme, then does every $F$-torsor admit a section over a finite etale $U$-scheme? The answer is yes, simply because $F$ is itself finite etale over $k$ (since we are in characteristic $0$). $\endgroup$ Apr 13, 2017 at 3:36

1 Answer 1


This is true, yes. More generally, if $X$ is a scheme and $F$ is a locally constant étale sheaf of finite abelian groups on $X$, then $$ H^1_{et}(X,F) = H^1(\Pi_1^{et}(X), \tilde F), $$ where $\Pi_1^{et}(X)$ is the étale fundamental pro-groupoid of $X$ and $\tilde F$ a certain local system on $\Pi_1^{et}(X)$ corresponding to $F$. This follows from Proposition 5.9 in Friedlander's book on étale homotopy [the assumption there that $X$ is locally noetherian is not needed when $F$ is finite], together with the fact that $H^1$ of a (pro-)space is the same as $H^1$ of its 1-truncation.

When $X$ is connected and $x$ is a geometric point of $X$, $\Pi_1^{et}(X)\simeq B\pi_1^{et}(X,x)$, so that local systems on $\Pi_1^{et}(X)$ can be identified with $\pi_1^{et}(X,x)$-modules. Under this identification, the local system $\tilde F$ is just $F(x)$ with its $\pi_1^{et}(X,x)$-action.


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