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examples of computations of etale cohomology
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For a smooth connected complete curve \(X\) over an algebraically closed field of characteristic coprime to \(n\): \begin{align*} { {H}^{\scriptscriptstyle \bullet}} (X; C_n) \cong C_n \oplus C_n{ {}^{ \scriptscriptstyle\times^{2g} } }[1] \oplus C_n[2]\end{align*} and \begin{align*} { {H}^{\scriptscriptstyle \bullet}} (X; {\mathbf{G}}_m{}_{/ {k}} ) \cong k^{\times}\oplus \operatorname{Pic}(X)[1] \end{align*}
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For any sheaf \({\mathcal{F}}\) and \(x= \operatorname{Spec}k\) a geometric point, \begin{align*}\tau_{\geq 1} { {H}^{\scriptscriptstyle \bullet}} _\text{ét}(x; {\mathcal{F}}) = 0.\end{align*}
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\(H^i_{\text{ét}, c}({\mathbf{A}}^1_{/ {k}} \otimes_k k^{ {}^{ \operatorname{sep} } }; { {\mathbf{Q}}_\ell }) = { {\mathbf{Q}}_\ell }(-1)[2]\).
Etale fundamental groups
Elliptic curves
See https://people.math.ethz.ch/~kowalski/deligne.pdf#page=12
