ℓ-adic cohomology

Remarks

Example: let $k$ be algebraically closed and $X = \mathbb{P}^1_k$. Then $H^0_{\text{ét}}(X, \Z/\ell^n\Z) = \Z/\ell^n\Z$, $H^1_{\text{ét}} = 0$ and $H^2_{\text{ét}} \cong \Z/\ell^n\Z$, compatibly in $n$. So $H^i(X, \Q_\ell)$ is $\Q_\ell$ for $i = 0, 2$ and $0$ otherwise, as for the sphere $\mathbb{P}^1(\C)$. Non-example: the condition $\ell \ne \operatorname{char} k$ is needed. Let $k$ have characteristic $p > 0$ and let $E$ be an elliptic curve over $k$. Then $H^1(E, \Q_p)$ has dimension $1$ if $E$ is ordinary and $0$ if $E$ is supersingular, not the expected dimension $2$. For $X$ proper over $k$, the groups $H^i(X, \Q_\ell)$ are finite-dimensional and vanish for $i > 2 \dim X$.