Definition 64.18.8. If X is a separated scheme of finite type over an algebraically closed field k and \mathcal{F} = \left\{ \mathcal{F}_ n\right\} _{n\geq 1} is a \mathbf{Z}_\ell -sheaf on X, then we define
H^ i(X, \mathcal{F}) := \mathop{\mathrm{lim}}\nolimits _ n H^ i(X, \mathcal{F}_ n) \quad \text{and}\quad H_ c^ i(X, \mathcal{F}) := \mathop{\mathrm{lim}}\nolimits _ n H_ c^ i(X, \mathcal{F}_ n).
If \mathcal{F} = \mathcal{F}'\otimes \mathbf{Q}_\ell for a \mathbf{Z}_\ell -sheaf \mathcal{F}' then we set
H_ c^ i(X , \mathcal{F}) := H_ c^ i(X, \mathcal{F}')\otimes _{\mathbf{Z}_\ell }\mathbf{Q}_\ell .
We call these the \ell -adic cohomology of X with coefficients \mathcal{F}.
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