Definition 63.18.8. 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 $\mathcal{F}$.

Comment #75 by Keenan Kidwell on

I couldn't find the definition of $H^1_c(X,F)$ for a sheaf $F$ on the \'{e}tale site of $X$ anywhere else in the chapter. Did I just miss it?

Comment #82 by on

Fixed by adding a definition in Remark 78.2. But of course this needs a lot more work. Thanks.

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• 2 comment(s) on Section 63.18: On l-adic sheaves

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