Remark 63.29.2. Thus we conclude that if $X$ is also projective then we have functorially in the representation $\rho $ the identifications

\[ H^0(X_{\overline{k}}, \mathcal{F}_\rho ) = M^{\pi _1(X_{\overline{k}}, \overline\eta )} \]

and

\[ H_ c^2(X_{\overline{k}}, \mathcal{F}_\rho ) = M_{\pi _1(X_{\overline{k}}, \overline\eta )}(-1) \]

Of course if $X$ is not projective, then $H^0_ c(X_{\overline{k}}, \mathcal{F}_\rho ) = 0$.

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