The Stacks project

Proposition 64.29.3. Let $X/k$ as before but $X_{\overline{k}}\neq \mathbf{P}^1_{\overline{k}}$ The functors $ (M, \rho )\mapsto H_ c^{2-i}(X_{\overline{k}}, \mathcal{F}_\rho ) $ are the left derived functor of $(M, \rho )\mapsto H_ c^2(X_{\overline{k}}, \mathcal{F}_\rho )$ so

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

Moreover, there is a derived version, namely

\[ R\Gamma _ c(X_{\overline{k}}, \mathcal{F}_\rho ) = LH_0(\pi _1(X_{\overline{k}}, \overline\eta ), M(-1)) = M(-1) \otimes _{\Lambda [[\pi _1(X_{\overline{k}}, \overline\eta )]]}^\mathbf {L} \Lambda \]

in $D(\Lambda [[\widehat{\mathbf{Z}}]])$. Similarly, the functors $(M, \rho )\mapsto H^ i(X_{\overline{k}}, \mathcal{F}_\rho )$ are the right derived functor of $(M, \rho )\mapsto M^{\pi _1(X_{\overline{k}}, \overline\eta )}$ so

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

Moreover, in this case there is a derived version too.

Proof. (Idea) Show both sides are universal $\delta $-functors. $\square$

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