Lemma 59.99.3. Let $f : T \to S$ be a morphism of schemes. Then

For $K$ in $D(S_{\acute{e}tale})$ we have $H^ n_{\acute{e}tale}(S, \pi _ S^{-1}K) = H^ n(S_{\acute{e}tale}, K)$.

For $K$ in $D(S_{\acute{e}tale}, \mathcal{O}_ S)$ we have $H^ n_{\acute{e}tale}(S, L\pi _ S^*K) = H^ n(S_{\acute{e}tale}, K)$.

For $K$ in $D(S_{\acute{e}tale})$ we have $H^ n_{\acute{e}tale}(T, \pi _ S^{-1}K) = H^ n(T_{\acute{e}tale}, f_{small}^{-1}K)$.

For $K$ in $D(S_{\acute{e}tale}, \mathcal{O}_ S)$ we have $H^ n_{\acute{e}tale}(T, L\pi _ S^*K) = H^ n(T_{\acute{e}tale}, Lf_{small}^*K)$.

For $M$ in $D((\mathit{Sch}/S)_{\acute{e}tale})$ we have $H^ n_{\acute{e}tale}(T, M) = H^ n(T_{\acute{e}tale}, i_ f^{-1}M)$.

For $M$ in $D((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O})$ we have $H^ n_{\acute{e}tale}(T, M) = H^ n(T_{\acute{e}tale}, i_ f^*M)$.

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