Lemma 82.5.4. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Then

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

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

For $K$ in $D(X_{\acute{e}tale})$ we have $H^ n_{\acute{e}tale}(Y, \pi _ X^{-1}K) = H^ n(Y_{\acute{e}tale}, f_{small}^{-1}K)$.

For $K$ in $D(X_{\acute{e}tale}, \mathcal{O}_ X)$ we have $H^ n_{\acute{e}tale}(Y, L\pi _ X^*K) = H^ n(Y_{\acute{e}tale}, Lf_{small}^*K)$.

For $M$ in $D((\textit{Spaces}/X)_{\acute{e}tale})$ we have $H^ n_{\acute{e}tale}(Y, M) = H^ n(Y_{\acute{e}tale}, i_ f^{-1}M)$.

For $M$ in $D((\textit{Spaces}/X)_{\acute{e}tale}, \mathcal{O})$ we have $H^ n_{\acute{e}tale}(Y, M) = H^ n(Y_{\acute{e}tale}, i_ f^*M)$.

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