Lemma 80.5.3. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$.

For $K$ in $D((\textit{Spaces}/Y)_{\acute{e}tale})$ we have $ (Rf_{big, *}K)|_{X_{\acute{e}tale}} = Rf_{small, *}(K|_{Y_{\acute{e}tale}}) $ in $D(X_{\acute{e}tale})$.

For $K$ in $D((\textit{Spaces}/Y)_{\acute{e}tale}, \mathcal{O})$ we have $ (Rf_{big, *}K)|_{X_{\acute{e}tale}} = Rf_{small, *}(K|_{Y_{\acute{e}tale}}) $ in $D(\textit{Mod}(X_{\acute{e}tale}, \mathcal{O}_ X))$.

More generally, let $g : X' \to X$ be an object of $(\textit{Spaces}/X)_{\acute{e}tale}$. Consider the fibre product

Then

For $K$ in $D((\textit{Spaces}/Y)_{\acute{e}tale})$ we have $i_ g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$ in $D(X'_{\acute{e}tale})$.

For $K$ in $D((\textit{Spaces}/Y)_{\acute{e}tale}, \mathcal{O})$ we have $i_ g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$ in $D(\textit{Mod}(X'_{\acute{e}tale}, \mathcal{O}_{X'}))$.

For $K$ in $D((\textit{Spaces}/Y)_{\acute{e}tale})$ we have $g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)$ in $D((\textit{Spaces}/X')_{\acute{e}tale})$.

For $K$ in $D((\textit{Spaces}/Y)_{\acute{e}tale}, \mathcal{O})$ we have $g_{big}^*(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^*K)$ in $D(\textit{Mod}(X'_{\acute{e}tale}, \mathcal{O}_{X'}))$.

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