The Stacks project

Lemma 84.5.3. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$.

  1. 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 $D(X_{\acute{e}tale})$.

  2. 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

\[ \xymatrix{ Y' \ar[r]_{g'} \ar[d]_{f'} & Y \ar[d]^ f \\ X' \ar[r]^ g & X } \]

Then

  1. 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 $D(X'_{\acute{e}tale})$.

  2. 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 $D(\textit{Mod}(X'_{\acute{e}tale}, \mathcal{O}_{X'}))$.

  3. 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 $D((\textit{Spaces}/X')_{\acute{e}tale})$.

  4. 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 $D(\textit{Mod}(X'_{\acute{e}tale}, \mathcal{O}_{X'}))$.

Proof. Part (1) follows from Lemma 84.5.2 and (84.5.2.1) on choosing a K-injective complex of abelian sheaves representing $K$.

Part (3) follows from Lemma 84.5.2 and Topologies, Lemma 34.4.19 on choosing a K-injective complex of abelian sheaves representing $K$.

Part (5) is Cohomology on Sites, Lemma 21.21.1.

Part (6) is Cohomology on Sites, Lemma 21.21.2.

Part (2) can be proved as follows. Above we have seen that $\pi _ X \circ f_{big} = f_{small} \circ \pi _ Y$ as morphisms of ringed sites. Hence we obtain $R\pi _{X, *} \circ Rf_{big, *} = Rf_{small, *} \circ R\pi _{Y, *}$ by Cohomology on Sites, Lemma 21.19.2. Since the restriction functors $\pi _{X, *}$ and $\pi _{Y, *}$ are exact, we conclude.

Part (4) follows from part (6) and part (2) applied to $f' : Y' \to X'$. $\square$

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