Lemma 84.5.4 (0DGB)—The Stacks project

Lemma 84.5.4. 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 $H^ n_{\acute{e}tale}(Y, M) = H^ n(Y_{\acute{e}tale}, i_ f^*M)$.

Proof. To prove (5) represent $M$ by a K-injective complex of abelian sheaves and apply Lemma 84.5.2 and work out the definitions. Part (3) follows from this as $i_ f^{-1}\pi _ X^{-1} = f_{small}^{-1}$. Part (1) is a special case of (3).

Part (6) follows from the very general Cohomology on Sites, Lemma 21.37.5. Then part (4) follows because $Lf_{small}^* = i_ f^* \circ L\pi _ X^*$. Part (2) is a special case of (4). $\square$

The Stacks project

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