The Stacks project

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

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

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

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

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

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

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

Proof. To prove (5) represent $M$ by a K-injective complex of abelian sheaves and apply Lemma 59.99.1 and work out the definitions. Part (3) follows from this as $i_ f^{-1}\pi _ S^{-1} = f_{small}^{-1}$. 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 $Lf_{small}^* = i_ f^* \circ L\pi _ S^*$. Part (2) is a special case of (4). $\square$

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