The Stacks project

Lemma 21.37.5. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the corresponding morphism of topoi. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

  1. For $M$ in $D(\mathcal{D})$ we have $R\Gamma (U, g^{-1}M) = R\Gamma (u(U), M)$.

  2. If $\mathcal{O}_\mathcal {D}$ is a sheaf of rings and $\mathcal{O}_\mathcal {C} = g^{-1}\mathcal{O}_\mathcal {D}$, then for $M$ in $D(\mathcal{O}_\mathcal {D})$ we have $R\Gamma (U, g^*M) = R\Gamma (u(U), M)$.

Proof. In the bounded below case (1) and (2) can be seen by representing $K$ by a bounded below complex of injectives and using Lemma 21.37.1 as well as Leray's acyclicity lemma. In the unbounded case, first note that (1) is a special case of (2). For (2) we can use

\[ R\Gamma (U, g^*M) = R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}(j_{U!}\mathcal{O}_ U, g^*M) = R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(j_{u(U)!}\mathcal{O}_{u(U)}, M) = R\Gamma (u(U), M) \]

where the middle equality is a consequence of Lemma 21.37.2. $\square$

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