Lemma 21.20.5. Let $f : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Then $R\Gamma (\mathcal{D}, -) \circ Rf_* = R\Gamma (\mathcal{C}, -)$ as functors $D(\mathcal{O}_\mathcal {C}) \to D(\Gamma (\mathcal{O}_\mathcal {D}))$. More generally, for $V \in \mathcal{D}$ with $U = u(V)$ we have $R\Gamma (U, -) = R\Gamma (V, -) \circ Rf_*$.
Proof. Consider the punctual topos $pt$ endowed with $\mathcal{O}_{pt}$ given by the ring $\Gamma (\mathcal{O}_\mathcal {D})$. There is a canonical morphism $(\mathcal{D}, \mathcal{O}_\mathcal {D}) \to (pt, \mathcal{O}_{pt})$ of ringed topoi inducing the identification on global sections of structure sheaves. Then $D(\mathcal{O}_{pt}) = D(\Gamma (\mathcal{O}_\mathcal {D}))$. The assertion $R\Gamma (\mathcal{D}, -) \circ Rf_* = R\Gamma (\mathcal{C}, -)$ follows from Lemma 21.19.2 applied to
\[ (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D}) \to (pt, \mathcal{O}_{pt}) \]
The second (more general) statement follows from the first statement after applying Lemma 21.20.4. $\square$
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