The Stacks project

Lemma 20.32.5. Let $f : X \to Y$ be a morphism of ringed spaces. Then $R\Gamma (Y, -) \circ Rf_* = R\Gamma (X, -)$ as functors $D(\mathcal{O}_ X) \to D(\Gamma (Y, \mathcal{O}_ Y))$. More generally for $V \subset Y$ open and $U = f^{-1}(V)$ we have $R\Gamma (U, -) = R\Gamma (V, -) \circ Rf_*$.

Proof. Let $Z$ be the ringed space consisting of a singleton space with $\Gamma (Z, \mathcal{O}_ Z) = \Gamma (Y, \mathcal{O}_ Y)$. There is a canonical morphism $Y \to Z$ of ringed spaces inducing the identification on global sections of structure sheaves. Then $D(\mathcal{O}_ Z) = D(\Gamma (Y, \mathcal{O}_ Y))$. Hence the assertion $R\Gamma (Y, -) \circ Rf_* = R\Gamma (X, -)$ follows from Lemma 20.28.2 applied to $X \to Y \to Z$.

The second (more general) statement follows from the first statement after applying Lemma 20.32.4. $\square$

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