Lemma 20.13.1. Let $f : X \to Y$ be a morphism of ringed spaces. There is a commutative diagram
\[ \xymatrix{ D^{+}(X) \ar[rr]_-{R\Gamma (X, -)} \ar[d]_{Rf_*} & & D^{+}(\mathcal{O}_ X(X)) \ar[d]^{\text{restriction}} \\ D^{+}(Y) \ar[rr]^-{R\Gamma (Y, -)} & & D^{+}(\mathcal{O}_ Y(Y)) } \]
More generally for any $V \subset Y$ open and $U = f^{-1}(V)$ there is a commutative diagram
\[ \xymatrix{ D^{+}(X) \ar[rr]_-{R\Gamma (U, -)} \ar[d]_{Rf_*} & & D^{+}(\mathcal{O}_ X(U)) \ar[d]^{\text{restriction}} \\ D^{+}(Y) \ar[rr]^-{R\Gamma (V, -)} & & D^{+}(\mathcal{O}_ Y(V)) } \]
See also Remark 20.13.2 for more explanation.
Proof.
Let $\Gamma _{res} : \textit{Mod}(\mathcal{O}_ X) \to \text{Mod}_{\mathcal{O}_ Y(Y)}$ be the functor which associates to an $\mathcal{O}_ X$-module $\mathcal{F}$ the global sections of $\mathcal{F}$ viewed as an $\mathcal{O}_ Y(Y)$-module via the map $f^\sharp : \mathcal{O}_ Y(Y) \to \mathcal{O}_ X(X)$. Let $restriction : \text{Mod}_{\mathcal{O}_ X(X)} \to \text{Mod}_{\mathcal{O}_ Y(Y)}$ be the restriction functor induced by $f^\sharp : \mathcal{O}_ Y(Y) \to \mathcal{O}_ X(X)$. Note that $restriction$ is exact so that its right derived functor is computed by simply applying the restriction functor, see Derived Categories, Lemma 13.16.9. It is clear that
\[ \Gamma _{res} = restriction \circ \Gamma (X, -) = \Gamma (Y, -) \circ f_* \]
We claim that Derived Categories, Lemma 13.22.1 applies to both compositions. For the first this is clear by our remarks above. For the second, it follows from Lemma 20.11.10 which implies that injective $\mathcal{O}_ X$-modules are mapped to $\Gamma (Y, -)$-acyclic sheaves on $Y$.
$\square$
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