The Stacks project

Lemma 20.28.1. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. The functor $Rf_*$ defined above and the functor $Lf^*$ defined in Lemma 20.27.1 are adjoint:

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(Lf^*\mathcal{G}^\bullet , \mathcal{F}^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ Y)}(\mathcal{G}^\bullet , Rf_*\mathcal{F}^\bullet ) \]

bifunctorially in $\mathcal{F}^\bullet \in \mathop{\mathrm{Ob}}\nolimits (D(\mathcal{O}_ X))$ and $\mathcal{G}^\bullet \in \mathop{\mathrm{Ob}}\nolimits (D(\mathcal{O}_ Y))$.

Proof. This follows formally from the fact that $Rf_*$ and $Lf^*$ exist, see Derived Categories, Lemma 13.30.3. $\square$

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