Lemma 21.19.1 (07A6)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.19: Cohomology of unbounded complexes
Lemma 21.19.1 (cite)

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Lemma 21.19.1. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi. The functor $Rf_*$ defined above and the functor $Lf^*$ defined in Lemma 21.18.2 are adjoint:

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

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

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

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